j. differential geometry 51 (1999) 149-202

INSTANTON FLOER HOMOLOGY FOR KNOTS VIA 3-ORBIFOLDS

OLIVIER COLLIN & BRIAN STEER

Abstract

In this article we construct an instanton Floer Homology for knots in S^{3}. Rather than working on knot complements, the invariant is constructed using 3-orbifolds singular along a knot, (S^{3}, K, n). The resulting knot invariant consists of four graded abelian groups HF_{i}^{(k)}(S^{3}, K, n)(0 ≤ i ≤ 3). We give some properties of the invariant and also provide some examples.

1. Introduction

For more than 15 years now, Gauge Theory has provided Lowdimensional Topology with many interesting invariants, sometimes leading to solutions of important problems, but up to now, relatively little has been done to apply Gauge Theory to Knot Theory. The primary goal of this article is to contribute to the development of a proper gauge theoretic setting for the study of knots.

Floer Homology, as initiated by Floer in his seminal paper [22], is by now a well-known gauge theoretic 3-manifold invariant which has been quite useful when related to Donaldson’s constructions in 4-dimensional Gauge Theory. This invariant non-trivially generalises Casson’s invariant, and therefore is of interest for 3-manifold Topology. One may think of applying Floer’s ideas to develop a Floer Theory for knot complements. In the 4-dimensional setting, Kronheimer and Mrowka have partially opened the way in [32] and [33], where they used Gauge Theory to study smoothly embedded surfaces in 4-manifolds. This somehow provides a model for the objects to be considered in developing a Floer Theory for knots.

Received March 27, 1998, and, in revised form, September 30, 1998. The first author acknowledges support from NSERC Canada 1967 Scholarship and British Council ORS funds during his doctoral studies. Both authors wish to thank David Austin and the University of British Columbia for their hospitality during the period of preparation of this article.

When trying to transpose Floer’s ideas from 3-manifolds to knots, there are many advantages in working with 3-dimensional orbifolds singular along a knot rather than with knot complements. The most substantial gains are on the analytical side. As knot complements are either incomplete manifolds (S^{3}−K) or have a boundary (removing the interior of a tubular neighbourhood of K in S^{3}) the analytical setting for Gauge Theory is much more complicated than in the case of closed manifolds (see for example work in [26] and comments in [32]). By contrast, if one works with an orbifold, one may encode information about the knot and at the same time work with a compact object, hence simplifying greatly the Analysis required for a Floer Theory.

Here is an outline of the article. In Section 2, we are concerned with introducing 3-orbifolds and set-up the gauge theoretic framework which will be needed to do a Floer Theory. Section 3 gives a detailed construction of the Floer Homology for 3-orbifolds, and is the central part of the article. In Section 4, various properties of the Floer Homology for 3-orbifolds are studied while Section 5 gives some examples. Section 6 is a conclusion, hinting at further developments of the theory and its applications.

Acknowledgments. Both authors would like to thank Simon Donaldson for his generosity and advice regarding this work, and for various suggestions about [11], containing an earlier version of this article. The first author would also like to thank David Austin and Chris Herald for various conversations.

2. 3-orbifolds and gauge theory

2.1 3-orbifolds

Suppose K is a knot in S^{3}, that is, a smoothly embedded circle in S^{3}. Denote by N_{K} a tubular neighbourhood of K in S^{3}. The fundamental group of the knot complement, π_{1}(S^{3} − N_{K}) is called the knot group, and in it are two distinguished elements, the meridian, μ, well-defined up to conjugation, and the (trivially framed) longitude, λ. Let n be an integer.

In this article, we shall use 3-orbifolds introduced by Thurston (see [46]) and commonly used in 3-dimensional Topology (see [5] and [6]). We shall be very brief here and refer the reader unfamiliar with such objects to [46] or [6].

Definition 2.1. A 3-dimensional orbifold is given by a triple (S^{3}, K, n), where S^{3} is the underlying topological space, the knot K ↪ S^{3} is the singular locus, and the isotropy group around K is ℤ_{n}.

There is an alternative approach to (S^{3}, K, n) which will be useful. Let V_{n}(K) be the n-fold cyclic branched covering of S^{3} along K. It is easy to prove that (S^{3}, K, n) is the quotient of V_{n}(K) by the ℤ_{n}-action coming from meridional rotations on N_{K} and covering transformations on the n-fold unbranched covering of S^{3} − N_{K}. See, for example, [6] for details.

For 3-orbifolds (S^{3}, K, n), there is a practical definition of the orbifold fundamental group, via loops, as in the manifold case (see [24] in the case of 2-orbifolds). It is clear from the above construction that the orbifold fundamental group π_{1}^{V} (S^{3}, K, n) is

π_{1}^{V} (S^{3}, K, n) = π_{1}(S^{3} − N_{K})/ < μ^{n} >,

where μ is a meridian in π_{1}(S^{3}−N_{K}). The abelianization of the orbifold fundamental group, H_{1}^{V} (S^{3}, K, n) is always isomorphic to ℤ_{n}. It follows from the construction of V_{n}(K) and (S^{3}, K, n) that there is a short exact sequence

1 ⟶ π_{1}(V_{n}(K)) ⟶ π_{1}^{V} (S^{3}, K, n) ⟶ ℤ_{n} ⟶ 1,

called the orbifold exact sequence.

2.2 Gauge theory for orbifolds

We now proceed to introduce the basic gauge theoretical objects that will be needed and developed in the orbifold setting. After introducing rapidly orbifold bundles and connections (for more information on standard material the reader is referred to [43], [28] or [24]), we define a Chern-Simons functional for orbifolds via the Chern-Weil theory of Kronheimer and Mrowka.

The definition of an orbifold vector bundle over orbifolds is very similar to that of a genuine vector bundle, the only difference being that one needs to specify some data near the singular locus. A vector bundle π: E → (S^{3}, K, n) is given locally by trivializations

θ_{i}: E_{|U_{i}} → Ũ_{i} × ℂ^{r}/G_{i},

where G_{i} acts on ℂ^{r} by some representation τ: ℤ_{n} → Aut(𝔸^{r}). The standard compatibility (patching) must hold. We will only be concerned with SU(2)-bundles. Over the knot complement (a 3-manifold with boundary), an SU(2)-bundle is trivial, while over the tubular neighbourhood, an orbifold bundle is determined by its isotropy τ: ℤ_{n} → SU(2). When we glue the two together to have a bundle over the orbifold, the class will depend on the isotropy and a gluing map

∂(S^{3} − N_{K}) × ℂ^{2} → ∂(S^{3} − N_{K}) × ℂ^{2}

i.e., a map T^{2} → SU(2). Since π_{i}(SU(2)) is trivial for i = 1, 2, we have the following:

Proposition 2.2. The SU(2)-bundles over (S^{3}, K, n) are classified by their isotropy τ: ℤ_{n} → SU(2).

We will denote by E_{k} the orbifold SU(2)-vector bundle over (S^{3}, K, n) whose isotropy is given by

τ(1) = (\begin{matrix}e^{i2kπ/n} & 0 \\ 0 & e^{−i2kπ/n} \end{matrix}) .

Let Ω^{i}(𝔤_{E_{k}}) denote the space of differential i-forms on (S^{3}, K, n) with values in the associated bundle 𝔤_{E_{k}}. On E_{k}, one can define connections as in the usual way, and form the space of connections on E_{k}, A(E_{k}). The gauge group G(E_{k}) is the group of bundle isomorphisms g of E_{k} respecting the structure of the fibres and covering the identity. The group G(E_{k}) acts on A(E_{k}) by

g · (d + A) = gdg^{−1} + gAg^{−1},

and one can consider the quotient B(E_{k}) = A(E_{k})/G(E_{k}) the moduli space of connections. This space is infinite dimensional and is not necessarily a manifold as it has some singular points, where the action of G(E_{k}) on A(E_{k}) is not free. A reducible flat connection A on E_{k} has a non-central stabilizer in the gauge group, I_{A} = {g ∈ G(E_{k}) | g · A = A}. If E_{k} is E_{0} or E_{n/2} then for a reducible connection, I_{A} ≃ SU(2). Otherwise, I_{A} ≃ S^{1}. Denote by A^{∗}(E_{k}) the subset in A(E_{k}) where the action is free, and let B^{∗}(E_{k}) = A^{∗}(E_{k})/G(E_{k}). Inside the space B(E_{k}), we are particularly interested in the moduli space of flat connections M_{flat}(E_{k}), in which the set of irreducible flat connections is denoted M_{flat}^{∗}(E_{k}).

Remark 2.3. As (S^{3}, K, n) is expressible as a global quotient V_{n}(K)/ℤ_{n}, the orbifold setting corresponds to a ℤ_{n}-equivariant one over V_{n}(K). An orbifold bundle E_{k} over (S^{3}, K, n) is equivalent to a ℤ_{n}- equivariant bundle E′_{k} over V_{n}(K), and orbifold gauge transformations are then seen as ℤ_{n}-equivariant gauge transformations on E′_{k}. On the other hand, the ℤ_{n}-action on E′_{k} induces one on forms and hence on connections. Orbifold connections on E_{k} correspond to invariant connections on E′_{k} under this induced ℤ_{n}-action. For more details on this, see [20], for example.

Let χ(S^{3}, K, n) be the SU(2)-character variety of the group π_{1}^{V} (S^{3}, K, n). Parallel transport yields a homeomorphism between gauge equivalence classes of flat connections on an SU(2)-bundle and their corresponding conjugacy classes of holonomy SU(2)-representations. See [44, Proposition 4.5] for more details. Contrary to the 3-manifold situation, the orbifold bundles E_{k} → (S^{3}, K, n) are not necessarily trivial, but depend on the isotropy representation τ: ℤ_{n} → SU(2). Given a representation ρ: π_{1}^{V} (S^{3}, K, n) → SU(2), which bundle(s) carries the corresponding flat connection? The reader will have no difficulty in proving the following:

Proposition 2.4. On a bundle E_{k} over (S^{3}, K, n) (1 ≤ k ≤ n/2), M_{flat}(E_{k}) is homeomorphic to χ^{k}(S^{3}, K, n), where

χ^{k}(S^{3}, K, n) = \{ρ ∈ χ(S^{3}, K, n) | tr ρ(μ) = 2cos (2\frac{k}{n}π)\}.

Corollary 2.5. Each SU(2)-bundle E_{k} over (S^{3}, K, n) carries one and only one reducible flat connection.

We now need to outline rapidly relevant aspects of Chern-Weil theory for singular connections in order to define the Chern-Simons functional. Consider Σ ↪ X^{4} a closed embedded surface in a smooth 4manifold. In [32] and [33], Kronheimer and Mrowka study singular connections over the non-compact manifold X^{4} − Σ. These depend on a holonomy parameter α ∈ (0, 1/2), corresponding to the holonomy of the connection along a linking circle of Σ in X^{4}. There are, moreover, two topological invariants in this situation. Consider an SU(2)-bundle E over X^{4}. Such a bundle is determined by the integer

\frac{1}{4π^{2}} ∫_{X} c_{2}(E).

Let N_{Σ} be a tubular neighbourhood of Σ. Then over N_{Σ}, E is trivial and there is a decomposition E = L ⊕ L^{∗}, where L is a line bundle, and the holonomy is α on L, and −α on L^{∗}. There is no reason why L should be trivial, and this fact provides us with a second topological invariant: the integer

\frac{1}{2π} ∫_{Σ} c_{1}(L).

The Chern-Weil formula of Kronheimer and Mrowka for singular connections may now be given.

Theorem 2.6 ([32, Proposition 5.7]). Let 𝔸 be a singular connection with holonomy parameter α ∈ (0, 1/2). Then the following holds

\frac{1}{8π^{2}} ∫_{X−Σ} tr F_{𝔸} ∧ F_{𝔸} = ∫_{X} c_{2}(E) + 2α ∫_{Σ} c_{1}(L) − α^{2}Σ · Σ,

where Σ · Σ is the self-intersection number of Σ ↪ X^{4}.

Kronheimer and Mrowka explain in some detail how their work can be expressed in terms of orbifolds in the case where α ∈ ℚ. Consider an orbifold (X^{4}, Σ, n), and an orbifold SU(2)-bundle 𝔼_{k} over it. Such a bundle is determined by a bundle E over the underlying manifold X and the isotropy data near the singular locus Σ, say ξ ∈ ℤ_{n} acting on L by ξ^{k}. Then for a connection 𝔸 on 𝔼_{k}, by the Chern-Weil formula,

\frac{1}{8π^{2}} ∫_{(X^{4},Σ,n)} tr F_{𝔸} ∧ F_{𝔸} = ∫_{X^{4}} c_{2}(E) + 2\frac{k}{n} ∫_{Σ} c_{1}(L) − \frac{k^{2}}{n^{2}} Σ · Σ.

Let us denote a = ∫_{X^{4}} c_{2}(E) and b = ∫_{Σ} c_{1}(L). The number a is called

the instanton number while b is the monopole number.

We will apply this in our situation to define the Chern-Simons functional. Consider E_{k} over (S^{3}, K, n), and A ∈ B(E_{k}). We know that E_{k} carries one reducible flat connection, θ_{k}. One can consider the 4dimensional orbifold (S^{3} × I, K × I, n) and the pullback bundle of E_{k}. Let 𝔸 be a connection over (S^{3} × I, K × I, n) such that 𝔸_{|0} = θ_{k} and 𝔸_{|1} = A.

Definition 2.7. The Chern-Simons functional is a function

CS: B(E_{k}) → S^{1}

defined by

CS(A) = \frac{1}{8π^{2}} ∫_{(S^{3}×I,K×I,n)} tr F_{𝔸} ∧ F_{𝔸}.

Proposition 2.8. The Chern-Simons functional is well-defined on B(E_{k}) as a function to ℝ/\frac{1}{n}ℤ.

Proof. The definition of CS depends on the connection 𝔸 chosen to extend the connections θ_{k}, A ∈ B(E_{k}), so CS is really well-defined as a functional to a quotient of ℝ. Recall that K has trivial normal bundle in S^{3}. Identify together the ends (S^{3} × {0}, K × {0}, n) and (S^{3} ×{1}, K ×{1}, n) to yield (S^{3} ×S^{1}, K ×S^{1}, n). Consider an orbifold bundle over this orbifold, and apply the Chern-Weil formula of Theorem 2.12. The formula reads

\frac{1}{8π^{2}} ∫_{(S^{3}×S^{1},K×S^{1},n)} tr F_{𝔸} ∧ F_{𝔸} = ∫_{S^{3}×S^{1}} c_{2}(E) + 2\frac{k}{n} ∫_{K×S^{1}} c_{1}(L)

which is 0 modulo \frac{1}{n}ℤ and therefore the definition does not depend on the instanton and monopole numbers involved in choosing an extension of a connection in B(E_{k}). q.e.d.

While all SU(2)-bundles over V_{n}(K) are trivializable, and the trivializations are equivalent up to gauge transformation, there are various non-equivalent ways of equivariantly trivializing SU(2)-bundles over V_{n}(K). Each way is uniquely determined up to equivariant gauge transformation by an isotropy representation ρ_{k}: ℤ_{n} → SU(2), so as an equivariant bundle, E_{k} ≅ V_{n}(K) × ℂ^{2}/ρ_{k}. It will be useful to consider an expression of the Chern-Simons functional in terms of a trivialization of E_{k}:

CS(A) = \frac{1}{8π^{2}n} ∫_{V_{n}(K)} tr A ∧ dA + \frac{2}{3}A ∧ A ∧ A.

This is of course, up to a scalar factor, the usual Chern-Simons functional for the cyclic branched cover V_{n}(K), restricted to ℤ_{n}-invariant forms.

The critical points of CS are easily computed. These are the flat connections on E_{k}. At a critical point, the Hessian of CS is

∗d_{A}: Ker d_{A}^{∗} → Ker d_{A}^{∗}.

For our purposes, it will be more useful to consider the following operator:

L_{A}: Ω^{0}(𝔤_{E_{k}}) ⊕ Ω^{1}(𝔤_{E_{k}}) → Ω^{0}(𝔤_{E_{k}}) ⊕ Ω^{1}(𝔤_{E_{k}}),

expressed in matrix notation as

L_{A} = (\begin{matrix}0 & d_{A}^{∗} \\ d_{A} & ∗d_{A} \end{matrix}) .

It may be shown that L_{A} is an elliptic operator and is self-adjoint. Such an operator has real and discrete spectrum. It will be important to identify the kernels of the operators involved. For ∗d_{A} the kernel is H_{A}^{1}, the first twisted cohomology group of the complex (Ω^{i}(𝔤_{E_{k}}), d_{A}) of E_{k}, while d_{A}: Ω^{0}(𝔤_{E_{k}}) → Ω^{1}(𝔤_{E_{k}}) has kernel the space of covariant sections of 𝔤_{E_{k}}, which is precisely H_{A}^{0}. One has H_{A}^{0} ≠ 0 if and only if A is a reducible flat connection; on the other hand, if H_{A}^{1} = 0, then A is an isolated flat connection. Notice that if H_{A}^{1} ≠ 0, then A may or may not be isolated in M_{flat}. By Corollary 2.5, on a bundle E_{k}, there is only one flat connection for which H_{A}^{0} ≠ 0, so this group will not play an important role. On the other hand, the condition H_{A}^{1} = 0 will be important to avoid manifolds of critical points for the functional CS.

Let (X^{4}, Σ, n) be a 4-dimensional orbifold and 𝔼_{k} a bundle over it. For a connection A ∈ A(𝔼_{k}) over a 4-dimensional orbifold, the curvature is split into F_{A} = F_{A}^{+} + F_{A}^{−}, and A is called an anti-self-dual connection or instanton if it satisfies the ASD equation F_{A}^{+} = 0. Near A, the linearization of the ASD equation is given by the operator

d_{A}^{∗} ⊕ d_{A}^{+} : Ω^{1}(𝔤_{𝔼_{k}}) → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}).

The case that will interest us is that of an orbifold of the form (S^{3}, K, n) × ℝ. In this situation we will see relations between the 3dimensional and 4-dimensional settings. Consider E_{k} over (S^{3}, K, n). Let A(t) be a path of connections in B(E_{k}), and denote by ∗_{3} the Hodge *-operator in dimension three. The proofs of the next three propositions are not difficult and entirely analogous those in 3-manifold Floer Homology.

Proposition 2.9 The gradient vector field of CS is given by the equation

\frac{dA}{dt}(t) = \frac{1}{4π^{2}} ∗_{3} F_{A(t)}.

Now extend the bundle E_{k} over (S^{3}, K, n) to a bundle 𝔼_{k} over (S^{3}, K, n) × ℝ and consider a connection 𝔸. Then, 𝔸 = A_{t} + B_{t} ∧ dt, for A_{t} ∈ Ω^{1}(𝔤_{E_{k}}) and B_{t} ∈ Ω^{0}(𝔤_{E_{k}}). This is done in A(𝔼_{k}). Say that 𝔸 is in temporal gauge if it has no dt component in this decomposition.

Proposition 2.10. Any connection 𝔸 on 𝔼_{k} is gauge-equivalent to a connection in temporal gauge.

In particular, when working on B(𝔼_{k}) it may be assumed that an instanton over (S^{3}, K, n) × ℝ has no dt component. Then the crucial observation is the following:

Proposition 2.11. A connection

𝔸 = A_{t} ∈ B(𝔼_{k})

over

(S^{3}, K, n) × ℝ

is an instanton if and only if it satisfies

\frac{dA_{t}}{dt} = ∗_{3}F_{A_{t}}.

Since the orbifold (S^{3}, K, n) × ℝ is not compact, one has to impose an extra condition to the ASD connections over the orbifold cylinder, and restrict to “finite energy” instantons. An instanton 𝔸 over M × ℝ is said to be of finite energy if its Yang-Mills action

Y M(𝔸) = ∫_{(S^{3},K,n)×ℝ} |F_{A}|^{2}dvol

is finite. This condition is necessary to have asymptotically flat instantons, that is, instantons that “connect” flat connections over (S^{3}, K, n).

3. Floer homology for 3-orbifolds

We now give the construction of a Floer Homology for 3-orbifolds (S^{3}, K, n). This was announced in [10] and developed in [11]. First we define the Floer index of critical points of CS, in order to define the chain groups for the Floer Homology. Then we define a boundary operator between chain groups in adjacent dimensions and obtain the desired Floer homology. Initially we shall impose non-degeneracy conditions in order to define the Floer Homology, and then we treat the problem of perturbations of the Chern-Simons functional so as to define the Homology in degenerate situations.

As we have seen in Section 2, the 3-orbifolds (S^{3}, K, n) are very much related to the cyclic branched covers V_{n}(K) of S^{3} along the knot K. Before trying to set up a Floer homology for 3-orbifolds, it is therefore legitimate to wonder about the need for such an invariant. Couldn’t one study knots using the Floer Homology of cyclic branched coverings? There are some obvious reasons for the preference of 3-orbifolds over cyclic branched coverings.

One good reason is that a Floer Homology for cyclic branched coverings is not always well-defined, as in general the 3-manifolds considered are not homology 3-spheres and therefore Floer’s original construction does not apply. In some instances, one could use generalizations of Floer Homology, but this would simply lead one away from Floer’s original theory and, at present state of knowledge, still leave some cases untreated; which is not advisable if one wishes to have an effective knot invariant.

Also, a Floer Homology for 3-orbifolds ties in well with other gauge theoretic constructions. For example, in the context of invariants developed in [33], it could play a similar role to the one played by Floer Homology with respect to Donaldson Theory of 4-manifolds. In another direction, various people have recently developed Casson-type and symplectic Floer-type invariants for knots ([3], [27] and [36]) and Floer Homology for 3-orbifolds is related in various ways to these constructions.

3.1 The Floer index

Here we develop the necessary analytical tools to define a relative index between critical points of the Chern-Simons functional. Floer’s original construction for homology 3-spheres makes use of the notion of spectral flow, but for the orbifold setting, the Analysis appeared clearer to us if we used the notion of adapted bundles as in [17]. For the most part, what follows is an adaptation to the orbifold case of the work done in [17].

Definition 3.1. An SU(2)-bundle 𝔼 over (S^{3}, K, n)×ℝ with a fixed flat connection over each end is called an adapted bundle. Adapted bundles are said to be equivalent if there is a bundle isomorphism preserving the flat structures over the ends.

When working over a cylinder (S^{3}, K, n) × ℝ, there is an identification of the operator which linearises the ASD equation, D_{𝔸} = d_{𝔸}^{∗} + d_{𝔸}^{+}, as

D_{𝔸} = \frac{d}{dt} + L_{A}.

First consider the pullback case. Given a bundle E_{k} over (S^{3}, K, n) with a flat, possibly reducible, connection A, which is possibly reducible, pullback E_{k} to a bundle 𝔼_{k} over (S^{3}, K, n) × ℝ, with an induced connection 𝔸. If A is acyclic then

D_{𝔸}: L_{1}^{2}(Ω^{1}(𝔤_{𝔼_{k}})) → L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}))

is invertible. However, in the case at hand, M_{flat}(E_{k}) contains one connection θ_{k} which is reducible and is therefore not acyclic, hence some difficulties arise. Indeed, 0 ∈ Spec L_{A} is an eigenvalue and over the cylinder, D_{𝔸} is not invertible or even Fredholm. To handle this problem we use weighted Sobolev spaces L_{1}^{2,α}, L^{2,α} which are completions of the smooth compactly supported sections over (S^{3}, K, n) × ℝ with respect to norms

∥ f ∥_{L^{2,α}}=∥ e^{αt}f ∥_{L^{2}}

and

∥ f ∥_{L_{1}^{2,α}}=∥ e^{αt}f ∥_{L_{1}^{2}} .

These definitions extend to an orbifolds with cylindrical ends and a weight α associated to each end.

Proposition 3.2.

D_{𝔸}: L_{1}^{2,α}(Ω^{1}(𝔤_{𝔼_{k}})) → L^{2,α}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}))

is an invertible operator over (S^{3}, K, n)×ℝ for any weight α ∉ Spec L_{A}.

Proof. First notice that the problem may be reduced to considering another operator between the corresponding unweighted Sobolev spaces: using the isometries L_{1}^{2,α} → L_{1}^{2} and L^{2,α} → L^{2} given by multiplying by e^{αt}, one can define

e^{αt}D_{𝔸}e^{−αt}: L_{1}^{2}(Ω^{1}(𝔤_{𝔼_{k}})) → L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})),

which is equivalent to D_{𝔸} on weighted spaces. Since

e^{αt}D_{𝔸}e^{−αt}(f) = e^{αt}(\frac{d}{dt} + L_{A})e^{−αt}(f) = \frac{d}{dt}f + (L_{A} − α)f,

the problem is reduced to show that for 0 ∉ Spect L_{A}, \frac{d}{dt} + L_{A} is invertible. The proof of this follows from the next elementary analytical lemma which may be proved using the spectral decomposition of the operator L_{A} along with elliptic regularity.

Lemma 3.3. Let δ ∈ ℝ be such that |λ| ≥ δ > 0 for any λ ∈ Spec L_{A}, and ρ a compactly supported section of Ω^{0}(𝔤_{𝔼_{k}})⊕Ω^{+}(𝔤_{𝔼_{k}}) over (S^{3}, K, n) × ℝ. Then there is a section f of Ω^{1}(𝔤_{𝔼_{k}}) such that D_{𝔸}f = ρ and ∥ f ∥_{L_{1}^{2,α}}≤ \frac{1}{δ} ∥ ρ ∥_{L^{2,α}}.

To set up an index and establish some of its elementary properties, it is essential to consider not only cylinders, but more generally orbifolds with cylindrical ends. For such orbifolds the invertibility of the operator D_{𝔸} in appropriate weighted spaces is lost, but one can still prove a Fredholm property. We shall need only particular orbifolds with cylindrical ends: they will be orbifolds (X^{4}, Σ, n) with underlying space X^{4}, singular locus Σ with locally a ℤ_{n} isotropy around Σ, such that the ends of the orbifold are modeled on some (S^{3}, K, n)×ℝ. Although these orbifolds with cylindrical ends are not necessarily globally quotients of some cyclic branched covering by a ℤ_{n}-action, they are at least global quotients on the cylinder part.

Let (X^{4}, Σ, n) be such an orbifold with m cylindrical ends, and consider weighted Sobolev norms with weight α_{i} on the i^{th} end of the orbifold. Let α = (α_{1}, ..., α_{m}) and let L_{1}^{2,α}, L^{2,α} be the completions with respect to the weighted norms with weights α_{1}, ..., α_{m} over the ends. Consider the operator D_{𝔸} for an adapted bundle over (X^{4}, Σ, n) with limiting data (A_{1}, ..., A_{m}), as before. Over the ends of the orbifold, D_{𝔸} splits. The following is very important for the construction:

Theorem 3.4. If none of the α_{i} lies in the spectrum of L_{A_{i}}, then

D_{𝔸}: L_{1}^{2,α}(Ω^{1}(𝔤_{𝔼_{k}})) → L^{2,α}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}))

is a Fredholm operator.

Proof. By an analogous reasoning to that of Proposition 3.2, using the isometries e^{α_{i}t} one may reduce the problem to an operator which is given on each end by \frac{d}{dt} + L_{A} − α_{i}. For simplicity, we will restrict ourselves to the case where the orbifold has only one cylindrical end. The essential ingredients can be found in Section 2 of [39], but here we follow [17].

First we wish to show that Ker D_{𝔸} is finite dimensional. Take a sequence {f_{i}} ∈ Ker D_{𝔸} such that ∥ f_{i} ∥≤ 1. Consider a decomposition of (X^{4}, Σ, n) as

(X_{0}, Σ_{0}, n) ∪ (S^{3}, K, n) × ℝ^{+}

into a compact set and a cylinder. Over (X_{0}, Σ_{0}, n) the solutions {f_{i}} such that D_{𝔸}(f_{i}) = 0 converge in L^{2} to a limit f_{∞}, as D_{𝔸} is an elliptic operator on a compact orbifold. ∥ f_{∞} ∥≤ 1 and D_{𝔸}(f_{∞}) = 0. The problem is thus reduced to showing that {f_{i}} also converges in L^{2} over the cylinder (S^{3}, K, n) × (0, ∞). For this, the following analytical lemma whose proof, again, uses the spectral decomposition of L_{A} and elementary ODE results, is needed.

Lemma 3.5. Any solution to D_{𝔸}(f) = 0 over (S^{3}, K, n) × (0, ∞) satisfies the following inequality

∫_{(S^{3},K,n)×(0,∞)} |f|^{2} ≤ C_{δ} ∫_{(S^{3},K,n)×(0.1)} |f|^{2}

for some constant C_{δ} depending on δ, mentioned in Lemma 3.3.

With this lemma, one gets

∫_{(S^{3},K,n)×(0,∞)} |f_{i} − f_{∞}| ≤ C_{δ} ∫_{(S^{3},K,n)×(0,1)} |f_{1} − f_{∞}|,

where the last integral is taken over a finite volume domain, and therefore this gives L^{2}-convergence of the {f_{i}} over the cylinder also. This implies that Ker D_{𝔸} is finite dimensional as otherwise one could construct a sequence of solutions without any convergent subsequence.

Next one establishes that Im D_{𝔸} has finite codimension. Decompose again (X^{4}, Σ, n) into a compact part and a cylindrical part. Over the cylinder D_{𝔸} has an inverse by Proposition 3.2. For any

ρ ∈ L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}))

let Q̃_{0}(ρ) be given by Q̃_{0}(β_{0}ρ) where β_{0} is a cut-off function which is identically zero on (X_{0}, Σ_{0}, n) and takes the value 1 on U_{0}, the cylinder. Then on the cylinder

D_{𝔸}Q̃_{0}(ρ) = ρ.

Now cover (X_{0}, Σ_{0}, n) with coordinate patches such that D_{𝔸} has a right inverse over each of the patches. This can be done as D_{𝔸} is elliptic and (X_{0}, Σ_{0}, n) is compact. Thus there is an open cover

(X^{4}, Σ, n) = ⋃_{i=0}^{n} U_{i}

such that U_{i} ∩ U_{j} is pre-compact and D_{𝔸}Q̃_{i}(ρ) = ρ on a neighbourhood of the closure of U_{i}. This is the local part of the argument.

Now for a partition of unity subordinate to this cover, {β_{i}}, define the operator P: L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})) → L_{1}^{2}(Ω^{1}(𝔤_{𝔼_{k}})) by

P (ρ) = ∑_{i=0}^{n} β_{i}Q̃_{i}(ρ).

By the construction of P, the operator D_{𝔸}P − I is supported on the overlaps U_{i} ∩ U_{j} for i ≠ j, which is fixed. The Arzelà-Ascoli Theorem implies that D_{𝔸}P −I is a compact operator. As D_{𝔸}P −I is compact an elementary result from Functional Analysis (see (11.3.2) p. 315 in [14]) implies that the image of D_{𝔸}P is closed and has finite codimension in L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})). Then D_{𝔸} also has finite codimension in that space as Im D_{𝔸}P ⊂ Im D_{𝔸}. Like Im D_{𝔸}P, Im D_{𝔸}P − I is also closed and therefore Im D_{𝔸} is closed. D_{𝔸} is Fredholm as required. q.e.d.

Therefore we can define an index

ind^{(α)} D_{𝔸} = dimKer D_{𝔸} − dimCoker D_{𝔸}.

As the index is a deformation invariant, it will be independent of the particular 𝔸 chosen on the adapted bundle over (X^{4}, Σ, n). Also, though D_{𝔸} was constructed using a metric on (X^{4}, Σ, n), the index is easily seen to be independent of that metric.

One of the fundamental properties of the index for an adapted bundle is the additivity under the process of gluing. Let (X_{1}, Σ_{1}, n) and (X_{2}, Σ_{2}, n) be two orbifolds with cylindrical ends such that one of the ends is (S^{3}, K, n) × ℝ^{+} and (S̄^{3}, K̄, n) × ℝ^{+} respectively. For simplicity assume that the two orbifolds have no other ends. Take two adapted bundles over these, each with the same flat limits over the ends, and use weighted Sobolev spaces of weight α and −α respectively (to have a compatibility condition for the gluing), with α and −α not in the spectrum of the relevant operators. Consider an adapted bundle over the glued orbifold (X_{1}♯X_{2}, Σ_{1}♯Σ_{2}, n). Then the additivity is expressed as follows.

Theorem 3.6. The index is additive:

ind D_{𝔸_{1}♯𝔸_{2}} = ind^{(α)} D_{𝔸_{1}} + ind^{(−α)} D_{𝔸_{2}}.

Proof. Assume first that both operators D_{𝔸_{1}} and D_{𝔸_{2}} have no cokernel so that

ind D_{𝔸_{i}} = dimKer D_{𝔸_{i}},

and there are bounded right inverses

Q_{i}: L^{2,α}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})) → L_{1}^{2,α}(Ω^{1}(𝔤_{𝔼_{k}})),

D_{𝔸_{i}}Q_{i} = I such that

∥ Q_{i}(ρ) ∥_{L_{1}^{2,α}}≤ C ∥ ρ ∥_{L_{1}^{2,α}} .

By conformal invariance, one can stretch the gluing region in (X_{1}♯X_{2}, Σ_{1}♯Σ_{2}, n) to obtain an orbifold denoted (X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) for a real parameter T, gluing a cylinder of length 2T in the middle.

We first prove that for T large enough, there is an injection

σ: Ker D_{𝔸_{1}♯𝔸_{2}} → Ker D_{𝔸_{1}} ⊕ Ker D_{𝔸_{2}}.

Indeed, let σ(f) = (f_{1}, f_{2}) = (ϕ_{1}f − Q_{1}D_{𝔸_{1}}(ϕ_{1}f), ϕ_{2}f − Q_{2}D_{𝔸_{2}}(ϕ_{2}f)) for functions on (X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) such that ϕ_{1}^{2} + ϕ_{2}^{2} = 1, where ϕ_{i} is supported on

(X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) − (X_{j}♯_{T} (S^{3}, K, n) × (0, T/2)),

for i ≠ j and ∥ ∇ϕ_{i} ∥= ϵ(T) for ϵ(T) → 0 as T → ∞. Then using the fact that f ∈ Ker D_{𝔸_{1}♯𝔸_{2}}, we obtain

∥ f_{i} − f_{∞} ∥_{L_{1}^{2,α}} = ∥ Q_{i}D_{𝔸_{i}}(ϕ_{i}f) ∥
≤ C ∥ D_{𝔸_{i}}(ϕ_{i}f) ∥
≤ C ∥ ∇ϕ_{i}f ∥
≤ Cϵ(T) ∥ f ∥ .

As ϕ_{1}^{2} + ϕ_{2}^{2} = 1, ∥ f ∥_{L^{2,α}(X_{1}♯X_{2})}=∥ (ϕ_{1}f, ϕ_{2}f) ∥_{L^{2,α}(X_{1})⊕L^{2,α}(X_{2})} and

| ∥ σ(f) ∥ − ∥ f ∥ | = | ∥ σ(f) ∥ − ∥ (ϕ_{1}f, ϕ_{2}f) ∥ |
≤ ∥ (f_{1} − ϕ_{1}f, f_{2} − ϕ_{2}f) ∥
≤ \sqrt{2}Cϵ(T) ∥ f ∥ .

If σ(f) = 0 and ∥ f ∥≠ 0, then ∥ f ∥≤ \sqrt{2}Cϵ(T) ∥ f ∥ so for ϵ(T) ≤ 1/\sqrt{2}C, σ has to be an injection.

Also D_{𝔸_{1}♯𝔸_{2}} is surjective if the D_{𝔸_{i}} are. As above, there are right inverses Q_{i} to D_{𝔸_{i}}. Define

Q: L^{2,α}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})) → L_{1}^{2,α}(Ω^{1}(𝔤_{𝔼_{k}}))

by

Q(ρ) = ϕ_{1}^{2}Q_{1}(ρ_{1}) + ϕ_{2}^{2}Q_{2}(ρ_{2}),

for ρ_{i} the restriction of ρ to the support of ϕ_{i}. Then

(D_{𝔸_{1}♯𝔸_{2}}Q)ρ = ρ + ∇(ϕ_{1}^{2})Q_{1}(ρ_{1}) + ∇(ϕ_{2}^{2})Q_{2}(ρ_{2})

as D_{𝔸_{i}}Q_{i} = I and D_{𝔸_{1}♯𝔸_{2}} = D_{𝔸_{i}} over the support of ϕ_{i}. It follows that

∥ (D_{𝔸_{1}♯𝔸_{2}}Q − I)ρ ∥_{L^{2,α}}≤ 4C ∥ ρ ∥_{L^{2,α}}∥ ∇(ϕ_{i}^{2}) ∥_{L^{∞}} .

Hence choosing ϕ_{i} such that ∇(ϕ_{i}^{2}) = ϵ′(T), where ϵ′(T) → 0 as T → ∞, for T so large that ϵ′(T) ≤ 1/(4C) there is an inverse.

The final step, supposing surjectivity of the D_{𝔸_{i}}, is to show that for T large, there exists an injection

σ′: Ker D_{𝔸_{1}} ⊕ Ker D_{𝔸_{2}} → Ker D_{𝔸_{1}♯𝔸_{2}}.

This map is given by

σ′(f_{1}, f_{2}) = ϕ_{1}^{2}f_{1} + ϕ_{2}^{2}f_{2} − QD_{𝔸_{1}♯𝔸_{2}}(ϕ_{1}^{2}f_{1} + ϕ_{2}^{2}f_{2}).

As for the case of the construction of σ, QD_{𝔸_{1}♯𝔸_{2}}(ϕ_{1}^{2}f_{1}+ϕ_{2}^{2}f_{2}) is bounded by an arbitrarily small multiple of ∥ (f_{1}, f_{2}) ∥. Then, as above,

| ∥ σ′(f_{1}, f_{2}) ∥ − ∥ (f_{1}, f_{2}) ∥ | ≤ C′ϵ′(T) ∥ (f_{1}, f_{2}) ∥,

and anything in Ker σ′ is forced to be trivial if T is large enough. Therefore σ′ is injective as required.

Using σ, one gets

dim Ker D_{𝔸_{1}♯𝔸_{2}} ≤ dim Ker D_{𝔸_{1}} + dim Ker D_{𝔸_{2}},

while using σ′ one obtains

dim Ker D_{𝔸_{1}♯𝔸_{2}} ≥ dim Ker D_{𝔸_{1}} + dim Ker D_{𝔸_{2}}.

So the additive property is verified in the case where the operators are surjective.

In the case where the D_{𝔸_{i}} are not necessarily surjective, choose maps

F_{i}: ℝ^{n} → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})

over each (X_{i}, Σ_{i}, n) with images supported in the interior so that

D̃_{𝔸_{i}} = D_{𝔸_{i}} ⊕ F_{i}: Ω^{1}(𝔤_{𝔼_{k}}) ⊕ ℝ^{n_{i}} → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})

are surjective. Forming D̃_{𝔸_{1}♯𝔸_{2}} = D_{𝔸_{1}♯𝔸_{2}} ⊕ F_{1} ⊕ F_{2}, the gluing formula for the index in the surjective case may be used to get

ind D̃_{𝔸_{1}♯𝔸_{2}} = ind^{(α)} D̃_{𝔸_{1}} + ind^{(−α)} D̃_{𝔸_{2}}.

Now for

D_{𝔸_{i}}: Ω^{1}(𝔤_{𝔼_{k}}) ⊕ ℝ^{n_{i}} → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}),

the index will be

ind^{(α)} D_{𝔸_{i}} − n_{i},

where ind^{(α)} D_{𝔸_{i}} is computed for

D_{𝔸_{i}}: Ω^{1}(𝔤_{𝔼_{k}}) → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}).

Moreover, by a homotopy D_{𝔸_{i}} ⊕ tF_{i}, the operator

D_{𝔸_{i}}: Ω^{1}(𝔤_{𝔼_{k}}) ⊕ ℝ^{n_{i}} → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})

and D̃_{𝔸_{i}} have the same index. A similar argument applies to D_{𝔸_{1}♯𝔸_{2}} and D̃_{𝔸_{1}♯𝔸_{2}} to give finally

ind D_{𝔸_{1}♯𝔸_{2}} = ind D̃_{𝔸_{1}♯𝔸_{2}} − (n_{1} + n_{2}) = ind^{(α)} D_{𝔸_{1}} + ind^{(−α)} D_{𝔸_{2}}.

q.e.d.

If a variation of α crosses the spectrum of L_{A}, then not all the operators in the homotopy are Fredholm, and the index could change. For simplicity, suppose that there is only one end, and let α^{+} and α^{−} be respectively positive and negative such that |α^{±}| < δ, two weights for Sobolev spaces over the cylindrical end. The two indices ind^{(α^{+})} D_{𝔸} and ind^{(α^{−})} D_{𝔸} are well-defined and the relation between them is given by:

Theorem 3.7.

ind^{(α^{−})} D_{𝔸} − ind^{(α^{+})} D_{𝔸} = dimKer L_{A}.

Proof. The gluing result just proved may be used to reduce the computation of this difference to a computation of an index over the cylinder (with positive weight α^{+} at one end and negative weight α^{−} at the other). Over the cylinder there is separation of variables as usual, and using an isometry argument as in Proposition 3.2 and Theorem 3.4, one reduces the argument to an operator

\frac{d}{dt} + (L_{A} + σ(t)): L_{1}^{2}(Ω^{1}(𝔤_{𝔼_{k}})) → L^{2}(Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}})),

where σ(t) → α^{+} as t → +∞ and σ(t) → α^{−} as t → −∞. By splitting according to the spectrum of L_{A}, this operator may be reduced to an ODE operator on ℝ: \frac{d}{dt} + k(t). For k(t) = λ + σ(t), eigenvalues λ ∈ Spec L_{A}, the index contributions of this operator come only from the zero eigenspace. Therefore

ind (\frac{d}{dt} + L_{A} + σ(t)) = dimKer L_{A}.

q.e.d.

We now carry the discussion on the index of an adapted bundle over to the Floer setting: given an SU(2)-bundle E_{k} over a 3-orbifold (S^{3}, K, n), for each critical point of the Chern-Simons functional over E_{k} we wish to define an index. To do this we exploit the fact that there is only one reducible flat connection over E_{k} and, as in the case of Floer Homology for oriented homology 3-spheres, this will serve as a reference connection.

E_{k} can be pulled back to give a bundle over (S^{3}, K, n) × ℝ, and let 𝔼_{k} be an adapted bundle over (S^{3}, K, n) × ℝ with a connection 𝔸 such that the limiting flat data are the flat connections θ_{k} and A, where A is irreducible here. If 𝔼k^{′} is another such bundle, it was seen earlier that they are always equivalent as orbifold bundles but may be different as adapted bundles. For 𝔼_{k} and 𝔼k^{′} we have operators D_{𝔸} and D_{𝔸′} and, by the index theory developed, ind^{−} D_{𝔸} and ind^{−} D_{𝔸′}. We should like to define the index of A to be ind^{−}D_{𝔸}, but this a priori depends not only on the limiting data θ_{k} and A, but also on the chosen connection on the cylinder. Moreover, the following removes any ambiguity if we reduce modulo 4.

Theorem 3.8. The index of an adapted 𝔼_{k} over (S^{3}, K, n)×ℝ with limiting data θ_{k} and A depends only on A if it is reduced modulo 4.

Proof. This essentially follows the approach in usual Floer Homology, by the gluing results introduced before, combined with the index theory of [32]. In the orbifold setting, the index formula of Kronheimer and Mrowka is as follows:

Theorem 3.9 ([32, Section 6]). Let Σ ↪ X be a closed embedded surface of genus g in a compact 4-manifold. Consider singular connections 𝔸 with rational holonomy parameter α ∈ (0, 1/2) on a bundle with instanton and monopole numbers a and b. Then

ind D_{𝔸} = 8a + 4b − 3(b^{+}(X) − b_{1}(X) + 1) − (2g − 2).

By this index formula for D_{𝔸} on a closed orbifold, the standard argument yields the result. q.e.d.

This result partitions the set M_{flat}^{∗}(E_{k}) of critical points of the Chern-Simons functional over a 3-orbifold (S^{3}, K, n) into four classes. We impose the following non-degeneracy condition in order to make the definition below.

Condition 3.10. For every A ∈ M_{flat}(E_{k}), A is a non-degenerate critical point of CS: B(E_{k}) → ℝ/ℤ.

Definition 3.11. Let 𝔸 be a connection on an adapted bundle over (S^{3}, K, n) × ℝ with limits θ_{k} and A.

1. The Floer index of an element A ∈ M_{flat}^{∗}(E_{k}) is

μ(A) ≡ ind^{−} D_{𝔸} (mod 4).

2. The Floer chain groups for E_{k} over (S^{3}, K, n),

C_{i}^{(k)}(S^{3}, K, n) for i = 0, 1, 2, 3

are defined to be the free abelian groups generated by elements A ∈ M_{flat}^{∗}(E_{k}) such that μ(A) = i.

3.2 The Floer boundary operator

In analogy with Morse Theory, we wish to define a boundary operator between Floer chain groups, in order to define a homology complex, the Floer complex. This operator will, in fact, increase indices by 1:

∂: C_{∗}^{(k)}(S^{3}, K, n) → C_{∗+1}^{(k)}(S^{3}, K, n).

Just as Floer did, we define ∂ using finite energy instantons on the orbifold cylinder (S^{3} × ℝ, K × ℝ, n). Our luck is that these orbifold instantons correspond bijectively to singular instantons with holonomy parameter k/n over S^{3}×ℝ−(K×ℝ), and therefore we can avail ourselves the detailed analysis of Kronheimer and Mrowka of the moduli spaces of such instantons. The new feature is that we are dealing with orbifolds which are not compact, but have cylindrical ends. However, analytically, the transition from the compact to the non-compact case in the orbifold setting corresponds exactly to the transition in the manifold case, as treated by Floer in [22].

The definition of the boundary operator in Floer Theory requires a great deal of analytical techniques. It is not clear how much detail should be given as we adapt such techniques to our setting. In many cases, results proved for manifolds in [16], [17] or [22] carry over directly to orbifolds. We shall largely follow the unpublished manuscript [17], essentially observing that the analytical results there extend to orbifolds, often by restricting to invariant connections and operators on a cyclic branched cover. As some of those are perhaps not readily available, we give them and the appropriate definitions, focusing on the extension to orbifolds as it is essential in generalising the Floer Homology. For more details on standard techniques, the reader is referred to [17], [40] or [45]. For clarity, we shall first state the main results of the section, then make a few comments on the techniques needed for the proofs of these results, and finally proceed to the actual proofs. As the reducible flat connections play a minor role here, in this section, unless we mention otherwise, we shall be working only with irreducible flat connections.

Let 𝔼_{k} be an SU(2)-bundle over (S^{3}, K, n) × ℝ (or more generally over an orbifold with such cylindrical ends). As usual, fix a metric on (S^{3}, K, n) and pull it back to the cylinder. We are interested in the following moduli spaces:

Definition 3.12. Let M_{ASD}(A, A′, i) denote the moduli space of finite energy instantons 𝔸 over on 𝔼_{k} with limits A, A′ ∈ M_{flat}^{∗}(E_{k}) such that i = ind D_{𝔸}.

For convenience, we shall provisionally make the following transversality assumption:

Condition 3.13. For every 𝔸 in M_{ASD}(A, A′, i), D_{𝔸} is surjective.

When this condition is satisfied, M_{ASD}(A, A′, i) is said to be regular. Notice that there is an ℝ-action on this moduli space induced from translations along the cylinder. Denote by M̂_{ASD}(A, A′, i) the quotient M_{ASD}(A, A′, i)/ℝ. The main aim of this section is to prove the following theorem.

Theorem 3.14. The following assertions hold if M_{ASD}(A, A′, i) is regular.

1. M_{ASD}(A, A′, i) is a smooth oriented manifold of dimension i, and the reduced moduli space M̂_{ASD}(A, A′, i) = M_{ASD}(A, A′, i)/ℝ is a smooth manifold of dimension i − 1.

2. For μ(A′) − μ(A) ≡ 1 (mod 4), M̂_{ASD}(A, A′, 1) is compact and therefore the algebraic number of points in M̂_{ASD}(A, A′, 1), m(A, A′), may be defined.

3. For μ(A′) − μ(A) ≡ 2 (mod 4), M̂_{ASD}(A, A′, 2) has an oriented compactification given by

∂M̂_{ASD}(A, A′, 2) = ⨆_{A′′} M̂_{ASD}(A, A′′, 1) × M̂_{ASD}(A′′, A′, 1),

where the disjoint union is taken over all A′′’s such that μ(A′′) = μ(A) − 1.

4. Define ∂: C_{i}^{(k)}(S^{3}, K, n) → C_{i+1}^{(k)}(S^{3}, K, n) by

∂(A) = ∑_{μ(A′)=μ(A)+1} m(A, A′)A′.

Then by (3) ∂ ∘ ∂ = 0 and so the Floer Homology groups of (S^{3}, K, n) may be defined as the homology groups of the complex (C_{∗}^{(k)}(S^{3}, K, n), ∂).

The manifold property in clause (1) is obtained by proving the Fredholm property of the linearised ASD operator D_{𝔸} and by showing that for 𝔸 ∈ M_{ASD}(A, A′, i), ind D_{𝔸} = i. For the orientability, following [15], we need to show that the associated determinant line bundle is trivial. For clause (2), two distinct features come into play. The first one is present already in finite dimensional Morse Theory, where gradient lines between two critical points can be extended to broken trajectories, when the lines go through intermediate critical points. The second one is that in our gauge theoretic setting, “bubbles” may appear along the cylinder, so care is necessary. To prove the compactness of M̂_{ASD}(A, A′, 1), we need to exclude this possibility, and do so by adapting a general compactness principle given in [17] and, under another form, in [22]. For (3), to show the compactification property of M̂_{ASD}(A, A′, 2), the general compactness principle is again applied and, more to the point, a gluing construction for instantons is required. The latter is done using the Taubes construction.

Before we start, we mention two technical points. The first concerns the instanton and monopole numbers we shall be using. Recall that in the case of a closed orbifold, one has the instanton and monopole numbers a and b respectively. In dealing with an adapted bundle over an orbifold with cylindrical ends, things are slightly modified. This modification was explained in Section 3 of [31]. For an adapted bundle 𝔼_{k} over (S^{3}, K, n) × ℝ with limiting data A and A′, the resulting instanton and monopole numbers will be denoted ā and b̄, in order to avoid confusion with the case of a closed orbifold.

The second technical point concerns the Sobolev spaces in which we work. We follow Floer and work with L_{1}^{4}-connections and L_{2}^{4}-gauge transformations. The independence of the construction with respect to this particular Sobolev setting is given by the following lemmas.

Lemma 3.15. Given an instanton 𝔸 ∈ L_{1}^{p}(Ω^{1}(𝔤_{𝔼_{k}})) such that F_{𝔸} ∈ L^{p}(Ω^{2}(𝔤_{𝔼_{k}})), (p ≥ 2), there exist unique flat connections A_{−∞}, A_{∞} such that modulo gauge 𝔸 = A_{t} is C^{∞} convergent to A_{−∞} as t → −∞ and A_{∞} as t → ∞.

Proof. Let B_{T} be a band of width 1, (S^{3}, K, n)×[T −1, T], and denote by 𝔸_{T} the restriction of 𝔸 to B_{T}. As 𝔸 is a finite energy instanton, ∥ F_{𝔸_{T}} ∥→ 0 as T goes to infinity. By Uhlenbeck’s weak compactness for orbifolds (see e.g. [32, Proposition 7.4]), for any sequence T_{i} → ∞ and a flat connection A over B_{T}, after gauge transformation, 𝔸_{T′_{i}} converges to A in the C^{∞} sense over compact subsets. In particular 𝔸_{T −1/2} converges in C^{∞} to A. By Condition 3.10, A is isolated, and therefore the target flat connection A_{±∞} in the statement above does not depend on the choices of {T_{i}} and its converging subsequence {T′_{i}}. q.e.d.

Lemma 3.16. For 𝔸 as in the previous lemma and δ smallest positive eigenvalue of D_{𝔸}, there exists a constant C such that

|F_{y,t}(𝔸)| ≤ Ce^{δt}.

The proof is readily adapted from the manifold case (see Theorem 1 Section 4.1 in [17]) to the orbifold case. It is quite long but now well-known (see e.g. [40], [23]).

Lemma 3.17. For any 𝔸 ∈ L_{1}^{p}(Ω^{1}(𝔤_{𝔼_{k}})) with irreducible flat limits, D_{𝔸} is Fredholm and has index ind D_{𝔸}.

Proof. This is a straightforward adaptation of the material used to define the Floer index in Section 3.1. Notice that the result is true even if the flat limits are not irreducible, as long as weighted spaces are introduced. q.e.d.

Now we can proceed to the proof of (1). Exactly as in the standard Floer setting, in the Sobolev completions chosen, G(𝔼_{k}) has a Banach Lie group structure and acts smoothly on A^{∗}(𝔼_{k}), yielding B^{∗}(𝔼_{k}) = A^{∗}(𝔼_{k})/G(𝔼_{k}) which has a Banach manifold structure locally modeled on T_{𝔸,ϵ} = {𝔸 + a|d_{𝔸}^{∗}a = 0, ∥ a ∥_{L_{1}^{p}}≤ ϵ}, provided p > 2. For manifolds, proofs of such facts are by now very standard, and as A(𝔼_{k}) and B(𝔼_{k}) can be seen in a ℤ_{n}-invariant setting of objects defined for the cyclic branched cover V_{n}(K), the proofs in Section 2 of [22] adapt readily.

Proposition 3.18. Inside the Banach manifold B^{∗}(𝔼_{k}) lies the instanton moduli space M_{ASD}(A, A′, i), which is a manifold of virtual dimension ind D_{𝔸}.

Proof. By Lemma 3.17 and the implicit function theorem, Condition 3.13 implies directly that the irreducible part in M_{ASD}(A, A′, i) is a manifold whose dimension is indeed ind D_{𝔸}. The only obstacle is the possible presence of singularities corresponding to reducible instantons in the moduli space M_{ASD}(𝔼_{k}). For such points the local model given by the slices T_{𝔸,ϵ} is not valid. But given a connection 𝔸 on the cylinder whose limits are A, A′, by Arondzan’s Theorem (see Remark 3.2 in [23]) the following relation among the various stabilizers in the relevant gauge groups holds: Γ_{𝔸} ⊂ Γ_{A} ∩ Γ_{A′}. So for each element 𝔸 ∈ M_{ASD}(A, A′, i), Γ_{𝔸} is central in SU(2) by construction and therefore 𝔸 is irreducible. q.e.d.

The proof of (1) is completed if M_{ASD}(A, A′, i) is shown to be oriented. Here no differences arise in the method when we compare to ordinary Floer Homology. One essentially follows the construction given in [15]. The crucial fact to obtain is that the determinant line bundle det(Ker D_{𝔸}) be trivial, as this clearly implies the orientability of M_{ASD}(A, A′, i).

Proposition 3.19. The determinant line bundle det(Ker D_{𝔸}) over B(𝔼_{k}) is trivial.

Proof. This is proved in Appendix I of [33], and we refer to this article for a detailed discussion. q.e.d.

To prove clause (2) of Theorem 3.14, we first have to derive general compactness properties of the moduli spaces of ASD connections. There is an adequate notion of weak convergence yielding desired compactness properties for the moduli spaces. With appropriate restrictions on the index of critical points, the compactness of M̂_{ASD}(A, A′, 1) may then be derived. In this we follow the exposition of [17].

First of all we gain some idea about which instantons over the cylinder (S^{3}, K, n) × ℝ belong to the spaces M_{ASD}(A, A′, i) used to define ∂. The following lemma is a technical result and the proof, again, is a mild adaptation of material found in [17] (Lemma 2 in Section 5.1). Alternatively, one may refer to [40] (Theorem 4.0.1), where much greater generality is achieved.

Lemma 3.20. Let E_{k} be a bundle over (S^{3}, K, n). There are C, δ ≥ 0 such that if {𝔸_{α}} is a sequence of ASD connections on 𝔼_{k} over (S^{3}, K, n) × [0, ∞) with

∫_{0}^{∞} |F_{𝔸_{α}}|^{2} ≤ C,

then there is a subsequence {𝔸_{α′} }, a flat connection A and an instanton 𝔸 = A + 𝔞 over the half-cylinder such that, up to gauge equivalence, 𝔸_{α′} = A + 𝔞_{α′} and for each l ≥ 0 and h, p > 0

∫_{h}^{∞} |∇_{𝔸}^{(l)}(𝔞_{α′} − 𝔞)|^{p}e^{pδt} → 0

as α′ → ∞.

Corollary 3.21. The sequence of instantons {𝔸_{α}} converges in L_{l}^{p}, for all l, to a limit 𝔸 if and only if {𝔸_{α}} converges on compact subsets and

∫_{h}^{∞} |F_{𝔸_{α}}|^{2} → ∫_{h}^{∞} |F_{𝔸}|^{2}.

Proof. One way is easy: from convergence in L_{l}^{p}, in particular one has convergence on compact subsets and also

∫_{h}^{∞} |F_{𝔸_{α}}|^{2} − ∫_{h}^{∞} |F_{𝔸}|^{2} ≤ ∫_{h}^{∞} |F_{𝔸_{α}} − F_{𝔸}|^{2} → 0.

The converse statement follows from the previous lemma. q.e.d.

If we now consider moduli spaces on the whole cylinder, the previous results give that L_{l}^{p} convergence and convergence on compact subsets are equivalent:

Corollary 3.22. The sequence {𝔸_{α}} on 𝔼_{k} over (S^{3}, K, n) × ℝ converges to 𝔸 in the ASD moduli space of 𝔼_{k} if and only if {𝔸_{α}} converges to 𝔸 in C^{∞} on compact subsets of (S^{3}, K, n) × ℝ.

Proof. By elliptic regularity of the ASD equation, convergence in L_{l}^{p} gives C^{∞} convergence on compact subsets. Conversely, by Chern-Weil applied to {𝔸_{α}} converging in the C^{∞} sense to 𝔸 on compact subsets,

∫_{(S^{3},K,n)×ℝ} |F_{𝔸_{α}}|^{2} = ∫_{(S^{3},K,n)×ℝ} |F_{𝔸}|^{2}

and, using the previous corollary, get convergence in the L_{l}^{p} sense. q.e.d.

On the orbifold cylinder (S^{3}, K, n) × ℝ, we can define translations

c_{T} : (S^{3}, K, n) × ℝ → (S^{3}, K, n) × ℝ

by c_{T} (x, t) = (x, t + T). These maps pullback on the level of forms and connections and we will be interested in the effect on the level of connections: c_{T}^{∗} (𝔸). A translation vector T⃗ is a sequence of real numbers

T (1) ≤ T (2) ≤ ... ≤ T (N)

for some integer N. For a sequence {𝔸_{α}} on an adapted bundle 𝔼_{k} and a sequence T⃗_{α} of translation vectors with T⃗_{α}(i)−T⃗_{α}(i−1) → ∞ as α → ∞, we say that 𝔸_{α} is T⃗_{α}-convergent to connections 𝔸(1), 𝔸(2), ..., 𝔸(N) if each of the translates c_{T_{α}(i)}^{∗}(𝔸_{α}) converges on compact subsets to 𝔸(i). It should be pointed out that, because of bubbling, the ASD connections 𝔸(1), ..., 𝔸(N) over (S^{3}, K, n)×ℝ may live on adapted bundles different from 𝔼_{k}. However, by virtue of the flat limits providing the limiting data of these bundles, they must have the same isotropy as 𝔼_{k}. Let A_{−∞}, A_{+∞} be limiting flat connections on the adapted bundle 𝔼_{k}. A chain of bundles 𝔼⃗ = (𝔼(1), ..., 𝔼(N)) from A_{−∞} to A_{+∞} is a sequence of adapted bundles with limiting flat connections A_{−}(i), A_{+}(i) such that

A_{−∞} = A_{−}(1), A_{+}(1) = A_{−}(2), ..., A_{+}(N−1) = A_{−}(N), A_{+}(N) = A_{+∞}.

For such a chain 𝔼⃗, we can define the instanton and monopole numbers to be

ā(𝔼⃗) = ∑_{i=1}^{N} ā(𝔼(i)),

b̄(𝔼⃗) = ∑_{i=1}^{N} b̄(𝔼(i)).

A sequence {𝔸_{α}} is said to be chain convergent if there is a sequence T⃗_{α} of translation vectors, a chain of bundles 𝔼(1), ..., 𝔼(N) as above, and connections 𝔸(i) on 𝔼(i) such that {𝔸_{α}} is T⃗_{α}-convergent to 𝔸(1), ..., 𝔸(N). Similarly, weak chain convergence may be defined. The important feature is that there is a notion of convergence for any sequence. This is given by the following, which like the ideas is adapted from [17]:

Theorem 3.23. Any sequence {𝔸_{α}} of connections on 𝔼_{k} has a weak chain convergent subsequence. If the limit E⃗ has ā(E⃗) = ā(𝔼_{k}) and b̄(E⃗) = b̄(𝔼_{k}), then the subsequence is chain convergent.

Proof. The first thing to do is to find a positive “minimal energy number” which would be analogous to the energy of a “single instanton” in Floer’s setting (see Proposition 3b.2 in [22]). By the Chern-Weil formula, a non-flat instanton must have a strictly positive instanton number or monopole number (and maybe both). As there are two invariants in our setting, there is no “single instanton”, but in any case the energy of such an instanton is at least 8π^{2}(\frac{2k}{n}) again by Chern-Weil (recall that 0 < 2k/n < 1). This will be the “minimal energy number”.

It may be supposed that any non-flat instanton over the cylinder (S^{3}, K, n) × ℝ has energy greater than C and that C < 8π^{2}(\frac{2k}{n}), where C is as in Lemma 3.20. We wish to construct a limiting chain for a sequence {𝔸_{α}}. As the energy of 𝔸_{α} is greater than C, there is a unique T_{α} = T_{α}(1) such that

∫_{−∞}^{T_{α}} |F_{𝔸_{α}}|^{2} = C.

By replacing the 𝔸_{α} by their translates, one may suppose that T_{α} = 0. We know that we have weak convergence on compact subsets of the cylinder. Say that 𝔸_{α} → 𝔸(1) on compact subsets, for 𝔸(1) on a bundle 𝔼(1). What is needed is to verify that no energy is lost at the ends of the cylinder (outside the compact subsets) when we do the limiting process. This is given by the following equality:

Lemma 3.24. For any h > 0,

lim_{α→∞} ∫_{−∞}^{−h}|F_{𝔸_{α}}|^{2} = ∫_{−∞}^{−h}|F_{𝔸(1)}|^{2}.

Proof. As C < 8π^{2}(\frac{2k}{n}), so the 𝔸_{α} are converging strongly over (−∞, −h). Now suppose that the equality to be found does not hold:

C^{∗} = lim_{α→∞} ∫_{−∞}^{−h}|F_{𝔸_{α}}|^{2} − ∫_{−∞}^{−h}|F_{𝔸(1)}|^{2} ≠ 0.

First C^{∗} ≤ C by definition of C, and on the other hand by Lemma 3.16 on exponential decay along the tube, C^{∗} has to be positive. Define S_{α} ≤ −h by

∫_{−∞}^{S_{α}} |F_{𝔸_{α}}|^{2} = C^{∗}.

By strong convergence on compact subsets of 𝔸_{α}, one has S_{α} → −∞. Let 𝔸′_{α} be the translates of 𝔸_{α} by S_{α}. Possibly after extracting a subsequence, {𝔸′_{α}} converges. But then the limit is an instanton with energy less or equal to C^{∗} ≤ C, which contradicts the minimality of C. q.e.d.

Now Corollary 3.21 may be applied, to get that {𝔸_{α}} converges in L_{l}^{p} over the negative end. Also it follows that the limiting flat connection for 𝔸(1) on 𝔼(1) is A_{−∞}. If the equality in Lemma 3.24 extends to the whole cylinder, the other limiting flat connection for 𝔸(1) is A_{+∞}. In that situation {𝔸_{α}} is chain convergent to a chain with only one term, 𝔼(1). The convergence is strong if moreover the terms in the Chern-Weil formula agree. Using the additivity of Chern classes under gluing of connections, this equality means that ā(𝔼(1)) = ā(𝔼_{k}) and b̄(𝔼(1)) = b̄(𝔼_{k}). That is, the convergence is strong if and only if 𝔼(1) is isomorphic to 𝔼_{k}.

If the equality does not extend, suppose that

ā(𝔼_{k}) = ā(𝔼(1)) + ν,
b̄(𝔼_{k}) = b̄(𝔼(1)) + η,

where ν + \frac{2k}{n}η is positive. Let C′ = min(C, ν + \frac{2k}{n}η) and define T_{α}(2) by

∫_{−∞}^{T_{α}(2)}|F_{𝔸_{α}}|^{2} = C′.

In fact it turns out that C′ = C in the above. With T_{α}(2) define 𝔸_{α}(2) = c_{T_{α}(2)}^{∗}(𝔸_{α}). As above, T_{α}(2) → +∞ as α → +∞ and therefore taking a subsequence, {𝔸_{α}(2)} converges weakly on compact subsets to 𝔸(2) on a 𝔼(2), with ā(𝔼_{2}) ≤ ν and b̄(𝔼(2)) ≤ η.

The process may be finished at this stage, if the energies of 𝔸(1) and 𝔸(2) use up all the energy of the 𝔸_{α}. Else there is a remainder which is larger than C. Iterating the process, after a finite number of steps, all the energy must be used up, as at each stage at least C units are lost, and the energy along the cylinder is bounded by definition.

The only thing left to verify is that one has indeed a chain as required. For this it suffices to show that A_{+}(1) of 𝔼(1) is equal to A_{−}(2) of 𝔼(2). By Lemma 3.16 and elliptic regularity, there is a C′′ > 0 such that for any T > 1, any instanton over (−T, T) with energy less than C′′ may be represented as A + 𝔞 where A is a flat connection and 𝔞 satisfies the exponential decay conditions |∇^{(l)}𝔞| ≤ C_{l}e^{−δ(t−T)} for |t| ≤ T − 1. Then, as A is isolated, the exponential convergence of 𝔸_{α,t} and 𝔸_{α,−t} is towards the same connection; namely A above. Therefore

A_{+}(1) = A = A_{−}(2)

as required. q.e.d.

To prove compactness of M̂_{ASD}(A, A′, 1) = M_{ASD}(A, A′, 1)/ℝ and therefore complete the proof of clause (2) in Theorem 3.14, what we need is to strengthen Theorem 3.23, by restricting the index appropriately.

Theorem 3.25. If μ(A) − μ(A′) ≤ 4, any sequence {𝔸_{α}} in M_{ASD}(A, A′, i) has a chain convergent subsequence. Moreover if μ(A) − μ(A′) ≤ 2, then {𝔸_{α}} has a subsequence converging on a chain of length smaller or equal to 2 and with irreducible flat limits A_{−}(j), A_{+}(j).

Proof. For the first part, all that is left to do is to show that bubbling off may be avoided. Applying the index formula to a chain of length N, one has

μ(A) − μ(A′) = ∑_{i=1}^{N} μ(A_{+}(i)) − μ(A_{−}(i))

+ ∑_{i=1}^{N} dim I_{A_{±}}(i) + 8 ∑_{j=1}^{J} a_{j} + 4 ∑_{j=1}^{J} b_{j},

where a_{j}, b_{j} are instanton and monopole numbers coming from J concentrated instantons on (S^{4}, S^{2}) arising from bubbling off. There is some information about these: a_{j} ≥ 0 and a_{j} + b_{j} ≥ 0 for j = 1, ..., J. It follows that

4 ∑_{j=1}^{J}{2a_{j} + b_{j}} ≥ 4 ∑_{j=1}^{J} a_{j}

and hence by the index formula, if the index difference is less than or equal to 4, get J = 0, so that {𝔸_{α}} is not only weakly chain convergent, but chain convergent.

If, moreover, the index difference is less than or equal to 2, the index formula says that there is at most one intermediate flat limit A(1) between A_{−∞} and A_{+∞} and the isotropy group I_{A(1)} has to have dimension zero, implying the irreducibility of A(1). And if the index difference is 1, then there is none. q.e.d.

In order to prove (3) in Theorem 3.14, another technical ingredient is needed: Taubes’ gluing construction. This result is essential in defining the compactification of M_{ASD}(A, A′, 2), using moduli spaces M_{ASD}(A, A′′, 1) and M_{ASD}(A′′, A′, 1). We will deal mostly with points specific to Floer Homology, leaving out of the presentation some technical aspects which are well documented.

The first aspect to consider is the construction of an ASD connection 𝔸^{♯} over an orbifold connected sum (X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) from ASD connections 𝔸_{i} over the orbifolds with cylindrical ends (X_{i}, Σ_{i}, n). The connected sum is made with a cylinder of length T in the middle, for some large T, as discussed in [16], [17] or [23]. The solution involves three distinct steps which we outline. First estimates are used to show the existence of a connection 𝔸_{0} with small ∥ F_{𝔸_{0}}^{+} ∥ obtained by modifying 𝔸_{1} and 𝔸_{2} over the gluing region. Then one shows that for T large enough, the linearised ASD operator is invertible. Finally, the contraction mapping principle for Banach spaces is used to find a solution. The relevant result concerning this is Theorem 7.2.24 in [16] and the case of cylinders is more straightforward, as seen in [22] and [17]; for orbifolds the same estimates hold, as does the contraction mapping principle.

A slight generalisation needed is the case where we have compact subsets K_{i} ⊂ M_{ASD}(𝔼_{i}). Applying the construction above to each 𝔸_{i} ∈ K_{i}, for T > 0 large, there is a map

τ_{T} : K_{1} × K_{2} → M_{ASD}(𝔼^{♯}),

where 𝔼^{♯} is a bundle constructed over the connected sum. An important property of the map τ is the following.

Proposition 3.26. If T >> 0, τ_{T} is a diffeomorphism to its image and the image consists of regular points in M_{ASD}(𝔼^{♯}).

Proof. First show that the derivative dτ_{T} is an isomorphism if T >> 0. By the additivity of the index, it suffices to prove that dτ_{T} is injective. By definition of τ_{T}, one may restrict the problem to compact subsets of the (X_{i}, Σ_{i}, n) (around the gluing region), and then the result follows from the discussion in Section 7.2 of [16].

For regularity, 𝔸_{0}, has surjective d_{𝔸_{0}}^{+} with right inverse Q of bounded L^{2}-operator norm. Hence as τ_{T} (𝔸_{1}, 𝔸_{2}) = 𝔸 = 𝔸_{0} + 𝔞, with 𝔞 decaying exponentially with respect to T, the L^{2}-operator norm of d_{𝔸}^{+}Q − I can be made as small as desired so long as T is large enough. Consequently d_{𝔸}^{+} is surjective and 𝔸 is regular, for T >> 0. q.e.d.

One still needs to show a result of the following kind: there is a constant ϵ > 0 such that for T >> 0, any instanton 𝔸 = 𝔸_{0} + 𝔞 with ∥ 𝔞 ∥< ϵ, over a connected sum is gauge equivalent to a τ_{T} (𝔹_{1}, 𝔹_{2}), for 𝔹_{i} close to 𝔸_{i} in M_{ASD}(𝔼_{i}). This is proved in a standard way, using the method of continuity.

We still need to make more precise the notion of distance between 𝔸^{♯} and the 𝔸_{i}. For our purpose, we have used the distance

d(𝔸^{♯}, (𝔸_{i})) = inf_{g∈G(𝔼^{♯})}(∥ g · (𝔸^{♯} − 𝔸_{0}) ∥_{L^{4}} + ∥ d_{𝔸_{0}}^{+}(g(𝔸^{♯} − 𝔸_{0})) ∥_{L^{2}}).

Another notion of distance can be obtained by restricting our attention to the behaviour of connections on compact subsets C_{i}:

d_{C}(𝔸^{♯}, (𝔸_{i})) = ∑_{i=1}^{2} inf_{g∈G(𝔼^{♯})} ∥ g · 𝔸^{♯} − 𝔸_{i} ∥_{L^{4}(C_{i})} .

Proposition 3.27. Let 𝔸_{i} be instantons over (X_{i}, Σ_{i}, n) as above. Then there are compact subsets C_{i} ⊂ (X_{i}, Σ_{i}, n) and constants δ, C ≥ 0 such that for any instanton 𝔸^{♯} over (X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) for T > 0 large which satisfies d_{C}(𝔸^{♯}, (𝔸_{i})) ≤ δ, then

d(𝔸^{♯}, (𝔸_{i})) ≤ Cd_{C}(𝔸^{♯}, (𝔸_{i})).

Clearly d_{C}(𝔸^{♯}, (𝔸_{i})) is controlled by d(𝔸^{♯}, (𝔸_{i})). One has a partial converse using exponential decay; see Lemma 3.16. To do this, one chooses C_{i} and δ such that any instanton satisfying d_{C}(𝔸^{♯}, 𝔸_{i}) ≤ δ has very small energy on (X_{1}♯_{T} X_{2}, Σ_{1}♯_{T} Σ_{2}, n) − C_{1} ∪ C_{2}, and show that on the “neck” (S^{3}, K, n)×(−T/2, T/2)−C_{1} ∪C_{2} one has exponential decay |F_{A_{t}^{♯}}| ≤ C′e^{−δ(T −|t|)} (see [17] and also Proposition 2.d in [22]).

Then what was shown is that for ϵ > 0 small enough, T >> 0 and K_{i} compact subsets of regular elements in M_{ASD}(𝔼_{i}), there are neighbourhoods V_{i} ⊂ K_{i} and suitable compact subsets C_{i} of (X_{i}, Σ_{i}, n) for which

τ_{T} : V_{1} × V_{2} → M_{ASD}(𝔼^{♯})

is a diffeomorphism to its image and consists of regular points with the following properties:

- d_{C}(τ_{T} (𝔸_{1}, 𝔸_{2}), (𝔸_{i})) ≤ ϵ for all 𝔸_{i} in V_{i},

- any instanton 𝔸^{♯} with d_{C}(𝔸^{♯}, (𝔸_{i})) ≤ ϵ for some 𝔸_{i} ∈ K_{i} lies in the image τ_{T} (V_{1} × V_{2}).

We now have all the tools to complete the proof of (3) in Theorem 3.14. In general the moduli space M̂_{ASD}(A, A′, 2) is not compact. However, the ends of this moduli space may be identified with pairs (𝔸(1), 𝔸(2)) on a 2-term chain, by Theorem 3.23. And conversely, given a pair (𝔸(1), 𝔸(2)) by the gluing construction above, one can find an open subset N_{T} of M̂_{ASD}(A, A′, 2) with a single parameter T >> 0 such that any potentially divergent sequence in M̂_{ASD}(A, A′, 2) lies in N_{T}. This gives the compactification that was needed.

Thus what was achieved in this section is the construction of the 3-orbifold Floer groups HF_{i}^{(k)}(S^{3}, K, n) for i = 0, 1, 2, 3 associated to a knot K in S^{3} and depending on a rational in (0, 1/2). Up to now, nothing in our treatment allows us to claim that the Floer Homology is a knot invariant, as it may depend on the metric used on (S^{3}, K, n), upon which depends the construction of the Floer Homology. The invariance with respect to the metric will be proved in Section 4.3, when we discuss the functoriality property of HF_{∗}^{(k)}(S^{3}, K, n).

3.3 Perturbations

The goal of this section is to define an appropriate perturbation set-up to yield a well-defined Floer Homology for any knot in S^{3}. We will first perturb in order to remove degenerate elements in M_{flat}(E_{k}) and make sure that we have isolated points, to define the chain groups. Secondly, we will see how to use perturbations for the instanton equation to achieve the transversality for the definition of the boundary operator. This will remove the need for Conditions 3.10 and 3.13 in our construction. A discussion on the independence of the Floer Homology with respect to a generic perturbation concludes this section.

Before describing the construction, we make a digression. In the original case treated by Floer, when Y^{3} is a homology sphere, the perturbations used to define HF_{∗}(Y^{3}) did not really cause problems. A similar picture emerged from Taubes’ approach to Casson’s invariant for homology spheres.

On the other hand, when considering rational homology spheres, the picture is much more subtle. This was explained by Walker in [47], where an extension of Casson’s invariant was given. Here, one has to take into account non-trivial reducibles; as a result, this gives rise to Walker’s correction term, to account for the dependence upon the perturbations used. In [35], Lee and Li have extended Floer’s work to define the Floer Homology of 3-manifolds which are rational homology spheres, and the resulting Floer Homology depends on perturbations through Walker’s correction term.

The type of perturbations we will use were introduced by Herald in his work on Chern-Simons theory for 3-manifolds with boundary (see [26]). These are, with minor modifications, those used by Floer. Our task is to extend their approach to the setting of the 3-orbifolds (S^{3}, K, n). First we describe Herald’s construction and summarize the most important properties of the perturbed moduli space.

Consider an SU(2)-bundle E over a knot complement S^{3} − N_{K} and write Y = S^{3} − N_{K}. Take a collection {γ_{i}: S^{1} × D^{2} → Y }_{1≤i≤n} of embeddings of the solid torus in Y whose images are disjoint and also disjoint from the boundary torus of Y. Let η: D^{2} → ℝ be a bump function on D^{2} and define a function h: A(Y) → ℝ by

h(A) = ∑_{i=1}^{n} ∫_{D^{2}} h_{i}(trHol_{A}(γ_{i}(S^{1} × {x})))η(x)dx^{2},

where {h_{i}: ℝ → ℝ}_{1≤i≤n} is a collection of smooth functions. The function h is called an admissible perturbation function. Given an admissible perturbation function h, the perturbed flat moduli space of Y is

M_{h}(Y) = \{A ∈ B(Y) | − \frac{1}{2π} ∗ F_{A} + ∇h(A) = 0\}.

This moduli space is then the critical point set of the functional CS +h. The following theorem gives the main properties of M_{h}(Y) which will be needed.

Theorem 3.28 ([26, Theorem 15 (c)]). For a generic admissible perturbation function h, M_{h}(Y) is a compact stratified space

M_{h}(Y) = M_{h}^{SU(2)} ⨆ M_{h}^{U(1)} ⨆ M_{h}^{∗},

where M_{h}^{SU(2)} (the central perturbed flat connections) consists of two points. M_{h}^{U(1)} (the reducible perturbed flat connections) is a smooth compact 1-manifold except for two ends limiting to M_{h}^{SU(2)}. M_{h}^{∗} (the irreducible perturbed flat connections) is an oriented 1-manifold which is compact, except for open ends which limit to points in M_{h}^{U(1)}.

Let us now go to the orbifold setting. Given an SU(2)-bundle E_{k} over (S^{3}, K, n), it was already seen that M_{flat}(E_{k}) embeds in M_{flat}(Y) (in fact B(E_{k}) ↪ B(Y)). There is a natural way to adapt Herald’s perturbations to the orbifold setting. The important observation is that Herald perturbs the flatness equation along solid tori embedded away from the boundary of Y. When one considers the orbifold case, the only remaining operation is to glue in an orbifold solid torus. An admissible perturbation function can be defined for the orbifold as previously, by using a collection of solid tori embedded in (S^{3}, K, n), away from the orbifold solid torus. This last condition is required in order to keep a direct relation to Herald’s perturbations. Alternatively, one may consider the perturbed moduli space M_{h}(E_{k}) as the subset of elements A ∈ M_{h}(Y) such that trHol_{A}(μ) = 2cos(k/n2π). From this and Herald’s theorem, one may conclude the following about a functional CS + h.

Proposition 3.29. For a generic admissible perturbation function h, the set M_{h}^{∗}(E_{k}) of irreducible critical points of CS + h consists of finitely many points.

Let us concentrate now on the reducible flat connection θ_{k} on E_{k} over (S^{3}, K, n). For this, first recall the deformation theory along the reducible arc in M_{flat}(Y). This arc is parameterised by the holonomy of the connection along a meridian μ ∈ π_{1}(Y). Up to gauge equivalence Hol_{A}(μ) ⊂ S^{1} ⊂ SU(2). Let ξ ∈ S^{1}: ξ = e^{iα2π}. The ξ-equivariant signature of K, denoted σ_{α}(K), is the signature of the matrix B_{K}(ξ) = (1 − ξ)A_{K} + (1 − ξ̄)A_{K}^{T}, where A_{K} is an Alexander matrix for K and A_{K}^{T} its transpose. Also the Alexander polynomial of K is Δ_{K}(t) = det(A_{K}^{T} − tA_{K}).

Proposition 3.30 ([30, Theorem 19]). Let A be a reducible connection in M_{flat}(Y). If A satisfies Δ_{K}(Hol_{A}(μ)^{2}) ≠ 0, then H_{A}^{1}(Y) ≃ ℝ. Else, if Δ_{K}(Hol_{A}(μ)^{2}) = 0, then H_{A}^{1}(Y) is such that

dim H_{A}^{1}(Y) = 1 + 2dimKer B_{K}(Hol_{A}(μ)^{2}).

Notice that this implies in particular that if a reducible connection A_{0} is a limit point of some irreducible component of M_{flat}^{∗}(Y), then Δ_{K}(Hol_{A_{0}}(μ)^{2}) ≠ 0. A reducible flat connection, A, for which Hol_{A}(μ^{2}) is a root of the Alexander polynomial Δ_{K}(t) is called exceptional. And if an element β ∈ S^{1} satisfies Δ_{K}(β^{2}) = 0 we shall say, somewhat loosely, that β is a square root of the Alexander polynomial on S^{1}. We have the following obvious proposition.

Proposition 3.31. Along the abelian arc in M_{flat}(Y), there are only finitely many exceptional connections.

Therefore, for θ_{k} on E_{k} over (S^{3}, K, n), one has:

Proposition 3.32. If Δ_{K}(e^{i \frac{2k}{n} 2π}) ≠ 0, θ_{k} ∈ M_{flat}(E_{k}) is nondegenerate and isolated. Otherwise, θ_{k} is an exceptional connection.

A bundle E_{k} on which θ_{k} is exceptional we shall call an exceptional bundle for (S^{3}, K, n). We concentrate on the independence of perturbations of HF_{∗}^{(k)}(S^{3}, K, n) for non-exceptional bundles. In this case θ_{k} is isolated and non-degenerate and we are in a situation like that of Floer Homology for integral homology spheres: the critical point set may be perturbed compactly away from the reducible.

It is important to know whether there exists a pair (k, n) such that for all knots K ↪ S^{3}, E_{k} over (S^{3}, K, n) is not an exceptional bundle. For any knot K, the Alexander polynomial Δ_{K}(t) has the property that Δ_{K}(−1) is an odd integer. This implies the following:

Proposition 3.33. For any knot K in S^{3}, the orbifold bundle E_{1} over the 3-orbifold (S^{3}, K, 4) is not an exceptional bundle.

Proof. The reducible connection θ_{1} on E_{1} is such that

Hol_{θ_{1}}(μ) = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) ,

so Δ_{K}(e^{iπ}) = Δ_{K}(−1) ≠ 0 and therefore θ_{1} is not exceptional and hence neither is E_{1}. q.e.d.

Thus for the pair (1, 4), the Floer Homology HF_{∗}^{(1)}(S^{3}, K, 4) can always be defined and will be shown to be independent of perturbations.

More generally, the same will be true for a relatively prime pair (k, p^{n}), for any p prime and any n.

We now perturb so as to achieve the transversality of the instanton moduli spaces which we have imposed to state Theorem 3.14 (Condition 3.13). Using the interpretation of the instanton equation on the cylinder as the gradient flow of the Chern-Simons functional, we may perturb this instanton moduli space by simply perturbing the functional CS.

Given an admissible perturbation h, for any A, A′ ∈ M_{h}(E_{k}), let the perturbed instanton moduli space connecting corresponding to M_{ASD}(A, A′, i) be

M_{h}(A, A′, i) = \{𝔸 = A_{t} | \frac{dA_{t}}{dt} + ∗F_{A_{t}} + ∇h(A_{t}) = 0 , Y M(𝔸) < ∞\},

in analogy with what was done in Section 3.2. Say that M_{h}(A, A′, i) is regular if D_{𝔸} is surjective for all 𝔸 ∈ M_{h}(A, A′, i).

Now fix a metric on (S^{3}, K, n) (and hence on the cylinder as we take the product metric there) and an admissible perturbation h_{0}. We have to produce “enough” perturbations so that around h_{0}, a generic perturbation h will yield a regular moduli space M_{h}(A, A′, i), for all A, A′ ∈ M_{h}(E_{k}). As Herald’s perturbations are essentially the ones used by Floer, we can adapt Floer’s argument.

Proposition 3.34. Given a fixed metric on (S^{3}, K, n) and an admissible perturbation h_{0}, the set of admissible perturbations for which M_{h}(A, A′, i) is regular for all A, A′ ∈ M_{h}(E_{k}) is of first Baire category near h_{0}.

Proof. The proof follows that of Proposition 2c.2 in [22]. q.e.d.

Remark 3.35. The only point that could be unclear in the adaptation of Floer’s proof in the orbifold setting is the use of the Sard-Smale Theorem to achieve transversality. The important observation is that in the context of 3-orbifolds, as explained in Section 2.3, one deals with connections which are ℤ_{n}-invariant when seen on an equivariant bundle over the cyclic branched cover. This is a subspace of the equivariant space studied in [25] (see also Remark 2.12 in [25]). We do not require a ℤ_{n}-manifold structure, but the weaker structure of an ordinary manifold. A similar situation occurred under different forms in [20], [32] and [9]. There is no transversality obstruction in this context, and in particular the Sard-Smale Theorem applies.

We summarize the results of this section, in the following theorem.

Theorem 3.36. Given any knot K in S^{3}, there is a well-defined 3-orbifold Floer Homology HF_{∗}^{(k)}(S^{3}, K, n) for any pair (k, n), except for those pairs for which θ_{k} yields a square root of Δ_{K}(t). The Floer Homology is independent of perturbations when their use is required.

Proof. Given K in S^{3}, choose a non-exceptional bundle E_{k} over a (S^{3}, K, n). On E_{k}, the reducible θ_{k} is isolated and non-degenerate. Then

M_{flat}(E_{k}) = M_{flat}^{∗}(E_{k}) ⨆ {θ_{k}},

and one needs only to worry about M_{flat}(E_{k}). Using Herald’s perturbations adapted to our setting, for a generic admissible perturbation function h,

M_{h}(E_{k}) = M_{h}^{∗}(E_{k}) ⨆ {θ_{k}},

where M_{h}^{∗}(E_{k}) is finite. Each A ∈ M_{h}^{∗}(E_{k}) has a well-defined index μ(A) ∈ ℤ_{4}. Define the chain groups C_{[h],∗}^{(k)}(S^{3}, K, n) and the boundary operator as was previously done, possibly using perturbations to have regular instantons. The result is a Floer Homology HF_{[h],∗}^{(k)}(S^{3}, K, n). Floer’s cobordism argument will then be applied (in Section 4.3) to obtain independence of the construction with respect to the generic perturbation function and metric used. q.e.d.

4. Properties of Floer homology

We start this section by giving some elementary properties of the Floer Homology developed in the last section, in particular noting how it generalises Floer’s original construction. Then we develop a local deformation theory for the Floer Homology which enables us to generalise our construction from orbifold connections to singular connections with irrational holonomy parameter. Next, the functoriality of Floer Homology under orbifold cobordism is studied as, in particular, this is formally needed to prove invariance under change of metrics and perturbations. Finally, the Euler characteristic of the Floer Homology is related to a classical knot invariant, the equivariant signature of a knot.

4.1 Basic properties

The first example, a rather trivial one, is the unknot. Here there are no irreducible flat connections and is one reducible flat connection, θ_{k}, which is non-degenerate. Therefore the Floer Homology HF_{∗}^{(k)}(S^{3}, K, n) is always trivial. Similarly, we have

Proposition 4.1. For any knot K in S^{3}, HF_{∗}^{(0)}(S^{3}, K, n) ≡ 0; if n is even HF_{∗}^{( \frac{n}{2} )}(S^{3}, K, n) ≡ 0, and more generally there is an ϵ_{K} > 0 such that if 0 < k/n < ϵ_{K} or 1/2 − ϵ_{K} < k/n < 1/2, then HF_{∗}^{(k)}(S^{3}, K, n) ≡ 0.

Proof. As above, there are no irreducible flat SU(2)-connections in the case k = 0 or k = n/2. Using Theorem 3.31, it follows that r(M_{flat}^{∗}(S^{3} − N_{K})) ∩ S_{\frac{k}{n}} = ∅ for k/n close enough to 0 or 1/2. q.e.d.

Up to now, we have focused on knots in S^{3}, but more generally we may consider knots K ↪ Y^{3} in a homology sphere Y^{3}. There is no difficulty in developing a Floer Homology for 3-orbifolds (Y^{3}, K, n), as we have done for (S^{3}, K, n). Denote this Floer Homology HF_{∗}^{(k)}(Y^{3}, K, n). This Floer Homology is a generalisation of ordinary Floer Homology. Notice that for k = 0, there are not 4 but 8 Floer chain groups, and as E_{0} and 𝔼_{0} have trivial isotropy, the connections involved extend across the singular loci (K and K × ℝ) of the orbifolds ((S^{3}, K, n) and (S^{3}, K, n) × ℝ). This readily implies the following:

Proposition 4.2. HF_{∗}^{(0)}(Y^{3}, K, n) ≃ HF_{∗}(Y^{3}).

Next, we are interested in what happens when one applies an orientation reversing diffeomorphism to S^{3}. Given K in S^{3}, applying such a diffeomorphism changes K into its mirror image K̄. There is a relation between the Floer Homology of K and that of K̄, which is not difficult to obtain.

Proposition 4.3. HF_{∗}^{(k)}(S^{3}, K̄, n) = HF_{−∗−1}^{(k)}(S^{3}, K, n).

It follows that any knot K for which a Floer Homology satisfies

HF_{0}^{(k)}(S^{3}, K, n) ≠HF_{3}^{(k)}(S^{3}, K, n)
or HF_{1}^{(k)}(S^{3}, K, n) ≠HF_{2}^{(k)}(S^{3}, K, n)

will be distinguished from its mirror image.

4.2 Deformation theory

In this section we explore how the Floer Homology varies as one changes the pair (k, n) used to define HF_{∗}^{(k)}(S^{3}, K, n). One aim is to extend the construction of a Floer Homology for knots to cover the case of irrational holonomy parameter singular connections, which is excluded when working with orbifolds.

Start with a Floer Homology HF_{∗}^{(k)}(S^{3}, K, n), for a knot K ↪ S^{3} and a pair of integers (k, n) such that E_{k} is not an exceptional bundle, and suppose that the elements in M_{flat}^{∗}(E_{k}) which are generators of

C_{∗}^{(k)}(S^{3}, K, n) are non-degenerate. Otherwise simply use an appropriate perturbation as done in Section 3.3.

Theorem 4.4. There exists an ϵ_{K} > 0 depending on the knot K and the pair (k, n), such that if (k′, n′) is another pair of integers for which |k/n − k′/n′| < ϵ_{K}, then

HF_{∗}^{(k′)}(S^{3}, K, n′) ≃ HF^{(k)}(S^{3}, K, n).

Proof. By the non-degeneracy assumption, an element A ∈ M_{flat}^{∗}(E_{k}) may be seen as lying in a one-dimensional component of M_{flat}^{∗}(S^{3} − N_{K}), which is locally an arc A_{t} centred about A, which may be parameterised by the trace of

Hol_{A_{t}}(μ) = (\begin{matrix}e^{i2tπ} & 0 \\ 0 & e^{−i2tπ} \end{matrix}) ,

where t ∈ (k/n − ϵ/2, k/n + ϵ/2) for a small enough ϵ > 0 as to keep the local parameterisation. Then for any pair (k′, n′) such that k′/n′ is in the above interval, there exists a corresponding A′ ∈ M_{flat}^{∗}(E_{k′}) over the 3-orbifold (S^{3}, K, n′), and A′ is well-defined. As there are finitely many elements in M_{flat}^{∗}(E_{k}), take ϵ > 0 small enough to have a oneto-one correspondence A ↔ A′, as above, for all A ∈ M_{flat}^{∗}(E_{k}) and A′ ∈ M_{flat}^{∗}(E_{k′}).

Now assemble these critical points into C_{∗}^{(k)}(S^{3}, K, n) and

C_{∗}^{(k′)}(S^{3}, K, n′). For A and A′ given by the construction above, their Floer indices μ(A) and μ(A′) are equal. To prove this, one has to apply the deformation invariance of the index with respect to the variation in the holonomy parameter given in Section 6 of [32]. All connections along the arc A_{t} have to be non-degenerate: H_{A_{t}}^{1} = 0. Otherwise, the gluing formula for the index implies that there is a change in the index at an A_{t_{0}} where H_{A_{t_{0}}}^{1} is non-zero. By the choice of ϵ, such points are avoided, and hence one gets a chain group isomorphism

C_{∗}^{(k′)}(S^{3}, K, n′) ≃ C_{∗}^{(k)}(S^{3}, K, n).

Now take into account the boundary operator in the two Floer Homologies. For this, the extended moduli space of singular instantons, parameterised by the holonomy parameter is needed. Adapting Proposition 2.8 in [32] to our setting, start with M_{ASD}(A_{1}, A_{2}) for flat connections on E_{k}, A_{1}, A_{2}. Take ϵ > 0 as needed above to obtain corresponding A1^{′} and A2^{′} on E_{k′}. Consider M_{ASD}^{t}(A_{1}, A_{2}) as extended moduli space with holonomy parameter t such that at t = k/n and t = k′/n′, one has M_{ASD}(A_{1}, A_{2}) and M_{ASD}(A1^{′}, A2^{′}) respectively. This moduli space is a manifold of dimension μ(A_{1}) − μ(A_{2}) + 1, but taking into account the ℝ-translations along the cylinder, form the reduced moduli space M̂_{ASD}^{t}(A_{1}, A_{2}), whose dimension will be μ(A_{1}) − μ(A_{2}) and whose restriction to t = k/n and t = k′/n′ gives the reduced instanton moduli spaces between A_{1}, A_{2} and A2^{′}, A2^{′} respectively. The case of interest for the boundary operator is when μ(A_{1}) − μ(A_{2}) ≡ 1 (mod 4). Then M̂_{ASD}^{t}(A_{1}, A_{2}) is a 1-dimensional manifold whose restrictions at each end gives the boundary operator ∂ and ∂′ for (C_{∗}^{(k)}(S^{3}, K, n), ∂) and

(C_{∗}^{(k′)}(S^{3}, K, n′), ∂′). Let ϵ′ > 0 be small enough so that, if k/n ≤ t ≤ k/n + ϵ′, M̂_{ASD}^{t}(A_{1}, A_{2}) is a product M̂_{ASD}(A_{1}, A_{2}) × [k/n, k/n + ϵ′]. Now take ϵ_{K} = min{ϵ, ϵ′}. Then the Floer chain groups and boundary operators coincide, and thus the Floer Homologies are naturally isomorphic. q.e.d.

Theorem 4.4 may be used to give a model for a general Floer Homology for knots built from singular connections over knot complements with irrational holonomy parameter α: we have a well-defined notion of limit of Floer groups with respect to rational parameters converging to an irrational parameter and hence we set:

Definition 4.5. If α ∈ (0, 1/2) is such that Δ_{K}(e^{i2α2π}) ≠ 0, define the Floer Homology

HF_{∗}^{(α)}(S^{3}, K) = lim_{\frac{k}{n} →α} HF_{∗}^{(k)}(S^{3}, K, n).

We may now give a reformulation of Theorem 4.4 in light of the general construction HF_{∗}^{(α)}(S^{3}, K).

Corollary 4.6. Let Z = {α ∈ [0, 1/2] | Δ_{K}(e^{i2α2π}) = 0}. The Floer Homology HF_{∗}^{(α)}(S^{3}, K) is constant on connected components of [0, 1/2]\Z.

4.3 Functoriality property

In [22], Floer pointed out one of the novel aspects about his Homology Theory for homology spheres: this is the so-called functoriality property of HF_{∗}(Y). He considers a smooth cobordism X^{4} between two 3-manifolds Y_{1} and Y_{2} and shows that there is an induced group homomorphism

Ψ_{X}: HF_{i}(Y_{1}) → HF_{i+ξ}(Y_{2}),

where ξ = 3(b^{+}(X) + b_{1}(X)). Moreover, this construction yields the identity homomorphism if X is the product cobordism and is functorial with respect to composition of cobordisms: Ψ_{X∘X′} = Ψ_{X} ∘ Ψ_{X′}. On the other hand, in [35], where a Floer Homology for rational homology spheres was defined, Lee and Li pointed out that the properties described by Floer did not hold in general and gave an example where the map Ψ cannot be functorial. Therefore, in trying to understand the behaviour of Floer Homology for 3-orbifolds under cobordism, we have to worry about the functoriality property. First, a definition of orbifold cobordism is needed. Although there exists a general theory (see [18]), we shall be rather practical here: a 4-dimensional orbifold (X^{4}, Σ^{2}, n) is said to be a cobordism between 3-orbifolds (Y_{1}, K_{1}, n) and (Y_{2}, K_{2}, n) if X is a manifold cobordism between Y_{1} and Y_{2}, with a smoothly embedded surface Σ ↪ X such that Σ ∩ Y_{i} = K_{i} for i = 1, 2. We denote by χ(Σ) the Euler characteristic of the bordered surface Σ. Adjoin to (X, Σ, n) ends (S^{3}, K_{i}, n) × ℝ on each side to obtain an orbifold with cylindrical ends, which is also denoted (X, Σ, n). The situation is here identical to the one encountered when defining the boundary operator in the Floer Homology, except for the fact that we now allow Σ to be any surface of genus g rather than a standard product K × ℝ. We consider a bundle E_{k}^{(i)} (k ≠ 0) over (S^{3}, K_{i}, n), along with an adapted bundle 𝔼_{k} over (X, Σ, n). As in Section 3.2, using the appropriate analytical set-up, we can define M_{ASD}(A_{1}, A_{2}), the space of finite energy instantons on 𝔼_{k} limiting to A_{1} ∈ M_{flat}(E_{k}^{(1)}) and A_{2} ∈ M_{flat}(E_{k}^{(2)}). Then, generically, M_{ASD}(A_{1}, A_{2}) is a smooth oriented manifold whose dimension is computed by adapting the index computation mentioned in Theorem 3.9:

dim M_{ASD}(A_{1}, A_{2}) = μ(A_{1}) − μ(A_{2}) − 3(b^{+}(X) − b_{1}(X)) + χ(Σ).

We shall be concerned with knots in S^{3}, although a set-up for knots in homology spheres could be considered and our results extend to that situation. Moreover, as our interest is in the knots K_{1} and K_{2}, we will work with X = S^{3} × [0, 1]. Much in the same way as for the boundary operator, let

ψ_{(X,Σ,n)} : C_{∗}^{(k)}(S^{3}, K_{1}, n) → C_{∗−χ(Σ)}^{(k)}(S^{3}, K_{2}, n)

be defined by

ψ_{(X,Σ,n)}(A_{1}) = ∑_{A_{2}∈M_{flat}^{∗}(E_{k}^{(2)})} n(A_{1}, A_{2}) · A_{2},

where n(A_{1}, A_{2}) is the algebraic number of elements in M_{ASD}(A_{1}, A_{2}) (which is 0-dimensional by the above grading shift, and also compact following [22]). Notice that ψ_{(X,Σ,n)} may depend on the metric chosen to obtain instantons, but what is important is that this does not carry over to the Floer Homology. Indeed, if ψ_{(X,Σ,n)} induces Ψ_{(X,Σ,n)} in homology, then we have the following result.

Theorem 4.7.

Ψ_{(X,Σ,n)} : HF_{∗}^{(k)}(S^{3}, K_{1}, n) → HF_{∗−χ(Σ)}^{(k)}(S^{3}, K_{2}, n)

depends only on the orbifold cobordism (X, Σ, n). For a composite cobordism (X∘X′, Σ∘Σ′, n), one has Ψ_{(X∘X′,Σ∘Σ′,n)} = Ψ_{(X,Σ,n)}∘Ψ_{(X,Σ,n)}. For the standard product cobordism, the map Ψ_{(S^{3},K,n)×[0,1]} is the identity.

Proof. We shall briefly discuss here as there is nothing really new in the orbifold case compared to Floer’s case, and all the analytical techniques required were explained in the construction of Floer Homology. That Ψ_{(X,Σ,n)} depends only on (X, Σ, n) is similar to the boundary property ∂^{2} = 0 proved for ∂, and the argument given in Theorem 3 of [22] simply lifts to orbifolds.

As mentioned earlier when citing work of Lee and Li, the delicate point is the possibility that Ψ may not be functorial. First notice that the grading shift is compatible with functoriality as the Euler characteristic of a bordered surface is additive. The proof given in [22] could break down if one needs to perturb the Chern-Simons functional (and hence the ASD equation). Then it is possible that two different perturbations yield different critical points, or also the Floer index may be affected, as explained in [35]. However, in defining the Floer Homologies HF_{∗}^{(k)}(S^{3}, K_{i}, n), we have been careful to impose that each of the reducible connections θ_{k}^{(i)} ∈ M_{flat}(E_{k}^{(i)}) be non-degenerate. In this case, one can perturb compactly away from the reducible, and the situation described in [35, Example 5.1] cannot occur. Floer’s argument then applies. Finally, that Ψ_{(S^{3},K,n)×[0,1]} is the identity follows directly from the functoriality property. q.e.d.

With the functoriality property of the Floer Homology HF_{∗}^{(k)}(S^{3}, K, n), it is now a purely formal consequence that HF_{∗}^{(k)}(S^{3}, K, n) is independent of the metric used in its construction, as the product cobordism (S^{3}, K, n) × [0, 1] (with any metric) induces the identity map on Floer Homology.

4.4 Relation to cyclic branched covers

We suppose in this section that n is odd and that the cyclic branched cover V_{n}(K) is a homology sphere. The orbifold exact sequence yields, via holonomy, a pull-back map from M_{flat}(E_{k}) to M_{flat}(V_{n}(K)), preserving irreducibility when V_{n}(K) is a homology sphere. If each flat connection A ∈ M_{flat}^{∗}(E_{k}) does induce a flat connection Ã ∈ M_{flat}(V_{n}(K)), the converse is not true. Indeed, a flat connection over V_{n}(K) is not necessarily ℤ_{n}-invariant (and hence does not “push-down” to the quotient 3-orbifold). Our aim is to compare the relative index of generators in C_{∗}^{(k)}(S^{3}, K, n) with the relative index of the corresponding generators in C_{∗}(V_{n}(K)), as this will be used later on.

Let A ∈ C_{∗}^{(k)}(S^{3}, K, n) be a generator in the 3-orbifold Floer complex, and let Ã ∈ C_{∗}(V_{n}(K)) be the corresponding generator for the Floer complex of V_{n}(K). The reducible θ_{k} ∈ M_{flat}(E_{k}) corresponds to the trivial connection θ ∈ M_{flat}(V_{n}(K)). Choose an adapted connection 𝔸 on 𝔼_{k} over (S^{3}, K, n) × ℝ, limiting to θ_{k} and A at the ends. When seen as a ℤ_{n}-invariant connection over V_{n}(K)×ℝ, the connection 𝔸 induces an adapted connection 𝔸̃ on a bundle 𝔼̃ over V_{n}(K) × ℝ with limits θ and Ã at the ends. The Floer indices of A and Ã are given respectively as

μ(A) = ind D_{𝔸} (mod 4), μ(Ã) = ind D𝔸̃ (mod 8)

for operators

D_{𝔸}: Ω^{1}(𝔤_{𝔼_{k}}) → Ω^{0}(𝔤_{𝔼_{k}}) ⊕ Ω^{+}(𝔤_{𝔼_{k}}), D𝔸̃: Ω^{1}(𝔤𝔼̃) → Ω^{0}(𝔤𝔼̃) ⊕ Ω^{+}(𝔤𝔼̃)

as defined in Section 3.1 (here we have removed the weighted Sobolev subscripts to simplify the notation). The following relates the Floer indices of A and Ã.

Proposition 4.8. μ(A) ≡ μ(Ã) (mod 2).

Proof. A ℤ_{n}-action on V_{n}(K) induces an action on differential forms, yielding decompositions

Ω^{1}(𝔤𝔼̃) = ⨁_{i=0}^{n−1} Ω^{1}(ρ_{i}), Ω^{0}(𝔤𝔼̃) = ⨁_{i=0}^{n−1} Ω^{0}(ρ_{i}), Ω^{+}(𝔤𝔼̃) = ⨁_{i=0}^{n−1} Ω^{+}(ρ_{i}),

for representations ρ_{i} ∈ ℤ̂_{n}. In these decompositions, the contribution from the trivial representation ρ_{0} is precisely the ℤ_{n}-invariant subspaces. This gives the following expression for the operator D_{𝔸}: D_{𝔸} = D𝔸̃_{|Ω^{1}(ρ_{0})}. Thus,

ind D𝔸̃ = ∑_{i=0}^{n−1} ind D𝔸̃_{|Ω^{1}(ρ_{i})} = ind D_{𝔸} + ∑_{i=1}^{n−1} ind D𝔸̃_{|Ω^{1}(ρ_{i})}.

As n was supposed to be odd at the beginning of this section, it follows from basic Representation Theory that the finite dimensional subspaces Ker D𝔸̃_{|Ω^{1}(ρ_{i})} and Coker D𝔸̃_{|Ω^{1}(ρ_{i})} are complex vector spaces for i = 1, ..., n − 1. Consequently, ind D𝔸̃_{|Ω^{1}(ρ_{i})} is an even number if i = 1, ..., n − 1 and hence

μ(Ã) ≡ ind D𝔸̃ = ind D_{𝔸}+∑_{i=1}^{n−1} ind D𝔸̃_{|Ω^{1}(ρ_{i})} ≡ ind D_{𝔸} ≡ μ(A) (mod 2).

q.e.d.

4.5 Relation to equivariant signatures

Work of Taubes, by providing a gauge theoretic framework to Casson’s invariant, shows that the Euler characteristic of the Floer Homology of a homology sphere Y^{3} is twice the Casson invariant of Y^{3}. One is therefore naturally interested in a possible relation between the Euler characteristic of the Floer Homology HF_{∗}^{(k)}(S^{3}, K, n) and some known knot invariant. The appropriate invariant was introduced by Herald in [27], where he defined a Casson-type invariant for knots in homology spheres, generalizing ideas developed in [47]. The main result of [27] is to relate this Casson-type invariant to a classical knot invariant, the Tristram-Levine signature. We shall exploit this to interpret the Euler characteristic of HF_{∗}^{(k)}(S^{3}, K, n) in terms of such equivariant signatures of K, which were introduced in Section 3.3. The main result in this section is:

Theorem 4.9. The Euler characteristic of HF_{∗}^{(k)}(S^{3}, K, n) is given by:

χ(HF_{∗}^{(k)}(S^{3}, K, n)) = \frac{1}{2} · σ_{2 \frac{k}{n}} (K).

We shall prove this using Herald’s work, and therefore we should first explain that briefly. Throughout it will be understood that we suppose M_{flat}(S^{3} −N_{K}) to be non-degenerate (and we use a perturbation if this is not the case). M_{flat}(S^{3} − N_{K}) consists of a union of three smooth strata corresponding to irreducible, abelian and central flat connections. Recall there is a restriction map r: M_{flat}(S^{3} − N_{K}) → M_{flat}(T^{2}) and a flat connection A with tr Hol_{A}(μ) = 2cos(2απ) (α ∈ (0, 1/2)) corresponds to an intersection in M_{flat}(T^{2}) of M_{flat}(S^{3} − N_{K}) with a vertical slice S_{α}. Consider the 2-fold covers of M_{flat}(S^{3} − N_{K}), and M_{flat}(T^{2}) ⊃ S_{α} along the two central flat connections: M̃_{flat}(S^{3}−N_{K}), M̃_{flat}(T^{2}) ⊃ S̃_{α} respectively. Then r is covered by

r̃: M̃_{flat}(S^{3} − N_{K}) → M̃_{flat}(T^{2}).

Herald’s invariant, λ_{α}(K), is the oriented intersection number

r̃(M̃_{flat}^{∗}(S^{3} − N_{K})) · S̃_{α},

and he proves the following:

Theorem 4.10 ([27, Corollary 3]). λ_{α}(K) = \frac{1}{2} · σ_{2α}(K).

Proof. We give a brief outline and refer to [27] for more details. First Herald introduces another number, S(α). At a bifurcation point A along the abelian arc in M_{flat}(S^{3} − N_{K}), he shows that up to reparameterisation the Hessian of A + t satisfies H_{A+t} = ±tI. This gives a number associated to A, ς(A) = ±1. He defines

S(α) = ∑_{A} ς(A),

where the sum is taken over all bifurcation points A along the abelian arc such that tr Hol_{A}(μ) = 2cos(2α_{A}π) with α_{A} ∈ (0, α). Herald first shows (Lemma 8 in [27]) that

λ_{α}(K) = S(α).

Now reducible connections in M_{flat}(S^{3} −N_{K}) can be viewed as connections in M_{flat}(Ȳ), where Ȳ is the 3-manifold obtained by a 0-surgery on S^{3} along K. In particular, let Ā_{0} and Ā_{α} be such connections with holonomy parameter 0 and α respectively. There is a well-defined spectral flow SF (Ā_{0}, Ā_{α}) between the two connections (see [29] or [2] for details on spectral flow). Herald proves the following (Corollary 10 in [27]):

2S(α) − 4 = SF (Ā_{0}, Ā_{α}),

and completes the proof by comparing the spectral flow to an equivariant signature:

SF (Ā_{0}, Ā_{α}) = σ_{2α}(K) − 4.

q.e.d.

Proof. For the proof of Theorem 4.9 it suffices to show that

χ(HF_{∗}^{(k)}(S^{3}, K, n)) = S(k/n).

There are two types of critical points in M_{flat}^{∗}(E_{k}): those lying in closed components (circles) of M_{flat}(S^{3} − N_{K}) and those lying in irreducible arcs limiting to abelian connections in M_{flat}(S^{3} − N_{K}). The first type generically comes in pairs of critical points. The claim is that the contribution of such a pair to the Euler characteristic of HF_{∗}^{(k)}(S^{3}, K, n) is trivial. To show this, all there is to know is that the index of the two critical points A_{0} and A_{1} satisfies

μ(A_{0}) − μ(A_{1}) ≡ 1 (mod 2).

First notice that by invariance of the index under deformation of the holonomy parameter , as one goes along the circle in M_{flat}^{∗}(S^{3}−N_{K}) (by varying the holonomy along a meridian), the index of the corresponding critical points is constant if no degenerate critical point is crossed. On the other hand, generically, a degenerate critical point appears when the holonomy parameter along the closed component has a non-degenerate critical point (doubles back on itself). By the independence of Floer Homology under perturbations, such double points cancel out in Floer Homology. Combining these two observations yields the result.

The contribution of the second type of critical points to the Euler characteristic of HF_{∗}^{(k)}(S^{3}, K, n) is as follows. Generically two cases arise: if on an arc there is an odd number of such points, all except one cancel out in the Euler characteristic, while, if on an arc there is an even number of such points, all these cancel out in the Euler characteristic. This follows from the index argument explained above. Therefore for the Floer Homology HF_{∗}^{(k)}(S^{3}, K, n), the contribution to the Euler characteristic is an algebraic number of bifurcation points along the abelian arc in M_{flat}(S^{3} − N_{K}) with holonomy parameter in (0, k/n). We need to show that this number agrees with Herald’s S(k/n). First notice that Herald’s sign convention agrees, up to some universal sign, with the one used by Taubes in [44]. Indeed, this follows from Proposition 18 in [27] and the linearity of both Herald’s and Taubes’ invariants under the operation of connected sum. Also, the proof of Proposition 5.2 in [44] extends to the case of invariant connections (we shall not reproduce it here, as it is rather long). These two observations imply that our algebraic count may differ from Herald’s by a universal sign. The matter will be settled by computing an example. With its standard orientation, the Poincaré sphere P^{3} is realised as a 5-fold cyclic branched covering of S^{3} along a right-handed trefoil K_{2,3}. Casson’s invariant for P^{3} is λ(P^{3}) = 1/2 · χ(HF_{∗}(V_{5}(K_{2,3}))) = −1 (this differs from [21] as they used ind^{+} D_{𝔸} rather than ind^{−} D_{𝔸} for the Floer grading). Using Proposition 4.8, we have:

χ(HF_{∗}^{(1)}(S^{3}, K_{2,3}, 5)) = −1, χ(HF_{∗}^{(2)}(S^{3}, K_{2,3}, 5)) = −1.

As a right-handed trefoil K_{2,3} has non-positive equivariant signatures, the orientation conventions agree and this completes the proof that for any knot K,

χ(HF_{∗}^{(k)}(S^{3}, K, n)) = \frac{1}{2} · σ_{2 \frac{k}{n}} (K).

q.e.d.

5. Examples

The unknot was seen to have trivial Floer Homology (Section 4.1), and we first give here other examples of knots with trivial Floer Homology.

Proposition 5.1. If Δ_{K}(t) is trivial, then HF_{∗}^{(k)}(S^{3}, K, n) vanishes for any pair (k, n). More generally the same holds for knots whose Alexander polynomial has no roots on S^{1} ⊂ 𝔸.

Proof. Combine Corollary 4.6 and Proposition 4.1. q.e.d.

This gives quite a few examples of knots with trivial Floer Homology. The figure-8 knot has no roots of its Alexander polynomial along S^{1} ⊂ 𝔸, while (p,q,r)-pretzel knots (for p, q, r odd integers such that pr + rq + qp = −1) or the Conway and Kinoshita-Terasaka knots have trivial Alexander polynomial. More generally any connected sum of such knots will satisfy the conditions of Proposition 5.1 (as Δ_{K_{1}♯K_{2}}(t) = Δ_{K_{1}}(t) · Δ_{K_{2}}(t)), so will mutants of these knots (as mutation preserves the Alexander polynomial). These examples indicate that, like the knot signatures σ_{α}(K), the efficiency of HF_{∗}^{(k)}(S^{3}, K, n) as a knot invariant is in part limited by the Alexander polynomial.

The case we next study is that of torus knots. A torus knot K_{p,q} of type (p, q) may be seen as the link of singularity f^{−1}(0) ∩ S_{ϵ}^{3} of a polynomial f: 𝔸^{2} → 𝔸 given by f(x_{1}, x_{2}) = x_{1}^{p} + x_{2}^{q}. The n-fold cyclic branched covering of S^{3} along K_{p,q} is the Brieskorn sphere Σ(p, q, n). The Floer Homology of Brieskorn spheres has been computed by Fintushel and Stern in [21]. This was rendered easier by the fact that the Floer boundary operator is always trivial because the Floer chain groups vanish in one parity. The computation of HF_{∗}(Σ(p, q, n)) is therefore reduced to finding the Floer index of irreducible representations ρ: π_{1}(Σ(p, q, n)) → SU(2). For torus knots, we have:

Theorem 5.2. For a torus knot K_{p,q}, the Floer Homology HF_{∗}^{(k)}(S^{3}, K_{p,q}, n) has trivial boundary operator and therefore

HF_{∗}^{(k)}(S^{3}, K_{p,q}, n) = C_{∗}^{(k)}(S^{3}, K_{p,q}, n).

Proof. The case where n is odd and coprime with p and q follows directly from the result of Fintushel and Stern and Proposition 4.8. Otherwise, the argument is different. First notice that according to [30, Theorem 1], M_{flat}^{∗}(S^{3} − N_{K_{p,q}}) consists of \frac{1}{2}(p − 1)(q − 1) arcs limiting to abelian connections. Therefore any non-degenerate element in M_{flat}^{∗}(E_{k}) over (S^{3}, K_{p,q}, 2n) is on such a component in M_{flat}^{∗}(S^{3} − N_{K_{p,q}}). Suppose A_{1}, A_{2} ∈ M_{flat}^{∗}(E_{k}) are such that

μ(A_{1}) − μ(A_{2}) ≡ 1 (mod 2),

so that A_{1} and A_{2} are not in the same parity in the Floer chain groups. By the deformation argument in the proof of Theorem 4.4, one may find a pair (k′, n′) where n′ is odd and coprime with p and q, such that C_{∗}^{(k′)}(S^{3}, K_{p,q}, n′) ≃ C_{∗}^{(k)}(S^{3}, K_{p,q}, 2n). Take A′_{1}, A′_{2} ∈ M_{flat}^{∗}(E′_{k}) corresponding to A_{1}, A_{2} ∈ M_{flat}^{∗}(E_{k}). Then

μ(A′_{1}) − μ(A′_{2}) ≡ 1 (mod 2),

and this is a contradiction with what was just proved. Therefore,

HF_{∗}^{(k)}(S^{3}, K_{p,q}, 2n) = C_{∗}^{(k)}(S^{3}, K_{p,q}, 2n).

q.e.d.

More generally, given a knot K in S^{3} and a Floer chain complex (C_{∗}^{(k)}(S^{3}, K, n), ∂), the Chern-Simons functional CS is said to be perfect if the Floer chain groups vanish in even or odd dimensions. It would be be interesting to characterise classes of knots for which the Chern-Simons functional is perfect, as for such knots the computation of Floer Homology does not require knowledge of instantons used for the boundary operator. By Theorem 5.2 torus knots form such a class. It may be shown ([11]) that although torus knots happen to be fibred, non-amphichaeral and prime, these properties are not enough in general to yield perfectness of CS.

Let us then turn our attention to knots which arise as links of singularities, another property torus have. Recall that a knot K ↪ S^{3} is said to be algebraic if it is the link of an isolated singularity K = f^{−1}(0)∩S_{ϵ}^{3}, for a polynomial f: ℂ^{2} → ℂ. This class of knots has been studied extensively, and a lot is known about them (see [19] or [34]). Our first aim is to find an expression of the Euler characteristic of the Floer Homology of an algebraic knot K.

For this, we relate algebraic knots to so-called iterated torus knots, as this gives us a way of inductively building the algebraic knot starting from torus knots (which are well understood). Given an algebraic knot K which is the link of f: ℂ^{2} → ℂ, by taking a Puiseux expansion, one sees that K is an iterated torus knot, say of type ((p_{1}, q_{1}), ..., (p_{r}, q_{r})), satisfying the conditions p_{i} > 0 and q_{i+1} > p_{i}p_{i+1}q_{i}. And conversely, an iterated torus knot of type ((p_{1}, q_{1}), ..., (p_{r}, q_{r})) is algebraic if p_{i} > 0 and q_{i+1} > p_{i}p_{i+1}q_{i} (we refer the reader to [19] for details on this correspondence). Notice that, in particular, if K is algebraic then p_{i} > 0 and q_{i} > 0 for all i’s. Thus, expressing an algebraic knot K as an iteration of torus knots K_{1},...,K_{r} of type (p_{1}, q_{1}),..., (p_{r}q_{r}), we have:

Theorem 5.3. Let K be an algebraic knot of type ((p_{1}, q_{1}), ..., (p_{r}, q_{r})) and consider the Floer Homology HF_{∗}^{(k)}(S^{3}, K, n) and let k_{i} be an integer such that 0 ≤ k_{i}/n ≤ 1/2 and k_{i} ≡ kp_{1}...p_{i}(mod n). Then

χ(HF_{∗}^{(k)}(S^{3}, K, n)) = ∑_{i=1}^{r} χ(HF_{∗}^{(k_{i})}(S^{3}, K_{i}, n)),

where the K_{i} are torus knots of type (p_{i}, q_{i}).

Proof. We shall use inductively the equivariant signature computation for iterated torus knots provided by Litherland (Theorem 2 in [38]) and Theorem 4.9. Start with a torus knot K_{1} = (p_{1}, q_{1}) as above. Use another torus knot K_{2} = (p_{2}, q_{2}), and consider the iterated torus knot of type ((p_{1}, q_{1}), (p_{2}, q_{2})) K_{2}(p_{1}, q_{1}). Then by [38],

σ_{2 \frac{k}{n}} (K_{2}(p_{1}, q_{1})) = σ_{2 \frac{k}{n} p_{1}}(K_{2}) + σ_{2 \frac{k}{n}} (K_{1}).

For an iterated torus knot K^{1...r} = ((p_{1}, q_{1}), ..., (p_{r}, q_{r})), its equivariant signature σ_{2 \frac{k}{n}} (K^{1...r}) will be computed inductively

\begin{aligned}σ_{2 \frac{k}{n}} (K^{1...r}) & = σ_{2 \frac{k}{n} p_{1}p_{2}...p_{r−1}}(K_{r}) + σ_{2 \frac{k}{n} p_{1}p_{2}...p_{r−2}}(K_{r−1}) + ... + σ_{2 \frac{k}{n}} (K_{1}) \\ & = ∑_{i=0}^{r−1} σ_{2 \frac{k}{n} p_{1}...p_{i}}(K_{i}). \end{aligned}

The coefficients kp_{1}...p_{i} have to lie between 0 and n/2 in order to evaluate the equivariant signature, so one obtains the k_{i}. Then Theorem 4.9 gives

σ_{2 \frac{k}{n} p_{1}...p_{i}}((p_{i}, q_{i})) = 2 · χ(HF_{∗}^{(k_{i})}(S^{3}, (p_{i}, q_{i}), n).

Now if K is algebraic, it will be an iterated torus knot ((p_{1}, q_{1}), ...(p_{r}, q_{r})) with extra conditions p_{i} > 0 and q_{i} > 0. These conditions are essential for the argument, as then all the torus knots K_{i} (1 ≤ i ≤ r) used to obtain K are of the same “mirror type”. That is, they all are either right-handed torus knots or left-handed torus knots. Therefore the contribution of each of these to the signature of the algebraic knot K is either always positive (for left-handed torus knots) or negative (for right-handed torus knots), proving the result. q.e.d.

We conclude this section conjecturing the perfectness of CS for algebraic knots, generalising Theorem 5.2.

Question 5.4. For an algebraic knot K in S^{3}, do the chain groups C_{∗}^{(k)}(S^{3}, K, n) all vanish either in even or in odd dimensions and therefore is

HF_{∗}^{(k)}(S^{3}, K, n) = C_{∗}^{(k)}(S^{3}, K, n)?

This conjecture is the analogue of one in ordinary Floer Homology, for 3-manifolds which arise as 3-dimensional links of complex singularities (see for example [21] or [24] for the Seifert fibred manifolds case).

6. Conclusion

In order to push further the theory presented in this article, there are many tools one may wish to develop aiming at a better understanding of the Floer Homology for knots and its role in Knot Theory. The possibilities below are being investigated.

We have seen that the Floer Homology for 3-orbifolds appears as a generalisation of ordinary Floer Homology, so one aspect to be considered is to generalise properties developed by Floer, Donaldson and others to the orbifold setting. From work done in this article, it is reasonable to expect that, at least in some cases, the generalisations pose no serious problems.

First comes to mind the surgery exact triangle developed by Floer (see [7]) which is a powerful tool for computations. To carry this out using 3-orbifolds, one needs other Floer homologies for knots which were introduced in [11], and adapted the arguments in [7] to the case of orbifolds.

Similarly, recall that in the case of 3-manifolds the Atiyah-Floer conjecture relates the instanton Floer Homology of a homology sphere, HF_{∗}(Y^{3}), with the symplectic Floer Homology of Y^{3}. We have briefly alluded to a symplectic Floer Homology for knots developed by Li in [36], and this has been generalised by Austin. It seems natural to look for a correspondence between the symplectic and instanton theories for knots, in light of the Atiyah-Floer conjecture.

Also, an analog of the relative Donaldson polynomials of 4-manifolds with boundary taking values in the Floer Homology of the boundary homology sphere should be found. This would yield relative versions of the polynomial invariants developed by Kronheimer and Mrowka for embedded surfaces in 4-manifolds, where the polynomials take values in a 3-orbifold Floer Homology. This should help in generalising results of Kronheimer and Mrowka about the 4-ball genus and unknotting numbers of knots.

For many recently developed knot invariants, skein relations play an important role in their definition or in elucidating some of their properties. Skein relations for HF_{∗}^{(k)}(S^{3}, K, n) would be good to have, in particular for explicit computations. This involves the generalisation to links outlined in Section 4.1 and an approach towards this is to obtain a surgery sequence in Floer Homology when performing a +1-surgery on S^{3} along a small unknotted linking circle at a knot crossing. Ideas involved here are similar to those needed for the surgery exact triangle mentioned above. Notice that such skein relations could also yield the Atiyah-Floer conjecture for knots, generalising the original Atiyah-Floer conjecture.

As the 3-orbifolds (S^{3}, K, n) are closely related to cyclic branched covers of S^{3}, one expects some relationship between the ordinary Floer Homology HF_{∗}(V_{n}(K)) and HF_{∗}^{(k)}(S^{3}, K, n), when V_{n}(K) is a homology sphere. More generally one may compare HF_{∗}(V_{m}(K)) and HF_{∗}^{(k)}(S^{3}, K, n) for m and n possibly different. In [13], N. Saveliev and the first author used such an approach to provide a geometric proof of the Fintushel-Stern formula relating the Casson invariant of a Brieskorn homology sphere to the signature of its Milnor fibre. More generally, the first author developed in [12] an equivariant Casson invariant via Floer Homology for knots which provides obstructions for generalisations of the Neumann-Wahl formula (see [41]) to hold.

Finally, it is worth looking at Floer Homology in relation to knot concordance. Indeed, HF_{∗}^{(k)}(S^{3}, K, n) generalises the equivariant signature σ_{2 \frac{k}{n}} (K), and it is a classical result of Tristram that equivariant signatures are topological concordance invariants. Among other reasons, the work contained in [32], [33] and [31] lead the first author to conjecture in [11] that HF_{∗}^{(k)}(S^{3}, K, n) is a smooth concordance invariant. If this turns out to be the case, then the Floer Homology would appear as a good candidate to quantitatively measure the difference between topological and smooth concordance of knots.

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