The n-Category Café

Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

Posted by Urs Schreiber

A while ago I had visited Grégory Ginot in Paris. He is an expert on string topology which is, despite its name, the study of the real homology of loop spaces.

The deRham cohomology of a loop space $L X$ is is captured already by those differential forms on $L X$ which are Chen iterated integrals: transgressions of forms $\Omega^\bullet(X)^{\otimes k}$ through the correspondence $$\array{ && \Delta^k \times L X \\ & \swarrow & \searrow \\ X^k && & L X \\ (\omega_1,\cdots, \omega_n) & \mapsto && \int_{\Delta^k} (\omega_1 \wedge \cdots \wedge \omega_n) }$$ induced by the obvious inclusion $\Delta^k \hookrightarrow (S^1)^k$. (see page 13 of Getzler, Jones and Petrack).

Acting with the loop space differential on such an iterated integral form is the same as acting with the differential on $X$ on each of the $\omega_i$, and then wedging all subsequent pairs of $\omega_i$s.

This second operation happens to be nothing but the Hochschild differential for the algebra $\Omega^\bullet(X)$ with values in itself.

It is famously known that the Hochschild cohomology of $\Omega^\bullet(X)$ computes the homology of the loop space of $X$, and I had thought of Chen’s iterated integrals as a good explanation for why that is the case.

But there should be an even nicer and even more conceptual point of view.

As you can read summarized concisely in

Grégory Ginot
Higher order Hochschild cohomology
(pdf)

one can understand the Hochschild differential as being essentially the differential on certain simplicial differential forms, where the simplicial structure is obtained by choosing the standard simplicial model of the circle by a single 1-simplex:

As apparently Pirashvili pointed out first, from any simplicial set $$S : \Delta^op \to FinSet$$ and any functor $$F : FinSet \to Vect$$ we obtain a simplicial vector space $F \circ S$ and hence, by Dold-Kan, a complex. This is such that if we take $F$ to be the functor induced by an algebra $A$, which sends $[n] \mapsto A^\otimes n$ and uses product and unit of the algebra to reflect surjective and injective maps of sets, and if we take $S$ to be the standard simplicial model of the circle, then the complex obtained from $F \circ S$ is literally the Hochschild complex of $A$.

That’s nice, because it suggests that we can vastly generalize Hochschild cohomology by using Pirashvili’s method, but using for $S$ a simplicial model of some higher dimensional space $\Sigma$, instead.

And it works: the cohomology of $F \circ S$ one obtains for $A = \Omega^\bullet(X)$ this way does compute the homology of the mapping space $Maps(\Sigma,X)$. See Grégory’s article for history, background, references, results and proofs.

What are the higher order Chen-iterated integrals that correspond to this higher order Hochschild cohomology?