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Higher Hochschild homology and exponential functors


We study higher Hochschild homology evaluated on wedges of circles, viewed as a functor on the category of free groups. Coefficients arising from square-zero extensions are used, motivated by work of Turchin and Willwacher in relation to hairy graph cohomology. For the main case of interest, the action of automorphisms of free groups on Hochschild homology factors across outer automorphisms. As the appropriate underlying framework, we introduce and study the category of outer functors, the full subcategory of functors on free groups on which inner automorphisms act trivially. These provide a natural source of coefficient systems for outer automorphisms of free groups. The main results exploit exponential functors, which arise both in calculating higher Hochschild homology and in relation to polynomial functors on free groups. We prove that these are related, using twisting by an orientation module to introduce the necessary Koszul signs. Working over a field of characteristic zero, we then apply the structure theory of polynomial functors to achieve effective calculations of higher Hochschild homology, generalizing the initial results of Turchin and Willwacher. These methods also allow us to prove structure results for outer functors.

Dates and versions

hal-01715121 , version 1 (22-02-2018)



Geoffrey Powell, Christine Vespa. Higher Hochschild homology and exponential functors. 2018. ⟨hal-01715121⟩
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