The Euler characteristic of an endotrivial complex

TL;DR

This paper investigates the Lefschetz homomorphism from endotrivial complexes to the trivial source ring in modular representation theory, establishing conditions for its surjectivity and describing its kernel for various finite groups.

Contribution

It introduces the Lefschetz homomorphism in the context of endotrivial complexes and characterizes its surjectivity and kernel for different classes of finite groups.

Findings

Surjectivity of the Lefschetz homomorphism under specific group conditions.

Explicit description of the kernel of the Lefschetz homomorphism.

Examples where surjectivity fails for groups with higher p-rank.

Abstract

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy category of $p$-permutation modules $K^b({}_{kG}\mathbf{triv})$, to the orthogonal unit group of the Grothendieck group of $K^b({}_{kG}\mathbf{triv})$, i.e. the trivial source ring. When $p = 2$ and $k = \mathbb{F}_2$, $\Lambda$ is surjective when $G$ has a Sylow $2$-subgroup with fusion controlled by its normalizer, and when $G$ has dihedral Sylow $2$-subgroups. When $p$ is odd, $\Lambda$ is surjective if $G$ has a cyclic Sylow $p$-subgroup or is $p$-nilpotent, but we exhibit examples of groups of $p$-rank 2 or greater for which $\Lambda$ is not surjective. We also examine the kernel of the Lefschetz homomorphism, determining it for all groups when $p = 2$ and for groups with cyclic Sylow $p$-subgroups when $p$ is odd.