Controlled L-Theory

(Preliminary announcement)

A. Ranicki and M. Yamasaki

Introduction.

This is a preliminary announcement of a controlled algebraic surgery theory, of the type first proposed by Quinn [1]. We define and study the ϵ-controlled L-groups L_{n}(X, p_{X}, ϵ), extending to L-theory the controlled K-theory of Ranicki and Yamasaki [4].

The most immediate application of the algebra to controlled geometric surgery is the controlled surgery obstruction: a normal map (f, b) : K → L from a closed n-dimensional manifold to a δ-controlled Poincaré complex determines an element

σ_{∗}^{δ}(f, b) ∈ L_{n}(X, 1_{X}, 100δ) .

(The construction in Ranicki and Yamasaki [3] can be used to produce a 6ϵ ndimensional quadratic Poincaré structure on an (n + 1)-dimensional chain complex. There is a chain equivalence from this to an n-dimensional chain complex with a 100δ n-dimensional quadratic Poincaré structure, and σ_{∗}^{δ}(f, b) is the cobordism class of this complex in L_{n}(X, 1_{X}, 100δ) .) A relative construction shows that if (f, b) can be made into a δ-controlled homotopy equivalence by δ-controlled surgery then

σ_{∗}^{δ}(f, b) = 0 ∈ L_{n}(X, 1_{X}, 100δ) .

Conversely, if n ≥ 5 and (f, b) is such that

σ_{∗}^{δ}(f, b) = 0 ∈ L_{n}(X, 1_{X}, 100δ)

then (f, b) can be made into an ϵ-controlled homotopy equivalence by ϵ-controlled surgery, where ϵ = C × 100δ for a certain constant C > 1 that depends on n. Proofs of difficult results and the applications of the algebra to topology are deferred to the final account.

The algebraic properties required to obtain these applications include the controlled L-theory analogues of the homology exact sequence of a pair (3.1, 3.2) and the Mayer-Vietoris sequence (3.3, 3.4).

The limit of the controlled L-groups

L_{n}^{c}(X; 1_{X}) = \begin{matrix}lim \\ ⟵ \\ ϵ \end{matrix} \begin{matrix}lim \\ ⟵ \\ δ \end{matrix} im\{L_{n}(X, 1_{X}, δ) ⟶ L_{n}(X, 1_{X}, ϵ)\}

is the obstruction group for controlled surgery to ϵ-controlled homotopy equivalence for all ϵ > 0.

120 A. RANICKI AND M. YAMASAKI

Theorem. (5.4.) Fix a compact polyhedron X and an integer n(≥ 0). There exist numbers ϵ_{0} > 0 and 0 < μ_{0} ≤ 1 such that

L_{n}^{c}(X; 1_{X}) = im{L_{n}(X, 1_{X}, δ) ⟶ L_{n}(X, 1_{X}, ϵ)}

for every ϵ ≤ ϵ_{0} and every δ ≤ μ_{0}ϵ.

Throughout this paper all the modules are assumed to be finitely generated unless otherwise stated explicitly. But note that all the definitions and the constructions are valid also for possibly-infinitely-generated modules and chain complexes. Actually we heavily use finite dimensional but infinitely generated chain complexes in the later part of the paper. (That is where the bounded-control over ℝ comes into the game.) So we first pretend that everything is finitely generated, and later we introduce a possibly-infinitely-generated analogue without any details.

1. Epsilon-controlled L-groups.

In this section we introduce ϵ-controlled L-groups L_{n}(X, p_{X}, ϵ) and L_{n}(X, Y, p_{X}, ϵ) for p_{X} : M → X, Y ⊂ X, n ≥ 0, ϵ > 0. These are defined using geometric module chain complexes with quadratic Poincaré structures, which were discussed in Yamasaki [5].

We use the convention in Ranicki and Yamasaki [4] for radii of geometric morphisms, etc. The dual of a geometric module is the geometric module itself, and the dual of a geometric morphism is defined by reversing the orientation of paths. Note that if f has radius ϵ then so does its dual f^{∗} and that f ∼_{ϵ} g implies f^{∗} ∼_{ϵ} g^{∗}, by our convention. For a geometric module chain complex C, its dual C^{n−∗} is defined using the sign convention used in Ranicki [2].

For a subset S of a metric space X, S^{ϵ} will denote the closed ϵ neighborhood of S in X when ϵ ≥ 0. When ϵ < 0, S^{ϵ} will denote the set X − (X − S)^{−ϵ}.

Let C be a free chain complex on p_{X} : M → X. An n-dimensional ϵ quadratic structure ψ on C is a collection {ψ_{s}|s ≥ 0} of geometric morphisms

ψ_{s} : C^{n−r−s} = (C_{n−r−s})^{∗} → C_{r} (r ∈ ℤ)

of radius ϵ such that

(∗) dψ_{s} + (−)^{r}ψ_{s}d^{∗} + (−)^{n−s−1}(ψ_{s+1} + (−)^{s+1}T ψ_{s+1}) ∼_{3ϵ} 0 : C^{n−r−s−1} → C_{r},

for s ≥ 0. An n-dimensional free ϵ chain complex C on p_{X} equipped with an ndimensional ϵ quadratic structure is called an n-dimensional ϵ quadratic complex on p_{X}. (Here, a complex C is n-dimensional if C_{i} = 0 for i < 0 and i > n.)

CONTROLLED L-THEORY 121

Next let f : C → D be a chain map between free chain complexes on p_{X}. An (n + 1)-dimensional ϵ quadratic structure (δψ, ψ) on f is a collection {δψ_{s}, ψ_{s}|s ≥ 0} of geometric morphisms

δψ_{s} : D^{n+1−r−s} → D_{r} , ψ_{s} : C^{n−r−s} → C_{r} (r ∈ ℤ)

of radius ϵ such that the following holds in addition to (∗):

d(δψ_{s}) + (−)^{r}(δψ_{s})d^{∗} + (−)^{n−s}(δψ_{s+1} + (−)^{s+1}T δψ_{s+1}) + (−)^{n}fψ_{s}f^{∗} ∼_{3ϵ} 0
: D^{n−r−s} → D_{r} (s ≥ 0) .

An ϵ chain map f : C → D between an n-dimensional free ϵ chain complex C on p_{X} and an (n + 1)-dimensional free ϵ chain complex D on p_{X} equipped with an (n + 1)- dimensional ϵ quadratic structure is called an (n + 1)-dimensional ϵ quadratic pair on p_{X}. Obviously its boundary (C, ψ) is an n-dimensional ϵ quadratic complex on p_{X}.

An ϵ cobordism of n-dimensional ϵ quadratic structures ψ on C and ψ′ on C′ is an (n + 1)-dimensional ϵ quadratic structure (δψ, ψ ⊕ −ψ′) on some chain map C ⊕ C′ → D. An ϵ cobordism of n-dimensional ϵ quadratic complexes (C, ψ), (C′, ψ′) on p_{X} is an (n + 1)-dimensional ϵ quadratic pair on p_{X}

(( f f′) : C ⊕ C′ → D, (δψ, ψ ⊕ −ψ′))

with boundary (C ⊕ C′, ψ ⊕ −ψ′). The union of adjoining cobordisms are defined using the formula in Chapter 1.7 of Ranicki [2]. The union of adjoining ϵ cobordisms is a 2ϵ cobordism.

ΣC and ΩC will denote the suspension and the desuspension of C respectively, and C(f) will denote the algebraic mapping cone of a chain map f.

Definition. Let W be a subset of X. An n-dimensional ϵ quadratic structure ψ on C is ϵ Poincaré (over W) if the algebraic mapping cone of the duality 3ϵ chain map

D_{ψ} = (1 + T)ψ_{0} : C^{n−∗} ⟶ C

is 4ϵ contractible (over W). A quadratic complex (C, ψ) is ϵ Poincaré (over W) if ψ is ϵ Poincaré (over W). Similarly, an (n + 1)-dimensional ϵ quadratic structure (δψ, ψ) on f : C → D is ϵ Poincaré (over W) if the algebraic mapping cone of the duality 4ϵ chain map

D_{(δψ,ψ)} = ((1 + T)δψ_{0} f(1 + T)ψ_{0}) : C(f)^{n+1−∗} ⟶ D

is 4ϵ contractible (over W) (or equivalently the algebraic mapping cone of the 4ϵ chain map

D̄_{(δψ,ψ)} = ( \begin{matrix}(1 + T)δψ_{0} \\ (−)^{n+1−r}(1 + T)ψ_{0}f^{∗} \end{matrix} ) : D^{n+1−r} → C(f)_{r} = D_{r} ⊕ C_{r−1}

is 4ϵ contractible (over W)) and ψ is ϵ Poincaré (over W). A quadratic pair (f, (δψ, ψ)) is ϵ Poincaré (over W) if (δψ, ψ) is ϵ Poincaré (over W). We will also use the notation D_{δψ} = (1 + T)δψ_{0}, although it does not define a chain map from D^{n+1−∗} to D in general.

Definition. (1) A positive geometric chain complex C (C_{i} = 0 for i < 0) is ϵ connected if there exists a 4ϵ morphism h : C_{0} → C_{1} such that dh ∼_{8ϵ} 1_{C_{0}}.

(2) A chain map f : C → D of positive chain complexes is ϵ connected if C(f) is ϵ connected.

(3) A quadratic complex (C, ψ) is ϵ connected if D_{ψ} is ϵ connected.

(4) A quadratic pair (f : C → D, (δψ, ψ)) is ϵ connected if D_{ψ} and D_{(δψ,ψ)} are ϵ connected.

Now we define the ϵ-controlled L-groups. Let Y be a subset of X.

Definition. For n ≥ 0 and ϵ ≥ 0, L_{n}(X, Y, p_{X}, ϵ) is defined to be the equivalence classes of n-dimensional ϵ connected ϵ quadratic complexes on p_{X} that are ϵ Poincaré over X − Y. The equivalence relation is generated by ϵ connected ϵ cobordisms that are ϵ Poincaré over X − Y. For Y = ∅ write

L_{n}(X, p_{X}, ϵ) = L_{n}(X, ∅, p_{X}, ϵ) .

Remarks. (1) We use only n-dimensional complexes and not the complexes chain equivalent to n-dimensional ones in order to make sure we have size control on some constructions.

(2) The ϵ connectedness condition is automatic for complexes that are ϵ Poincaré over X. Connectedness condition is used to insure that the boundary ∂C = ΩC(D_{ψ}) is chain equivalent to a positive one. There is a quadratic structure ∂ψ for ∂C so that (∂C, ∂ψ) is Poincaré (Ranicki [2]).

(3) Using locally-finitely generated chain complexes on M, one can similarly define ϵ-controlled locally-finite L-groups L_{n}^{lf}(X, Y, p_{X}, ϵ). All the results in sections 1 – 3 are valid for locally-finite L-groups.

Proposition 1.1. The direct sum

(C, ψ) ⊕ (C′, ψ′) = (C ⊕ C′, ψ ⊕ ψ′)

induces an abelian group structure on L_{n}(X, Y, p_{X}, ϵ). Furthermore, if

[C, ψ] = [C′, ψ′] ∈ L_{n}(X, Y, p_{X}, ϵ) ,

then there is a 100ϵ connected 2ϵ cobordism between (C, ψ) and (C′, ψ′) that is 100ϵ Poincaré over X − Y^{100ϵ}.

Next we study the functoriality. A map between control maps p_{X} : M → X and p_{Y} : N → Y means a pair of continuous maps (f : M → N, f̄: X → Y) which makes the following diagram commute:

[figure: M; f; p_{X}; N; p_{Y}; X; f̄; Y.]

For example, given a control map p_{Y} : N → Y and a subset X ⊂ Y, let us denote the control map p_{Y} |p_{Y}^{−1}(X) : p_{Y}^{−1}(X) → X by p_{X} : M → X. Then the inclusion maps j : M → N, j̄ : X → Y form a map form p_{X} to p_{Y}.

Epsilon controlled L-groups are functorial with respect to maps and relaxation of control in the following sense.

Proposition 1.2. Let F = (f, f̄) be a map from p_{X} : M → X to p_{Y} : N → Y, and suppose that f̄ is Lipschitz continuous with Lipschitz constant λ, i.e., there exists a constant λ > 0 such that

d(f̄(x_{1}), f̄(x_{2})) ≤ λd(x_{1}, x_{2}) (x_{1}, x_{2} ∈ X).

Then F induces a homomorphism

F_{∗} : L_{n}(X, X′, p_{X}, δ) ⟶ L_{n}(Y, Y′, p_{Y} , ϵ)

if ϵ ≥ λδ and f̄(X′) ⊂ Y′. If two maps F = (f, f̄) and G = (g, ḡ) are homotopic through maps H_{t} = (h_{t}, h̄_{t}) such that each h̄_{t} is Lipschitz continuous with Lipschitz constant λ, ϵ ≥ λδ, ϵ′ > ϵ, and h̄_{t}(X′) ⊂ Y′, then the following two compositions are the same:

\begin{matrix}L_{n}(X, X′, p_{X}, δ) ⟶^{F_{∗}} L_{n}(Y, Y′, p_{Y} , ϵ) ⟶ L_{n}(Y, Y′, p_{Y} , ϵ′) \\ L_{n}(X, X′, p_{X}, δ) ⟶^{G_{∗}} L_{n}(Y, Y′, p_{Y} , ϵ) ⟶ L_{n}(Y, Y′, p_{Y} , ϵ′) \end{matrix}

Proof: The direct image construction for geometric modules and morphisms [4, p.7] can be used to define the direct images f_{#}(C, ψ) of quadratic complexes and the direct images of cobordism. And this induces the desired F_{∗}. The first part is obvious. For the second part, split the homotopy in small pieces to construct small cobordisms. The size of the cobordism may be slightly bigger than the size of the object itself.

Remark. The above is stated for Lipschitz continuous maps to simplify the statement. For a specific δ and a specific ϵ, the following condition, instead of the Lipschitz condition above, is sufficient for the existence of F_{∗} :

d(f̄(x_{1}), f̄(x_{2})) ≤ kϵ whenever d(x_{1}, x_{2}) ≤ kδ,

for a certain finite set of integers k (more precisely, for k = 1, 3, 4, 8)

and similarly for the isomorphism in the second part. When X is compact and ϵ is given, the continuity of f̄ implies that this condition is satisfied for sufficiently small δ’s. [Use the continuity of the distance function d : X × X → ℝ and the compactness of the diagonal set Δ ⊂ X × X.] And, in the second half of the proposition, there are cases when the equality F_{∗} = G_{∗} holds without composing with the relax-control map; e.g., see 4.1.

We are interested in the “limit” of ϵ-controlled L-groups.

Definition. Let p_{X} : M → X be a control map.

(1) Let ϵ, δ be positive numbers such that δ ≤ ϵ. We define:

L_{n}^{ϵ}(X, p_{X}, δ) = im{L_{n}(X, p_{X}, δ)⟶L_{n}(X, p_{X}, ϵ)}.

(2) For ϵ > 0, we define the stable ϵ-controlled L-group of X with coefficient p_{X} by:

L_{n}^{ϵ}(X; p_{X}) := ⋂_{0<δ<ϵ} L_{n}^{ϵ}(X, p_{X}, δ).

(3) The controlled L-group with coefficient p_{X} is defined by:

L_{n}^{c}(X; p_{X}) := \begin{matrix}lim \\ ⟵ \\ ϵ \end{matrix} L_{n}^{ϵ}(X; p_{X}),

where the limit is taken with respect to the obvious relax-control maps:

L_{n}^{ϵ′}(X; p_{X}) ⟶ L_{n}^{ϵ}(X; p_{X}), (ϵ′ < ϵ).

In section 5, we study a certain stability result for the controlled L-groups in some special case.

2. Epsilon-controlled projective L-groups.

Fix a subset Y of X, and let F be a family of subsets of X such that Z ⊃ Y for each Z ∈ F. In this section we introduce intermediate ϵ-controlled L-groups L_{n}^{F}(Y, p_{X}, ϵ), which will appear in the stable-exact sequence of a pair and also in the Mayer-Vietoris sequence. Roughly speaking, these are defined using “controlled projective quadratic chain complexes” ((C, p), ψ) with vanishing ϵ-controlled reduced projective class [C, p] = 0 ∈ K̃_{0}(Z, p_{Z}, n, ϵ) (Ranicki and Yamasaki [4]) for each Z ∈ F. Here p_{Z} denotes the restriction p_{X}|p_{X}^{−1}(Z) : p_{X}^{−1}(Z) → Z of p_{X} as in the previous section.

For a projective module (A, p) on p_{X}, its dual (A, p)^{∗} is the projective module (A^{∗}, p^{∗}) on p_{X}. If f : (A, p) → (B, q) is an ϵ morphism ([4]), then f^{∗} : (B, q)^{∗} → (A, p)^{∗} is also an ϵ morphism. For an ϵ projective chain complex on p_{X}

(C, p) : … ⟶ (C_{r}, p_{r}) ⟶^{d_{r}} (C_{r−1}, p_{r−1}) ⟶^{d_{r−1}} …

in the sense of [4], (C, p)^{n−∗} will denote the ϵ projective chain complex on p_{X} defined by:

… ⟶ (C^{n−r}, p_{n−r}^{∗}) ⟶^{(−)^{r}d_{n−r+1}^{∗}} (C^{n−r+1}, p_{n−r+1}^{∗}) ⟶ ….

An n-dimensional ϵ quadratic structure on a projective chain complex (C, p) on p_{X} is an n-dimensional ϵ quadratic structure ψ on C (in the sense of §1) such that ψ_{s} : (C^{n−r−s}, p^{∗}) → (C_{r}, p) is an ϵ morphism for every s ≥ 0 and r ∈ ℤ. Similarly, an (n + 1)-dimensional ϵ quadratic structure on a chain map f : (C, p) → (D, q) is an (n + 1)-dimensional ϵ quadratic structure (δψ, ψ) on f : C → D such that δψ_{s} : (D^{n+1−r−s}, q^{∗}) → (D_{r}, q) and ψ_{s} : (C^{n−r−s}, p^{∗}) → (C_{r}, p) are ϵ morphisms for every s ≥ 0 and r ∈ ℤ. An n-dimensional ϵ projective chain complex (C, p) on p_{X} equipped with an n-dimensional ϵ quadratic structure is called an n-dimensional ϵ projective quadratic complex on p_{X}, and an ϵ chain map f : (C, p) → (D, q) between an n-dimensional ϵ projective chain complex (C, p) on p_{X} and an (n + 1)-dimensional ϵ projective chain complex (D, q) on p_{X} equipped with an (n + 1)-dimensional ϵ quadratic structure is called an (n+1)-dimensional ϵ projective quadratic pair on p_{X}.

An ϵ cobordism of n-dimensional ϵ projective quadratic complexes ((C, p), ψ), ((C′, p′), ψ′) on p_{X} is an (n + 1)-dimensional ϵ projective quadratic pair on p_{X}

(( f f′) : (C, p) ⊕ (C′, p′)⟶(D, q), (δψ, ψ ⊕ −ψ′))

with boundary ((C, p) ⊕ (C′, p′), ψ ⊕ −ψ′).

An n-dimensional ϵ quadratic structure ψ on (C, p) is ϵ Poincaré if

∂(C, p) = ΩC((1 + T)ψ_{0} : (C^{n−∗}, p^{∗})⟶(C, p))

is 4ϵ contractible. ((C, p), ψ) is ϵ Poincaré if ψ is ϵ Poincaré. Similarly, an (n + 1)- dimensional ϵ quadratic structure (δψ, ψ) on f : (C, p) → (D, q) is ϵ Poincaré if ∂(C, p) and

∂(D, q) = ΩC(((1 + T)δψ_{0} f(1 + T)ψ_{0}) : C(f)^{n+1−∗}⟶(D, q))

are both 4ϵ contractible. A pair (f, (δψ, ψ)) is ϵ Poincaré if (δψ, ψ) is ϵ Poincaré.

Let Y and be a subset of X and F be a family of subsets of X such that Z ⊃ Y for every Z ∈ F.

Definition. Let n ≥ 0 and ϵ ≥ 0. L_{n}^{F}(Y, p_{X}, ϵ) is the equivalence classes of ndimensional ϵ Poincaré ϵ projective quadratic complexes ((C, p), ψ) on p_{Y} such that [C, p] = 0 in K̃_{0}(Z, p_{Z}, n, ϵ) for each Z ∈ F. The equivalence relation is generated by ϵ Poincaré ϵ cobordisms (( f f′) : (C, p) ⊕ (C′, p′) → (D, q), (δψ, ψ ⊕ −ψ′)) on p_{Y} such that [D, q] = 0 in K̃_{0}(Z, p_{Z}, n + 1, ϵ) for each Z ∈ F. When F = {X}, we omit the braces and write L_{n}^{X}(Y, p_{X}, ϵ) instead of L_{n}^{{X}}(Y, p_{X}, ϵ). When F = { }, then we use the notation L_{n}^{p}(Y, p_{Y} , ϵ), since it depends only on p_{Y}.

Proposition 2.1. Direct sum induces an abelian group structure on L_{n}^{F}(Y, p_{X}, ϵ). Furthermore, if

[(C, p), ψ] = [(C′, p′), ψ′] ∈ L_{n}^{F}(Y, p_{X}, ϵ) ,

then there is a 100ϵ Poincaré 2ϵ cobordism on p_{Y}

(( f f′) : (C, p) ⊕ (C′, p′) → (D, q), (δψ, ψ ⊕ −ψ′))

such that [D, q] = 0 in K̃_{0}(Z, p_{Z}, n + 1, 9ϵ) for each Z ∈ F.

A functoriality with respect to maps and relaxation of control similar to 1.2 holds for epsilon controlled projective L-groups.

Proposition 2.2. Let F = (f, f̄) be a map from p_{X} : M → X to p_{Y} : N → Y, and suppose that f̄ is Lipschitz continuous with Lipschitz constant λ, i.e., there exists a constant λ > 0 such that

d(f̄(x_{1}), f̄(x_{2})) ≤ λd(x_{1}, x_{2}) (x_{1}, x_{2} ∈ X).

If ϵ ≥ λδ, f̄(A) ⊂ B, and there exists a Z ∈ F satisfying f̄(Z) ⊂ Z′ for each Z′ ∈ F′, then F induces a homomorphism

F_{∗} : L_{n}^{F}(A, p_{X}, δ) ⟶ L_{n}^{F′}(B, p_{Y} , ϵ).

Remark. As in the remark to 1.2, for a specific δ and a ϵ, we do not need the full Lipschitz condition to guarantee the existence of F_{∗}.

There is an obvious homomorphism

ι_{ϵ} : L_{n}(Y, p_{Y} , ϵ) ⟶ L_{n}^{F}(Y, p_{X}, ϵ); [C, ψ] ↦ [(C, 1), ψ].

On the other hand, the controlled K-theoretic condition posed in the definition can be used to construct a homomorphism from a projective L-group to a free L-group:

Proposition 2.3. There exist a constant α > 1 such that the following holds true: for any control map p_{X} : M → X, any subset Y ⊂ X, any family of subsets F of X containing Y , any element Z of F, any number n ≥ 0, and any positive numbers δ, ϵ such that ϵ ≥ αδ, there is a well-defined homomorphism functorial with respect to relaxation of control:

(i_{Z})_{∗} : L_{n}^{F}(Y, p_{X}, δ) ⟶ L_{n}(Z, p_{Z}, ϵ)

such that the following compositions are equal to the maps induced from inclusion maps:

\begin{matrix}L_{n}^{F}(Y, p_{X}, δ) ⟶^{(i_{Z})_{∗}} L_{n}(Z, p_{Z}, ϵ) ⟶^{ι_{ϵ}} L_{n}^{\{Z\}}(Z, p_{Z}, ϵ) , \\ L_{n}(Y, p_{Y} , δ) ⟶^{ι_{δ}} L_{n}^{F}(Y, p_{X}, δ) ⟶^{(i_{Z})_{∗}} L_{n}(Z, p_{Z}, ϵ) . \end{matrix}

Remark. Actually α = 30000 works.

3. Stably-exact sequences.

In this section we describe two ‘stably-exact’ sequences. The first is the stably-exact sequence of a pair:

....▷^{∂} L_{n}^{X}(Y_{n}, p_{X}, ϵ) .....▷^{i_{∗}} L_{n}(X, p_{X}, ϵ) ⟶^{j_{∗}} L_{n}(X, Y, p_{X}, ϵ) ....▷^{∂} L_{n−1}^{X}(Y, p_{X}, ϵ) .....▷^{i_{∗}}

where the dotted arrows are only ‘stably’ defined. The precise meaning will be explained below. The second is the Mayer-Vietoris-type stably-exact sequence:

\begin{matrix}....▷^{∂} L_{n}^{F}(C, p_{X}, ϵ) .....▷^{i_{∗}} L_{n}(A, p_{A}, ϵ)⊕ & L_{n}(B, p_{B}, ϵ) ⟶^{j_{∗}} L_{n}(X, p_{X}, ϵ) \\ ....▷^{∂} L_{n−1}^{F}(C, p_{X}, ϵ) .....▷^{i_{∗}} \end{matrix}

where X = A ∪ B, C = A ∩ B, and F = {A, B}.

Fix an integer n ≥ 0, let Y_{n}, Z_{n} be subsets of X, and let γ_{n}, δ_{n}, ϵ_{n} be three positive numbers satisfying

ϵ_{n} ≥ δ_{n}, δ_{n} ≥ αγ_{n}

where α is the number (> 1) posited in 2.3. Then there is a sequence

L_{n}^{X}(Y_{n}, p_{X}, γ_{n}) ⟶^{i_{∗}=(i_{X})_{∗}} L_{n}(X, p_{X}, δ_{n}) ⟶^{j_{∗}} L_{n}(X, Z_{n}, p_{X}, ϵ_{n}),

where i_{∗} is the homomorphism given in 2.3 and j_{∗} is the homomorphism induced by the inclusion map and relaxation of control. (The subscripts are there just to remind the reader of the degrees of the relevant L-groups.)

Theorem 3.1. There exist constants κ_{0}, κ_{1}, κ_{2}, … (> 1) which do not depend on p_{X} such that

(1) if n ≥ 0, Z_{n} ⊃ Y_{n}^{κ_{n}δ_{n}}, and ϵ_{n} ≥ κ_{n}δ_{n}, then the following composition j_{∗}i_{∗} is zero:

j_{∗}i_{∗} = 0 : L_{n}^{X}(Y_{n}, p_{X}, γ_{n}) ⟶^{i_{∗}} L_{n}(X, p_{X}, δ_{n}) ⟶^{j_{∗}} L_{n}(X, Z_{n}, p_{X}, ϵ_{n}),

(2) if n ≥ 1, Y_{n−1} ⊃ Z_{n}^{κ_{n}ϵ_{n}} and γ_{n−1} ≥ κ_{n}ϵ_{n}, then there is a connecting homomor-

phism

∂ : L_{n}(X, Z_{n}, p_{X}, ϵ_{n}) ⟶ L_{n−1}^{X}(Y_{n−1}, p_{X}, γ_{n−1}),

such that the following composition ∂j_{∗} is zero:

∂j_{∗} = 0 : L_{n}(X, p_{X}, δ_{n}) ⟶^{j_{∗}} L_{n}(X, Z_{n}, p_{X}, ϵ_{n}) ⟶^{∂} L_{n−1}^{X}(Y_{n−1}, p_{X}, γ_{n−1}),

and, if δ_{n−1} ≥ αγ_{n−1} (so that the homomorphism i_{∗} is well-defined), the following composition i_{∗}∂ is zero:

i_{∗}∂ = 0 : L_{n}(X, Z_{n}, p_{X}, ϵ_{n}) ⟶^{∂} L_{n−1}^{X}(Y_{n−1}, p_{X}, γ_{n−1}) ⟶^{i_{∗}} L_{n−1}(X, p_{X}, δ_{n−1}).

Theorem 3.2. There exist constants λ_{0}, λ_{1}, λ_{2}, … (> 1) which do not depend on p_{X} such that

(1) if n ≥ 0, δ_{n} ≥ αγ_{n} (so that i_{∗} is well-defined), ϵ′_{n+1} ≥ λ_{n}δ_{n}, Z′_{n+1} ⊃ Y_{n}^{λ_{n}δ_{n}}, Y′_{n} ⊃ Z′_{n+1}^{κ_{n+1}ϵ′_{n+1}} and γ′_{n} ≥ κ_{n+1}ϵ′_{n+1} (so that ∂′ is well-defined), then the image of the kernel of i_{∗} in L_{n}^{X}(Y′_{n}, p_{X}, γ′_{n}) is in the image of ∂′:

[figure: L_{n}^{X}(Y_{n}, p_{X}, γ_{n}); i_{∗}; L_{n}(X, p_{X}, δ_{n}); L_{n+1}(X, Z′_{n+1}, p_{X}, ϵ′_{n+1}); ∂′; L_{n}^{X}(Y′_{n}, p_{X}, γ′_{n})]

(2) if n ≥ 0, ϵ_{n} ≥ δ_{n} (so that j_{∗} is well-defined), Y′_{n} ⊃ Z_{n}^{λ_{n}ϵ_{n}}, γ′_{n} ≥ λ_{n}ϵ_{n}, and δ′_{n} ≥ αγ′_{n} (so that i′_{∗} is well-defined), then the image of the kernel of j_{∗} in L_{n}(X, p_{X}, δ′_{n}) is in the image of i′_{∗}:

[figure: L_{n}(X, p_{X}, δ_{n}); j_{∗}; L_{n}(X, Z_{n}, p_{X}, ϵ_{n}); L_{n}^{X}(Y′_{n}, p_{X}, γ′_{n}); i′_{∗}; L_{n}(X, p_{X}, δ′_{n})]

(3) if n ≥ 1, γ_{n−1} ≥ κ_{n}ϵ_{n} (so that ∂ is well-defined), ϵ′_{n} ≥ λ_{n}γ_{n−1}, and Z′_{n} ⊃ Y_{n−1}^{λ_{n}γ_{n−1}}, then the image of the kernel of ∂ in L_{n}(X, Z′_{n}, p_{X}, ϵ′_{n}) is in the image of j′_{∗}:

[figure: L_{n}(X, Z_{n}, p_{X}, ϵ_{n}); ∂; L_{n−1}^{X}(Y_{n−1}, p_{X}, γ_{n−1}); L_{n}(X, p_{X}, ϵ′_{n}); j′_{∗}; L_{n}(X, Z′_{n}, p_{X}, ϵ′_{n})]

Here the vertical maps are the homomorphisms induced by inclusion maps and relaxation of control.

Next we investigate the Mayer-Vietoris-type stably-exact sequence. Fix an integer n ≥ 0, and assume that X is the union of two closed subsets A_{n} and B_{n} with intersection C_{n} = A_{n} ∩ B_{n}. Suppose three positive numbers γ_{n}, δ_{n}, ϵ_{n} satisfy

δ_{n} ≥ αγ_{n}, ϵ_{n} ≥ δ_{n},

and define a family F_{n} to be {A_{n}, B_{n}}. Then we have a sequence

L_{n}^{F_{n}}(C_{n}, p_{X}, γ_{n}) ⟶^{i_{∗}} L_{n}(A_{n}, p_{A_{n}}, δ_{n}) ⊕ L_{n}(B_{n}, p_{B_{n}}, δ_{n}) ⟶^{j_{∗}} L_{n}(X, p_{X}, ϵ_{n}).

Theorem 3.3. There exist constants κ_{0}, κ_{1}, κ_{2}, … (> 1) which do not depend on p_{X} such that

(1) if n ≥ 0 and ϵ_{n} ≥ κ_{n}δ_{n}, then the following composition j_{∗}i_{∗} is zero:

L_{n}^{F_{n}}(C_{n}, p_{X}, γ_{n}) ⟶^{i_{∗}} L_{n}(A_{n}, p_{A_{n}}, δ_{n}) ⊕ L_{n}(B_{n}, p_{B_{n}}, δ_{n}) ⟶^{j_{∗}} L_{n}(X, p_{X}, ϵ_{n}).

(2) if n ≥ 1, C_{n−1} ⊃ C_{n}^{κ_{n}ϵ_{n}}, γ_{n−1} ≥ κ_{n}ϵ_{n}, and if we set

F_{n−1} = {A_{n−1} = A_{n} ∪ C_{n−1}, B_{n−1} = B_{n} ∪ C_{n−1}},

then there is a connecting homomorphism

∂ : L_{n}(X, p_{X}, ϵ_{n}) ⟶ L_{n−1}^{F_{n−1}}(C_{n−1}, p_{X}, γ_{n−1}),

such that the following composition ∂j_{∗} is zero:

L_{n}(A_{n}, p, δ_{n}) ⊕ L_{n}(B_{n}, p, δ_{n}) ⟶^{j_{∗}} L_{n}(X, p_{X}, ϵ_{n}) ⟶^{∂} L_{n−1}^{F_{n−1}}(C_{n−1}, p_{X}, γ_{n−1}),

and, if δ_{n−1} ≥ αγ_{n−1} (so that the homomorphism i_{∗} is well-defined), the following composition i_{∗}∂ is zero:

\begin{matrix}L_{n}(X, p_{X}, ϵ_{n}) ⟶^{∂} L_{n−1}^{F_{n−1}}(C_{n−1}, & p_{X}, γ_{n−1}) \\ ⟶^{i_{∗}} & L_{n−1}(A_{n−1}, p, δ_{n−1}) ⊕ L_{n−1}(B_{n−1}, p, δ_{n−1}). \end{matrix}

Theorem 3.4. There exist constants λ_{0}, λ_{1}, λ_{2}, … (> 1) which do not depend on p_{X} such that

(1) if n ≥ 0, δ_{n} ≥ αγ_{n} (so that i_{∗} is well-defined), ϵ′_{n+1} ≥ λ_{n}δ_{n}, C′_{n} ⊃ C_{n}^{λ_{n}δ_{n}}, γ′_{n} ≥ κ_{n+1}ϵ′_{n+1} (so that ∂′ is well-defined), then the image of the kernel of i_{∗} in L_{n−1}^{F′_{n−1}}(C′_{n−1}, p_{X}, γ′_{n−1}) is in the image of ∂′:

[figure: L_{n}^{F_{n}}(C_{n}, p_{X}, γ_{n}); i_{∗}; L_{n}(A_{n}, p, δ_{n}) ⊕ L_{n}(B_{n}, p, δ_{n}); L_{n+1}(X, p_{X}, ϵ′_{n+1}); ∂′; L_{n}^{F′_{n}}(C′_{n}, p_{X}, γ′_{n})]

(2) if n ≥ 0, ϵ_{n} ≥ δ_{n} (so that j_{∗} is well-defined), C′_{n} ⊃ C_{n}^{λ_{n}ϵ_{n}}, γ′_{n} ≥ λ_{n}ϵ_{n}, δ′_{n} ≥ αγ′_{n} (so that i′_{∗} is well-defined), and F′_{n} = {A′_{n} = A_{n} ∪ C′_{n}, B′_{n} = B_{n} ∪ C′_{n}}, then the image of the kernel of j_{∗} in L_{n}(A′_{n}, p, δ′_{n}) ⊕ L_{n}(B′_{n}, p, δ′_{n}) is in the image of i′_{∗}:

[figure: L_{n}(A_{n}, p, δ_{n}) ⊕ L_{n}(B_{n}, p, δ_{n}); j_{∗}; L_{n}(X, p_{X}, ϵ_{n}); L_{n}^{F′_{n}}(C′_{n}, p_{X}, γ′_{n}); i′_{∗}; L_{n}(A′_{n}, p, δ′_{n}) ⊕ L_{n}(B′_{n}, p, δ′_{n})]

(3) if n ≥ 1, C_{n−1} ⊃ C_{n}^{κ_{n}ϵ_{n}}, γ_{n−1} ≥ κ_{n}ϵ_{n} (so that ∂ is well-defined), ϵ′_{n} ≥ λ_{n}γ_{n−1}, C′_{n} ⊃ C_{n−1}^{λ_{n}γ_{n−1}}, A′_{n} = A_{n} ∪ C′_{n}, and B′_{n} = B_{n} ∪ C′_{n}, then the image of the kernel of ∂ in L_{n}(X, p_{X}, ϵ′_{n}) is in the image of j′_{∗}:

[figure: L_{n}(X, p_{X}, ϵ_{n}); ∂; L_{n−1}^{F_{n−1}}(C_{n−1}, p_{X}, γ_{n−1}); L_{n}(A′_{n}, p, ϵ′_{n}) ⊕ L_{n}(B′_{n}, p, ϵ′_{n}); j′_{∗}; L_{n}(X, p_{X}, ϵ′_{n})]

Here the vertical maps are the homomorphisms induced by inclusion maps and relaxation of control.

Theorems 3.1 – 3.4 are all straightforward to prove.

4. Locally-finite analogues.

Up to this point, we considered only finitely generated modules and chain complexes. To study the behaviour of controlled L-groups, we need to use infinitely generated objects; such objects arise naturally when we take the pullback of a finitely generated object via an infinite-sheeted covering map.

Consider a control map p_{X} : M → X, and take the product with another metric space N:

p_{X} × 1_{N} : M × N⟶X × N.

Here we use the maximum metric on the product X × N.

Definition. (Ranicki and Yamasaki [4, p.14]) A geometric module on the product space M × N is said to be M-finite if, for any y ∈ N, there is a neighbourhood U of y in N such that M × U contains only finitely many basis elements; a projective module (A, p) on M × N is said to be M-finite if A is M-finite; a projective chain complex (C, p) on M × N is M-finite if each (C_{r}, p_{r}) is M-finite. [ In [4], we used the terminology “M-locally finite”, but this does not sound right and we decided to use “M-finite” instead. “N-locally M-finite” may be describing the meaning better, but it is too long.] When M is compact, M-finiteness is equivalent to the ordinary locally-finiteness.

Definition. Using this notion, one can define M-finite ϵ-controlled L-groups L_{n}^{M}(X × N, Y ×N, p_{X} ×1_{N}, ϵ), and M-finite ϵ-controlled projective L-groups L_{n}^{M,F}(Y ×N, p_{X} × 1_{N}, ϵ) by requiring that every chain complexes concerned are M-finite.

Consider the case when N = ℝ. We would like to apply the M-finite version of the Mayer-Vietoris-type stable exact sequence with respect to the splitting ℝ = (−∞, 0] ∪ [1, ∞). The following says that one of the three terms in the sequence vanishes.

Proposition 4.1. Let p_{X} : M → X be a control map. For any ϵ > 0 and r ∈ ℝ,

\begin{matrix}L_{n}^{M}(X × (∞, r], p_{X} × 1, ϵ) = L_{n}^{M}(X × [r, ∞), p_{X} × 1, ϵ) = 0. \\ K̃_{0}^{M}(X × (∞, r], p_{X} × 1, n, ϵ) = K̃_{0}^{M}(X × [r, ∞), p_{X} × 1, n, ϵ) = 0. \end{matrix}

Proof: This is done using repeated shifts towards infinity and the ‘Eilenberg Swindle’. Let us consider the case of L_{n}^{M}(X × [r, ∞), p_{X} × 1, ϵ). Let J = [r, ∞) and define T : M ×J → M ×J by T (x, t) = (x, t+ϵ). Take an element [c] ∈ L_{n}^{M}(X ×J, p_{X} ×1, ϵ). It is zero, because there exist M-finite ϵ Poincaré cobordisms:

\begin{aligned}c & ∼ c ⊕ (T_{#}(−c) ⊕ T_{#}^{2}(c)) ⊕ (T_{#}^{3}(−c) ⊕ T_{#}^{4}(c)) ⊕ … \\ & = (c ⊕ T_{#}(−c)) ⊕ (T_{#}^{2}(c) ⊕ T_{#}^{3}(−c)) ⊕ … ∼ 0 . \end{aligned}

Thus, the Mayer-Vietoris stably-exact sequence reduces to:

0 ⟶ L_{n}^{M}(X × ℝ, p_{X} × 1_{ℝ}, ϵ) ⟶^{∂} L_{n−1}^{p}(X × I, p_{X} × 1_{I}, γ) ⟶ 0,

where γ = κ_{n}ϵ, I = [−δ, δ], for some δ > 0. A diagram chase shows that there exists a well-defined homomorphism:

β : L_{n−1}^{p}(X × I, p_{X} × 1_{I}, γ) ⟶ L_{n}^{M}(X × ℝ, p_{X} × 1_{ℝ}, ϵ′),

where ϵ′′ = λ_{n}κ_{n}λ_{n−1}αγ. The homomorphisms ∂ and β are stable inverses of each other; the compositions

\begin{aligned}β∂ & : L_{n}^{M}(X × ℝ, p_{X} × 1_{ℝ}, ϵ) ⟶ L_{n}^{M}(X × ℝ, p_{X} × 1_{ℝ}, ϵ′) \\ ∂β & : L_{n−1}^{p}(X × I, p_{X} × 1_{I}, γ) ⟶ L_{n−1}^{p}(X × I, p_{X} × 1_{I}, κ_{n}ϵ′) \end{aligned}

are both relax-control maps.

Note that, for any γ, a projective L-group analogue of 1.2 gives an isomorphism:

L_{n−1}^{p}(X × I, p_{X} × 1_{I}, γ) ≅ L_{n−1}^{p}(X × {0}, p_{X}, γ).

In this case, no composition with relax-control map is necessary, because X × I is given the maximum metric. Thus, we have obtained:

Theorem 4.2. There is a stable isomorphism:

L_{n}^{M}(X × ℝ, p_{X} × 1_{ℝ}, ϵ) ⟶ L_{n−1}^{p}(X, p_{X}, γ).

Similarly, we have:

Theorem 4.3. There is a stable isomorphism:

L_{n}^{lf}(X × ℝ, p_{X} × 1_{ℝ}, ϵ) ⟶ L_{n−1}^{p,lf}(X, p_{X}, γ).

5. Stability in a special case.

In this section we treat the special case when the control map is the identity map. The following can be used to replace the controlled projective L-group terms in the previous section by controlled L-groups.

Proposition 5.1. Suppose that Y (⊂ X) is a compact polyhedron or a compact metric ANR embedded in the Hilbert cube and that p_{Y} is the identity map 1_{Y} on Y. Then for any ϵ > 0 and n, there exists a δ_{0} > 0 such that for any positive number δ satisfying δ ≤ δ_{0} there is a well-defined homomorphism functorial with respect to relaxation of control:

τ_{ϵ,δ} : L_{n}^{F}(Y, p_{X}, δ) ⟶ L_{n}(Y, 1_{Y} , ϵ)

such that the compositions

L_{n}^{F}(Y, p_{X}, δ) ⟶^{τ_{ϵ,δ}} L_{n}(Y, 1_{Y} , ϵ) ⟶^{ι_{ϵ}} L_{n}^{F}(Y, p_{X}, ϵ)

L_{n}(Y, 1_{Y} , δ) ⟶^{ι_{δ}} L_{n}^{F}(Y, p_{X}, δ) ⟶^{τ_{ϵ,δ}} L_{n}(Y, 1_{Y} , ϵ)

are both relax-control maps. In particular L_{n}^{p}(Y, 1_{Y} , δ) and L_{n}(Y, 1_{Y} , ϵ) are stably isomorphic.

Proof: Let δ_{1} = ϵ/α, where α is the positive number posited in 2.3. By 8.2 and 8.3 of [4], there exists a δ_{0} > 0 such that the following map is a zero map:

K̃_{0}(Y, 1_{Y} , n, δ_{0}) ⟶ K̃_{0}(Y, 1_{Y} , n, δ_{1}); [C, p] ↦ [C, p].

Therefore, if δ ≤ δ_{0}, there is a homomorphism

L_{n}^{F}(Y, p_{X}, δ) ⟶ L_{n}^{F∪{Y }}(Y, p_{X}, δ_{1}); [(C, p), ψ] ↦ [(C, p), ψ].

The desired map τ_{ϵ,δ} is obtained by composing this with the map

(i_{Y})_{∗} : L_{n}^{F∪{Y }}(Y, p_{X}, δ) ⟶ L_{n}(Y, 1_{X}, ϵ)

corresponding to the subspace Y.

Remark. If Y is a compact polyhedron, then there is a constant κ_{n}^{Y} > 1 which depends on n and Y such that δ_{0} above can be taken to be δ_{1}/κ_{n}^{Y}. For this we need to change the statement and the proof of 8.1 of [4] like those of 5.4 below.

Recall that in our Mayer-Vietoris-type stably-exact sequence, each piece of space tends to get bigger in the process. The following can be used to remedy this in certain cases. (It is stated here for the identity control map, but there is an obvious extension to general control maps.)

Proposition 5.2. Let r : X → A be a strong deformation retraction, with a Lipschitz continuous strong deformation of Lipschitz constant λ, and i : A → X be the inclusion map. Then r and i induce “stable” isomorphisms of controlled L-groups in the following sense: if ϵ > 0, then for any δ ( 0 < δ ≤ ϵ/λ) the compositions

L_{n}(X, 1_{X}, δ) ⟶^{r_{∗}} L_{n}(A, 1_{A}, ϵ) ⟶^{i_{∗}} L_{n}(X, 1_{X}, ϵ)

L_{n}(A, 1_{A}, δ) ⟶^{i_{∗}} L_{n}(X, 1_{X}, δ) ⟶^{r_{∗}} L_{n}(A, 1_{A}, ϵ)

are relax-control maps. Proof: Obvious from 1.2.

Theorem 5.3. Fix a compact polyhedron X and an integer n ≥ 0. Then there exist numbers ϵ_{1} > 0, κ ≥ 1 and λ ≥ 1 (which depend on n, X, and the triangulation) such that, for any subpolyhedrons A and B of X, any integer k ≥ 0, and any number 0 < ϵ ≤ ϵ_{1}, there exists a ladder:

[figure: L_{n}^{lf}(C × ℝ^{k}, 1, ϵ); i_{∗}; L_{n}^{lf}(A × ℝ^{k}, 1, ϵ) ⊕ L_{n}^{lf}(B × ℝ^{k}, 1, ϵ); j_{∗}; L_{n}^{lf}(K × ℝ^{k}, 1, ϵ); L_{n}^{lf}(C × ℝ^{k}, 1, λϵ); i_{∗}; L_{n}^{lf}(A × ℝ^{k}, 1, λϵ) ⊕ L_{n}^{lf}(B × ℝ^{k}, 1, λϵ); j_{∗}; L_{n}^{lf}(K × ℝ^{k}, 1, λϵ)]

[figure: ∂; L_{n}^{lf}(C × ℝ^{k+1}, 1, κϵ); i_{∗}; L_{n}^{lf}(A × ℝ^{k+1}, 1, κϵ) ⊕ L_{n}^{lf}(B × ℝ^{k+1}, 1, κϵ); ∂; L_{n}^{lf}(C × ℝ^{k+1}, 1, κλϵ); i_{∗}; L_{n}^{lf}(A × ℝ^{k+1}, 1, κλϵ) ⊕ L_{n}^{lf}(B × ℝ^{k+1}, 1, κλϵ)]

which is stably-exact in the sense that

(1) the image of a horizontal map is contained in the kernel of the next map, and (2) the relax-control image in the second row of the kernel of a map in the first row

is contained in the image of a horizontal map from the left, where C = A ∩ B and K = A ∪ B, and the vertical maps are relax-control maps.

Proof: This is obtained from the locally-finite versions of 3.3, 3.4 combined with 4.3, 5.1, and 5.2 (the strong deformations of the neighbourhoods of A and B in K can be chosen to be PL and hence Lipschitz). Since there are only finitely many subpolyhedrons of X (with a fixed triangulation), we may choose constants κ and λ independent of A and B.

Theorem 5.4. Suppose X is a compact polyhedron and n ≥ 0 is an integer. Then there exist numbers ϵ_{0} > 0 and 0 < μ_{0} ≤ 1 which depend on X and n such that

L_{n}^{ϵ}(X, 1_{X}, δ) = L_{n}^{c}(X; 1_{X})

for every ϵ ≤ ϵ_{0} and every δ ≤ μ_{0}ϵ.

Proof: We inductively construct sequences of positive numbers

ϵ_{1} ≥ ϵ_{2} ≥ ϵ_{3} ≥ … (> 0)
(1 ≥) μ_{1} ≥ μ_{2} ≥ μ_{3} ≥ … (> 0)

such that for any subcomplex K of X with the number of simplices ≤ l,

(1) if 0 < ϵ ≤ ϵ_{l}, 0 < δ ≤ μ_{l}ϵ, and k ≥ 0, then

L_{n}^{lf,ϵ}(K × ℝ^{k}, 1_{K} × 1, δ) = L_{n}^{lf,ϵ}(K × ℝ^{k}, 1_{K} × 1, μ_{l}ϵ),

and

(2) if 0 < ϵ ≤ ϵ_{l}, then the homomorphism

L_{n}^{lf,ϵ}(K × ℝ^{k}; 1_{K} × 1) ⟶ L_{n}^{lf,ϵ_{l}}(K × ℝ^{k}; 1_{K} × 1)

is injective. Here ℝ^{k} is given the maximum metric.

First suppose l = 1 (i.e. K is a single point). Any object with bounded control on ℝ^{k} can be squeezed to obtain an arbitrarily small control; therefore,

ϵ_{1} = the number posited in 5.2, μ_{1} = 1

works.

Next assume we have constructed ϵ_{i} and μ_{i} for i ≤ l. We claim that

ϵ_{l+1} = min\{\frac{μ_{l}}{λ} , \frac{1}{κ}\}ϵ_{l}, μ_{l+1} = \frac{μ_{l}^{2}}{λκ}

satisfy the required condition. Suppose the number of simplices of K is less than or equal to l + 1. Choose a simplex of K of the highest dimension, and call the simplex (viewed as a subpolyhedron) A, and let B = K − intA. Suppose 0 < ϵ ≤ ϵ_{l+1} and 0 < δ ≤ μ_{l}ϵ. A diagram chase starting from an element of

L_{n}^{lf}(K × ℝ^{k}, 1, μ_{l+1}ϵ)

in the following diagram establishes the property (1). Here the entries in each of the columns are

L_{n}^{lf}(A × ℝ^{k}, 1, γ) ⊕ L_{n}^{lf}(B × ℝ^{k}, 1, γ) , L_{n}^{lf}(C × ℝ^{k}, 1, γ)
L_{n}^{lf}(K × ℝ^{k}, 1, γ) , and L_{n}^{lf}(A × ℝ^{k+1}, 1, γ) ⊕ L_{n}^{lf}(B × ℝ^{k+1}, 1, γ) ,

CONTROLLED L-THEORY 135

for various γ’s specified in the diagram.

[figure: L_{n}^{lf}(A …) ⊕ L_{n}^{lf}(B …); L_{n}^{lf}(K …); L_{n}^{lf}(C …); L_{n}^{lf}(A …) ⊕ L_{n}^{lf}(B …); μ_{l} \frac{δ}{λ}; μ_{l} \frac{δ}{λ}; \frac{δ}{λ}; \frac{δ}{λ}; δ; δ; κδ; μ_{l+1}ϵ; κμ_{l+1}ϵ; \frac{μ_{l+1}}{μ_{l}} ϵ; κ\frac{μ_{l+1}}{μ_{l}} ϵ; κ\frac{μ_{l+1}}{μ_{l}} ϵ; λκ\frac{μ_{l+1}}{μ_{l}} ϵ = μ_{l}ϵ; μ_{l}ϵ; ϵ; ϵ]

Next suppose 0 < ϵ ≤ ϵ_{l+1}. A diagram chase starting from an element of L_{n}^{lf}(K × ℝ^{k}, 1, μ_{l+1}ϵ)

representing an element of

ker{L_{n}^{lf,ϵ}(K × ℝ^{k}; 1) ⟶ L_{n}^{lf,ϵ_{l}}(K × ℝ^{k}; 1)}

establishes (2).

\xymatrix{L_{n}^{lf}(C …) & L_{n}^{lf}(A …) ⊕ L_{n}^{lf}(B …) & L_{n}^{lf}(K …) & L_{n}^{lf}(C …) \\  &  & μ_{l+1}ϵ = \frac{μ_{l}^{2}}{λκ}ϵ \ar[r] \ar[d] & \frac{μ_{l}^{2}}{λ} ϵ \ar[d] \\  &  & \frac{μ_{l}}{λκ}ϵ \ar[r] \ar[d] & \frac{μ_{l}}{λ} ϵ \ar[ddd] \\ μ_{l}ϵ \ar[d] \ar[r] & μ_{l}ϵ \ar[d] \ar[r] & μ_{l}ϵ \ar[d] \\ ϵ \ar[r] \ar[dd] & ϵ \ar[d] \ar[r] & ϵ \ar[d] \\  & ϵ_{l+1} \ar[d] \ar[r] & ϵ_{l+1} \ar[r] & κϵ_{l+1} \\ λϵ_{l+1} \ar[d] \ar[r] & λϵ_{l+1} \ar[d] \\ \frac{1}{μ_{l}} λϵ_{l+1} \ar[r] & \frac{1}{μ_{l}} λϵ_{l+1}}

References.

[1] Quinn, F., Resolutions of homology manifolds, and the topological characterization of manifolds, Invent. Math. 72, 267 – 284 (1983).

[2] Ranicki, A. A.,Exact Sequences in the Algebraic Theory of Surgery, Math. Notes, vol. 26, Princeton Univ. Press, 1981.

[3] −− and Yamasaki, M., Symmetric and quadratic complexes with geometric control, Proc. of TGRC-KOSEF vol. 3, 139 – 152 (1993), available electronically on

    WWW from http://www.maths.ed.ac.uk/~aar/ and also from http://math.
    josai.ac.jp/~yamasaki/.

[4] −− and −−, Controlled K-theory, Topology Appl. 61, 1 – 59 (1995).

[5] Yamasaki, M., L-groups of crystallographic groups, Invent. Math. 88, 571 – 602 (1987).

Andrew Ranicki Department of Mathematics and Statistics University of Edinburgh Edinburgh EH9 3JZ Scotland

email : aar@maths.ed.ac.uk

Masayuki Yamasaki Department of Mathematics Josai University 1-1 Keyaki-dai, Sakado Saitama 350-02 Japan

email : yamasaki@math.josai.ac.jp