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Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

John Nicholson

Abstract

In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective $\mathbb{Z} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X \vee S^n \simeq Y \vee S^n$ implies $X \simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective $\mathbb{Z} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem.

Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

Abstract

In previous work, we related homotopy types of finite -complexes when has periodic cohomology to projective -modules representing the Swan finiteness obstruction. We use this to determine when implies for finite -complexes and , and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective -modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case , difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem.

Paper Structure

This paper contains 22 sections, 63 theorems, 104 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications to Wall's D2 problem
  4. Applications to the classification of 4-manifolds
  5. The action of $\mathop{\mathrm{Aut}}\nolimits(G)$ on general projective modules
  6. Organisation
  7. Groups with periodic cohomology
  8. Binary polyhedral quotients
  9. Quaternionic representations
  10. Vanishing of the Swan finiteness obstruction
  11. Induced representations and the action of $\mathop{\mathrm{Aut}}\nolimits(G)$
  12. Cancellation for the Swan finiteness obstruction
  13. The Eichler mass formula
  14. Orders in quaternionic components of $\mathbb{Q} G$
  15. Cancellation for quaternionic orders
  16. ...and 7 more sections

Key Result

Theorem 1

Let $G$ have $k$-periodic cohomology, let $n = ik$ or $ik-2$ for some $i \ge 1$ and, if $n=2$, suppose $G$ has the D2 property. Then the following are equivalent:

Theorems & Definitions (104)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 1.2
  • Corollary 3
  • Remark 1.3
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • ...and 94 more