Topology Seminar: Lifting G-stable endotrivial modules

Endotrivial modules are indecomposable kG-modules M for which End(M) splits as a trivial module direct sum a projective. Such modules play an important role in modular representation theory, and form an abelian group T(G) under tensor product. The structure of T(S) for S a finite p-group is known through many years of effort, culminating in the work of Carlson-Thévenaz. For an arbitrary finite group G, Balmer and Grodal have described the kernel of the restriction from T(G) to its Sylow T(S). In this talk we study the image of this restriction map. We provide a complete description when p = 2 and get partial information at odd primes. This verifies conjectures of Carlson-Mazza-Thévenaz when p = 2 but provides counterexamples at infinitely many odd primes. This is joint work with Tobias Barthel and Jesper Grodal.