Endotrivial modules for finite groups via homotopy theory

Classifying endotrivial kG-modules, i.e., elements of the Picard group stable module category for an arbitrary finite G, has been a long-running quest. By deep work Dade, Alperin, Carlson, Thevenaz, and others, it reduced to understanding subgroup consisting modular representations that split as trivial k direct sum projective when restricted Sylow p-subgroup. In this paper we identify first cohomology orbit on non-trivial p-subgroups with values in units k^x, viewed constant coefficient system. We then use homotopical techniques give number formulas terms abelianization normalizers centralizers particular verifying Carlson-Thevenaz conjecture--this reduces calculation algorithmic calculations local theory rather than representation theory. also provide strong restrictions such dimension greater one can occur, p-subgroup complex p-fusion systems. immediately recover extend large computational results literature, further illustrate potential by calculating other sample new cases, e.g., Monster at all primes.