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\begin{document}
\title{ The Tutte Group of Projective Spaces}

\author{Andreas Dress and \underbar{Walter Wenzel}, Bielefeld}
\date{}
\maketitle
\thispagestyle{empty}

The Tutte group of a matroid offers an algebraic approach to matroid theory.
In particular, this group reflects some important geometric properties of
matroids in algebraic terms.
\medskip

In the sequel let $M$ denote a matroid defined on some possibly infinite
set $E$ of finite rank $n$ with $\Cal H = \Cal H_M$ as its set of hyperplanes
and $\Cal L = \Cal L_M$ as its set of hyperlines.
\medskip

{\bf Definition of the extended Tutte group:}
\newline
Let $\Bbb F^{\Cal H}_M$ denote the free abelian group generated by the symbols
$\varepsilon $ and $X_{H,a}$ for\linebreak $H \in \Cal  H,\ a \in E \setminus H$,
and let $\Bbb K^{\Cal H}_M$ denote the subgroup of $\Bbb F^{\Cal H}_M$
generated by $\varepsilon ^2$ and all elements of the form
$$\varepsilon \cdot X_{H_1,a_2}\cdot X^{-1}_{H_1,a_3} \cdot X_{H_2,a_3}
\cdot X^{-1}_{H_2,a_1} \cdot X_{H_3,a_1} \cdot X^{-1}_{H_3,a_2}$$
for $H_1,H_2,H_3 \in \Cal H,\ L := H_1 \cap H_2 \cap H_3 = H_i \cap H_j
\in \Cal L$ for $i \ne j$ and $a_i \in H_i \setminus L$ for $1 \le i \le
3$.
\newline
Then the {\bf (extended) Tutte group} $\Bbb T^{\Cal H}_M$ of $M$ is defined
by
$$\Bbb T^{\Cal H}_M := \Bbb F^{\Cal H}_M / \Bbb K^{\Cal H}_M.$$
Let $\varepsilon _M$ denote the image of $\varepsilon $ and $H(a)$ the image
of $X_{H,a}$ in $\Bbb T^{\Cal H}_M$, respectively.
\bigskip

{\bf Definition of the inner Tutte group:}
\newline
Define the homomorphism $\Phi : \Bbb T^{\Cal H}_M \to \Bbb Z^{\Cal H} \times
\Bbb Z^E$ by
$$\begin{array}{ll}
\Phi (\varepsilon _M) &:= 0, \\
\Phi (H(a)) &:= (\delta_H,\delta_a) \mbox{ for } H \in \Cal H,\ a \in E
\setminus H.\end{array}$$
$\Bbb T^{(0)}_M := ker\Phi$ is the {\bf inner Tutte group} of $M$.
\bigskip

\goodbreak
{\bf Proposition 1:}
\newline
Assume $M$ is representable over the skewfield $K$; that is, there exists
a family of maps $f_H : E \to K\ (H \in \Cal H)$ with

\itemindent=1.3 true cm
\begin{itemize}
\item[(H1)] $f_H^{-1}(\{0\}) = H$ for all $H \in \Cal H$;
\item[(H2)] if $H_1,H_2,H_3 \in \Cal H$ are pairwise distinct and
$L := H_1 \cap H_2 \cap H_3 \in \Cal L$, then there exist $c_1,c_2,c_3 \in
K^{\star}$ such that for all $e \in E$ we have
\end{itemize}

\itemindent=0pt
$$f_{H_1}(e) \cdot c_1 + f_{H_2}(e) \cdot c_2 + f_{H_3}(e) \cdot c_3 = 0.$$
Put $K_C := K^{\star}/[K^{\star},K^{\star}]$.
\newline
Then the homomorphism $\psi : \Bbb F^{\Cal H}_M \to K^{\star}$ given by
$$\begin{array}{ll}
&\psi(\varepsilon ) := -1, \\
&\psi(X_{H,a}) := f_H(a) \mbox{ for }H \in \Cal H,\ a \in E \setminus H
\end{array}$$
induces a homomorphism $\varphi = \overline{\psi} : \Bbb T^{\Cal H}_M \to
K_C$.
\bigskip

\goodbreak
{\bf Proposition 2:}
\newline
Assume $H_1,H_2,H_3 \in \Cal H$ with $H_1 \cap H_2 \cap H_3 \in \Cal L$
and $H_3 \ne H_i$ for $i \in \{1,2\}$. Then for $a,b \in H_3 \setminus (H_1
\cup H_2)$ we have
$$H_1(a) \cdot H_2(a)^{-1} = H_1(b) \cdot H_2(b)^{-1}.$$
\bigskip

{\bf Definition of cross ratios in Tutte groups:}
\newline
If $H_1,H_2,H_3$ are as in Proposition 2, put
$$\left| \begin{array}{ccc} H_1 & { } & H_2 \\
{ } & H_3 & { } \end{array}\right| := H_1(a) \cdot H_2(a)^{-1} \mbox{ for
} a \in H_3 \setminus (H_1 \cup H_2).$$
Assume $H_1,H_2,H_3,H_4 \in \Cal H$ with $H_1 \cap H_2 \cap H_3 \cap H_4
\in \Cal L$ and $H_1,H_2 \ne H_3,H_4$, i.e. $\{H_1,H_2\} \cap \{H_3,H_4\}
= \emptyset$. The {\bf cross ratio} $\left[ \begin{array}{cc} H_1 & H_2 \\ H_3 & H_4
\end{array}\right] \in \Bbb T^{(0)}_M$ is defined by
$$\left[ \begin{array}{cc} H_1 & H_2 \\
H_3 & H_4 \end{array}\right] := \left| \begin{array}{ccc} H_1 & { } & H_2 \\
{ } & H_3 & { } \end{array}\right| \cdot \left| \begin{array}{ccc} H_2 & { } & H_1 \\
{ } & H_4 { } \end{array}\right| .$$
\bigskip

By applying the more elementary part of Tutte's homotopy theory one proves
\bigskip
\goodbreak

{\bf Proposition 3:}
\itemindent=1 true cm
\begin{itemize}
\item[i)] $\Bbb T^{(0)}_M$ is the subgroup of $\Bbb T^{\Cal H}_M$
generated by $\varepsilon _M$ and all cross ratios $\left[ \begin{array}{cc} H_1 &
H_2 \\ H_3 & H_4 \end{array}\right]$ for which $H_1,H_2,H_3,H_4$ are pairwise
distinct hyperplanes in $M$ with
$H_1 \cap H_2 \cap H_3 \cap H_4 \in \Cal L$.
\item[ii)] $\Bbb T^{(0)}_M$ is generated by all cross ratios $\left[
\begin{array}{cc} H_1 & H_2 \\ H_3 & H_4\end{array}\right]$ as in i) if and only if
$M$ is not regular.
\end{itemize}
\bigskip     

\itemindent=0pt

{\bf Definition:}
\newline
Put
$$\begin{array}{ll}
\Cal H^{(4)} := \{ (H_1,H_2,H_3,H_4) \in\Cal H^4 \mid &H_1 \cap
H_2 \cap H_3 \cap H_4 = H_i \cap H_j \in \Cal L \\
&\mbox{for } i \in \{1,2\},\ j \in \{3,4\}\ \}.\end{array}$$
\itemindent=1 true cm
\begin{itemize}
\item[i)] $(H_1,H_2,H_3,H_4),(H'_1,H'_2,H'_3,H'_4)\in \Cal H^{(4)}$
are called {\bf projective neighbours of each other}, if $\bigcap^4_{i=1}H_i
\ne \bigcap^4_{i=1}H'_i$ and there exists some $H \in \Cal H$ with
\begin{itemize}
\item[(I)\ ]$\bigcap^4_{i=1} H_i \subsetneq H,\ \bigcap^4_{i=1}H'_i 
\subsetneq H$
and
\item[(II)\ ] for $1 \le i \le 4$ we have $H_i = H'_i$ or $H_i \cap H'_i
= H_i \cap H = H'_i \cap H \in \Cal L$.
\end{itemize}
If this is the case, we write $(H_1,H_2,H_3,H_4) \land
(H_1',H_2',H_3',H_4')$.
\item[ii)] $G = (H_1,H_2,H_3,H_4)$ and $G'=(H_1',H_2',H_3',H_4') \in
\Cal H^{(4)}$ are called {\bf projectively equivalent}, if there exist $k
\ge 0$ and $G_0,G_1,\dots ,G_k \in \Cal H^{(4)}$ such that 
$$G_0 = G,\ G_k = G',\ G_{i-1} \land G_i \mbox{ for } 1 \le i \le k.$$
In this case we write $(H_1,H_2,H_3,H_4) {\buildrel {pr} \over {\sim}}
(H'_1,H'_2,H'_3,H'_4)$. 
\end{itemize}
\bigskip  

\itemindent=0pt
\goodbreak
{\bf Proposition 4:}
\newline
Assume $(H_1,H_2,H_3,H_4) {\buildrel {pr} \over {\sim}} (H'_1,H'_2,H'_3,H'_4)$.
Then we have
$$\left[ \begin{array}{cc} H_1 & H_2 \\ H_3 & H_4 \end{array}\right] = \left[\begin{array}{cc}
H'_1 & H'_2 \\ H'_3 & H'_4 \end{array}\right].$$
From now on we assume that $M$ is a projective space of dimension $m \ge
2$.
\bigskip

\goodbreak
{\bf Proposition 5:}
\newline
Assume $H_1,H_2 \in \Cal H$ are distinct, put
$$\Cal H_{1,2} := \{H \in \Cal H \mid H_1 \cap H_2 \subseteq H\}$$
and let $\Cal G_{(H_1,H_2)}$ denote the group of projectivities $p : \Cal
H_{1,2} \to \Cal H_{1,2}$ with $p(H_i) = H_i$ for $i \in \{1,2\}$. Then
$\eta : \Cal G_{(H_1,H_2)} \twoheadrightarrow \Bbb T^{(0)}_M$ defined by
$$\eta(p) := \left[ \begin{array}{cc} H_1 & H_2 \\ H_3 & p(H_3) \end{array}\right]
\mbox{ for } H_3 \in \Cal H_{1,2} \setminus \{H_1,H_2\}$$
is a well defined group epimorphism; that is, $\eta$ does not depend on
$H_3$.
\bigskip

\goodbreak
{\bf Proposition 6:}
\newline
If $H_1,H_2,H_3 \in \Cal H$ are pairwise distinct with $L := H_1 \cap H_2
\cap H_3 \in \Cal L$, then
$$\Bbb T^{(0)}_M = \left\{\left[ \begin{array}{cc} H_1 & H_2 \\ H_3 & H_4
\end{array}\right] \mid L \subseteq H_4 \in \Cal H \setminus
\{H_1,H_2\}\right\}.$$
\bigskip

By applying Proposition 1 and Proposition 6 one proves
\bigskip

\goodbreak
{\bf Proposition 7:}
\newline
If $M$ is the projective space over the skewfield $K$ of dimension $m \ge
2$, then $\Bbb T^{(0)}_M \cong K_C$.
\newline
If, in particular, $K$ is a field, then $\Bbb T^{(0)}_M \cong K^{\star}$.
\bigskip

In general it follows from [G], Theorem 2, that in case $M$ is a finite
non-desarguesian projective plane its inner Tutte group $\Bbb T^{(0)}_M$
is necessarily trivial.
\bigskip

\bigskip


\begin{thebibliography}{42}
\bibitem[B]{} R. Baer: Linear Algebra and Projective Geometry. New York:
Academic Press 1952.
\bibitem[DW1]{} A. W. M. Dress and W. Wenzel: Geometric Algebra for
Combinatorial Geometries; to appear in Advances in Mathematics 76 (1989).
\bibitem[DW2]{} A. W. M. Dress and W. Wenzel: On Combinatorial and Projective
Geometry; in preparation.
\bibitem[G]{} Th. Grundh\"ofer: The Groups of Projectivities of Finite
Projective and Affine Planes. Ars Combinatoria 25A (1988), 269 -- 275.
\bibitem[HP]{} D. R. Hughes and F. C. Piper: Projective Planes. Springer
Verlag New York, Heidelberg, Berlin 1972.
\bibitem[T]{} W. T. Tutte: Lectures on Matroids. National Bureau of Standards
Journal of Research 69B (1965), 1 -- 47.
\bibitem[W]{} W. Wenzel: A Group Theoretic Interpretation of Tutte's Homotopy
Theory; to appear in Advances in Mathematics 76 (1989). 
\end{thebibliography}

\vspace*{1cm}
{\small 
\noindent
Address:\\
Universit\"at Bielefeld\\
Fakult\"at f\"ur Mathematik\\
Postfach 8640\\*[.2cm]
D-4800 Bielefeld 1}

\end{document} 
