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April 18, 2008

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 LXL X is is captured already by those differential forms on LXL X which are Chen iterated integrals: transgressions of forms Ω (X) k\Omega^\bullet(X)^{\otimes k} through the correspondence Δ k×LX X k LX (ω 1,,ω n) Δ k(ω 1ω n) \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 Δ k(S 1) k\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 XX on each of the ω i\omega_i, and then wedging all subsequent pairs of ω i\omega_is.

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

It is famously known that the Hochschild cohomology of Ω (X)\Omega^\bullet(X) computes the homology of the loop space of XX, 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

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:Δ opFinSet S : \Delta^op \to FinSet and any functor F:FinSetVect F : FinSet \to Vect we obtain a simplicial vector space FSF \circ S and hence, by Dold-Kan, a complex. This is such that if we take FF to be the functor induced by an algebra AA, which sends [n]A n[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 SS to be the standard simplicial model of the circle, then the complex obtained from FSF \circ S is literally the Hochschild complex of AA.

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

And it works: the cohomology of FSF \circ S one obtains for A=Ω (X)A = \Omega^\bullet(X) this way does compute the homology of the mapping space Maps(Σ,X)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?

Posted at April 18, 2008 7:21 PM UTC

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7 Comments & 1 Trackback

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

this is not a comment to the blog post, but i was rushed and this was the quickest way to contact the blog owners.

i am so excited to have stumbled upon your blog… and while i only understand about 2% of it (and that 2% is mostly the fact that it’s written in english), i still can’t wait until i have time later this evening to look it all over.

i just recently returned back to school to further my education, as i have been studying philosophy and the history of mathematics on my own for the last year. what began as a simple reading of “death by blackhole” became me reading math textbooks and science journals for fun. i reached a point where i realised that in order to fully grasp the concepts i was reading about- i needed to be fully schooled in mathematics. so, while i already have a BA, i am returning to school to receive a BS in mathematics, and a simultaneous degree (BA) in philosophy. my goal is to enter the doctorate program in mathematics with a focus on pure mathematics.

you’d be surprised though, by the looks i’ve gotten from both the philosophy department and the math department when i say i want to study the ‘philosophy of mathematics’. i am basically having to work with the heads of the depts. to forge my own program… but i’m passionate about the subject (even if my passion outweighs my understanding).

kudos on the blog- and thank you, because it’s nice to know i’m not alone…

Posted by: amy on April 21, 2008 3:51 PM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

Good to hear about your project, Amy. Try not to forget that there’s much more to be done in philosophy of mathematics than most (Anglophone) practitioners can imagine.

Other than my posts for the Café, you can find other ideas here.

Posted by: David Corfield on April 22, 2008 9:50 AM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

Have you considered majoring in physics? I think the main reason for learning math is to be able to understand physics. The main reason why these mathematical concepts were invented in the first place was for use by physicists. Since the ancient world, the driving force behind the advancement of mathematics has been physics and astronomy. I also believe at when you reach an extremely advanced level, such as that represented by this blog, the distinction between “math” and “physics” disappears. If you’re planning to major in two things at once, my suggestion would be to major in both math and physics. What do you think?

Jeffery Winkler

Posted by: Jeffery Winkler on April 23, 2008 6:38 PM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

yes, i’ve thought on that too… and in truth, i love physics- especially anything involving the sun (not sure why, just do).

but, while i my current mantra has been, “i’m doing all of this because no one’s told me i can’t.”, i do know my limits. what i lack in math i make up for by being driven. and what is most important to me is to have an unshakable foundation in all the elements of math before i pursue anything else. as for the philosophy- i just find the study of the beauty of math fascinating, and also logic, paradoxes, studies of truth- etc…

with what little experience i have, i feel that i am drawn to studying set theory (the study of infinity is a particular draw) and the history of how our definitions of infinity has changed.

also, my first BA is in education, and while i love research- i know me. i’m the type who likes being with people and instilling a passion for learning. it still perplexes me at times that others DON’T find math and science exciting! so, my final goal (i believe) will be to keep building upon whatever i’ve gained, and at the same time teach and excite those who may not yet realise that they too love math…

oh, and knowing me- i’m likely to decide to take up some physics classes eventually anyway- at some point i’ll be stalled in learning about something and just decide to study it further.

yep, i’m one of those people

Posted by: amy on April 24, 2008 10:30 PM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

i just find the study of the beauty of math fascinating

And that I think is the main reason most of us in math want to learn math – not as a means to be able to understand physics better, necessarily, but just because it’s beautiful, and permanent.

If you’re drawn to thinking about infinity, there’s a blog A Dialogue on Infinity which you might be interested in, co-hosted by David Corfield (who is one of the hosts here as you know). As someone interested in philosophy of mathematics, you may find some of the discussions attractive.

Posted by: Todd Trimble on April 25, 2008 2:59 AM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

thanks for the link, i’ve looked at it and look forward to reading it soon!

it seems right up my alley.

Posted by: amywithlemon on April 29, 2008 4:20 AM | Permalink | Reply to this

Re: Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

I encourage you to read my work on my website.

Posted by: Jeffery Winkler on April 25, 2008 5:40 PM | Permalink | Reply to this
Read the post Teleman on Topological Construction of Chern-Simons Theory
Weblog: The n-Category Café
Excerpt: A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
Tracked: June 18, 2008 10:01 AM

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