The Infinite Seminar

Some of the recent preprints seen in the arXiv mailings include:

Mikael Johansson: A partial $A_{\mathrm{\infty }}$-structure on the cohomology of $C_n\times C_m$

Suppose k is a field of characteristic 2, and n,m≥2 powers of 2. Then the $A_{\mathrm{\infty }}$-structure of the group cohomology algebras $H^{*}(C_n,k)$ and $H^{*}(C_m,k)$ are well known. We give results characterizing an $A_{\mathrm{\infty }}$-structure on $H^{*}(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

Alastair Hamilton & Andrey Lazarev: Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_{\mathrm{\infty }},C_{\mathrm{\infty }}$ and $L_{\mathrm{\infty }}$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_{\mathrm{\infty }}$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

Alastair Hamilton & Andrey Lazarev: Symplectic $A_{\mathrm{\infty }}$-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\’e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for $C_{\mathrm{\infty }}$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Yes, this post includes a bit of shameless self-promotion.

There are more and more mathematical research group blogs entering the scene lately. I would like to try and start one more, this one with a focus on various closely related subjects - the ${*}_{\mathrm{\infty }}$ algebras.

We know of the $A_{\mathrm{\infty }}$ algebras. We know of the $L_{\mathrm{\infty }}$ algebras. And we know of $E_{\mathrm{\infty }}$ ring spectras. I most probably forget about several highly interesting similar constructions, which I however expect will end up being more than welcome here.

The tagline I expect the prevalent theme here to follow, though, will be “…up to homotopy”. This seems, to my mind, to be a unifying factor of most ${*}_{\mathrm{\infty }}$-theories I have seen so far. Thus, as the technical host of discussions, I wish you all welcome.

If you feel you can contribute more than just as a regular reader and commenter, and might even wish to write posts, then please create a user and drop me a note on mik@math.uni-jena.de. As soon as I see the user created, I will be able to grant further privileges. Hopefully, this way, we can get enough of the conversation going here to make this as vibrant and interesting a group blog as all the other examples out there.