Factorization homology and calculus $\textit{à la}$ Kontsevich Soibelman

Abstract

We use factorization homology over manifolds with boundaries in order to construct operations on Hochschild cohomology and Hochschild homology. These operations are parametrized by a colored operad involving disks on the surface of a cylinder defined by Kontsevich and Soibleman. The formalism of the proof extends without difficulties to a higher dimensional situation. More precisely, we can replace associative algebras by algebras over the little disks operad of any dimensions, Hochschild homology by factorization (also called topological chiral) homology and Hochschild cohomology by higher Hochschild cohomology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum giving interesting operations on topological Hochschild homology and cohomology. Contents 1. Colored operad 3 2. Homotopy theory of operads and modules 8 3. The little d-disk operad 11 4. Homotopy pullback in Top W 12 5. Embeddings between structured manifolds 15 6. Homotopy type of spaces of embeddings 17 7. Factorization homology 22 8. KS and its higher versions. 25 9. Action of the higher version of KS 27 References 28Let A be an associative algebra over a field k. A famous theorem by Hochschild Kostant and Rosenberg (see [HKR09]) suggests that the Hochschild homology of A should be interpreted as the graded vector space of differential forms on the non commutative space "SpecA". Similarly, the Hochschild cohomology of A should be interpreted as the space of polyvector fields on SpecA.If M is a smooth manifold, let Ω * (M ) be the (homologically graded) vector space of de Rham differential forms and V * (M ) be the vector space of polyvector fields (i.e. global sections of the exterior algebra on T M ). This pair of graded vector spaces supports the following structure:• The de Rham differential : d : Ω * (M ) → Ω * −1 (M ).• The cup product of vector fields : −.− :

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

“…This is the derived tensor product of the right module of discs in M with the algebra A, over the little discs operad. As in Section 7 of [28], we can model this as (D θ…”

Differential Calculus of Hochschild Pairs for Infinity-Categories

“…We abuse terminology by referring to it as Kontsevich-Soibelman operad. In a nutshell, KS ⊂ DCyl is the maximal simplicial subcomplex spanned by vertices correponding to those tuples which do not contain the color C. It is not difficult to check that KS is equivalent to that of [16] or [23,Section 11.2]. In [11], a version of KS is called the cylinder operad.…”

“…It contains two colors, D and C M , such that the full suboperad spanned by the color D is the E 2 -operad. In [11], it was shown that a combinatorial dg operad KS comb , which is a certain dg version of KS, acts on the pair of the Hochschild cochain complex and the Hochschild chain complex of an associative algebra (A ∞ -sense), and in [16] an algebraic structure on the Hochschild pair over KS was constructed for a ring spectrum by means of the Swiss-cheese operad conjecture [34].…”

“…In other words, our method provides a natural Morita invariant structure. This invariance has a fundamental importance in noncommutative geometry and is reasonable to expect, whereas the Morita invariant property of algebra structures in [11] and [16] remains an open problem. We would like to mention a recent work [1] in which the authors prove the Morita invariance of the calculus structure at the level of graded vector spaces of homology.…”

$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

“…Equivalently one can work in a suitable model category of E θ n -algebras, see [33], [26], [21], or [34]. A definition of E θ n -cell structures in terms of push outs is used to define cofibrant replacement for algebras over operads, e.g.…”