Fixed Point Sets and the Fundamental Group II: Euler Characteristics

TL;DR

This work extends Oliver's Euler-characteristic obstruction for fixed point sets to fixed sets of finite G-CW-complexes with prescribed homotopy types, revealing that the obstruction is governed by Euler-characteristic congruences even beyond contractible fixed sets. It introduces pseudo-equivalences and a local-to-global framework, proving a local cell-wise partition of Euler characteristics and a global criterion for nonempty fixed sets via $\chi(F)=\chi(Y^G)$ mod $n_G$, with a refined ANR variant using $m_G$. An obstruction group $N_Y$ is defined to encapsulate remaining local data needed for extending fixed sets, and the paper analyzes how the fundamental group, lifted to a universal cover through $\Gamma$, yields a $K_0(R[\Gamma])$-based perspective that imposes further, often component-wise, constraints. The results substantially clarify when and how fundamental group data influence fixed-point realizations, connect to trace methods like the Hattori–Stallings trace, and extend the theory to compact Lie groups, offering new tools for equivariant topology and manifold actions.

Abstract

For a group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$-CW-complex is the Euler characteristic $χ(F)$. He also has the similar results for compact Lie group actions. We show that the analogous problem for $F$ to be the fixed point set of a finite $G$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps in $K_0$, we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

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Theorems & Definitions (30)

• Theorem 1
• Theorem : Oliver
• Theorem 2
• Remark 1
• Theorem 3
• Definition
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 2