Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology

Hodge decomposition of string topology

“…For > 1, it has been shown that the cup product on the higher Hochschild cohomology HH , * ( , ) of a commutative DG algebra (with finite-dimensional cohomology in each degree) preserves the Hodge decomposition [29,Section 6]. This statement is, however, not true in general for HH 1 , * ( , ) = HH * ( , ).…”

“…The question of compatibility of Hodge decomposition with various natural operations, including string topology operations, has been studied by numerous authors (see, e.g., [24,28,29,33,60] and references therein). In particular, we should mention that [7,Theorem 4.3] can be deduced from results of Felix and Thomas [24].…”

A simplicial construction for noncommutative settings

“…When X • is modeled by S 1 with the usual simplicial structure, one recovers the complex that defines Hochschild homology. For more results concerning higher order Hochschild (co)homology we refer to [5], [6], [7], and [13].…”

Brane actions for coherent $\infty$-operads

“…The case of the underlying E n -algebra structure has been proven by Ginot-Tradler-Zeinalian in [GTZ12], under the assumption that X is an (n − 1)-connected Poincaré duality space, using Hochschild homology models (see also [Hu06] for related results). Their algebraic methods can be refined to give an E fr n−1 ⊗E 1 -algebra structure (see [Gin17,Corollary 5.25], but do not readily generalize to the fullyframed situation; one might therefore try to use the brane action instead. However, since the ∞-operad E fr n is not reduced, having the non-trivial group E fr n (1) SO(n) as unary operations, one cannot use Toën's or Mann-Robalo's theorems.…”