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Projective modules and the homotopy classification of $(G,n)$-complexes

John Nicholson

Abstract

A $(G,n)$-complex is an $n$-dimensional CW-complex with fundamental group $G$ and whose universal cover is $(n-1)$-connected. If $G$ has periodic cohomology then, for appropriate $n$, we show that there is a one-to-one correspondence between the homotopy types of finite $(G,n)$-complexes and the orbits of the stable class of a certain projective $\mathbb{Z} G$-module under the action of $\text{Aut}(G)$. We develop techniques to compute this action explicitly and use this to give an example where the action is non-trivial.

Projective modules and the homotopy classification of $(G,n)$-complexes

Abstract

A -complex is an -dimensional CW-complex with fundamental group and whose universal cover is -connected. If has periodic cohomology then, for appropriate , we show that there is a one-to-one correspondence between the homotopy types of finite -complexes and the orbits of the stable class of a certain projective -module under the action of . We develop techniques to compute this action explicitly and use this to give an example where the action is non-trivial.

Paper Structure

This paper contains 20 sections, 63 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Computing the action of $\mathop{\mathrm{Aut}}\nolimits(G)$
  4. Overview of the wider project
  5. Extensions of modules
  6. Projective $\mathbb{Z} G$-modules and the Swan finiteness obstruction
  7. Preliminaries on projective $\mathbb{Z} G$-modules
  8. Swan modules
  9. Projective extensions
  10. The Swan finiteness obstruction
  11. Classification of projective chain complexes
  12. General classification of projective $n$-complexes
  13. Projective $0$-complexes and the unstable Euler class
  14. Polarised homotopy classification of $(G,n)$-complexes
  15. Homotopy classification of $(G,n)$-complexes
  16. ...and 5 more sections

Key Result

Theorem A

Let $G$ have $k$-periodic cohomology and let $n=ik$ or $ik-2$ for some $i \ge 1$. Then there is an injective map of graded trees for any projective $\mathbb{Z} G$-module $P_{(G.n)}$ with $\sigma_{ik}(G) = [P_{(G.n)}] \in C(\mathbb{Z} G)/T_G$. Furthermore, $\Psi$ is bijective if and only if $n \ge 3$ or if $n=2$ and $G$ has the D2 property.

Figures (2)

  • Figure 1: A graded tree which is a fork
  • Figure 2: Minimal complexes for any $n$ even with $n \ne 2$

Theorems & Definitions (105)

  • Theorem A
  • Remark B
  • Theorem C
  • Theorem \ref{thm:main-example}
  • Lemma \ref{thm:main-example}: Shifting
  • Lemma \ref{thm:main-example}: Duality
  • Lemma \ref{thm:main-example}: Shifting
  • proof
  • Lemma \ref{thm:main-example}: Duality
  • Lemma \ref{thm:main-example}
  • ...and 95 more