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.
Categories: math.RT math.GR math.KT