What does Hochschild (co)homology mean

Question

What does hochschild (co)homology mean. Is it a statement of a topological invariant? What is a way of picturing what it is doing. I am only asking this question because my experience with other homology is that things like betti numbers come up and one can make clever statements about boundaries and holes. What can one say after probing such (co)homologies.

Answer 1

Hochschild (co)homology is not a "topological invariant", because it doesn't even deal with topological spaces at all. Not all things called "homology" are about topology. In this case, Hochschild (co)homology is an invariant of associative algebras and their bimodules. Hochschild homology (resp. cohomology) takes as input an associative algebra $A$ and an $(A,A)$-bimodule $M$, and it spits out a graded module $HH_*(A;M)$ (resp. $HH^*(A;M)$). As you can see, there isn't a single topological space appearing.

It has a variety of uses. I'm not 100% sure about the history, but as far as I know it was introduced to study deformation theory of associative algebras. First-order deformations of an algebra $A$ are in bijection with $HH^2(A;A)$, and more generally the Hochschild cohomology groups $HH^k(A;A)$ provide obstructions to getting higher order deformations of $A$. The Hochschild cochain complex $CH^*(A;A)$ has a dg-Lie algebra structure, and the Maurer–Cartan elements of this dg-Lie algebra are in bijection with formal deformations of $A$.
You now have a bunch of keywords that you can look up if you want more information about all this; there's the book Deformation theory of algebras and their diagrams by Markl that I know deals with that.

It is the operadic (co)homology theory associated to the operad governing associative algebras, and since this operad is Koszul it's also the André–Quillen (co)homology associated to this operad. So in this sense it's the "natural" (co)homology theory that one can use for associative algebras. A good reference is the book Algebraic operads by Loday and Vallette (and they also explain in section 12.2 all the stuff about deformation theory).

It turns out to have some uses in topology, too, so not all is lost. For example if $S^*(X)$ is the singular cochain complex of a topological space, then $S^*(X)$ has the structure of a dg-algebra. Then the Hochschild homology $HH_*(S^*(X))$ is isomorphic to the homology of the free loop space $LX = \operatorname{Map}(S^1, X)$ (these are maps, not pointed maps). See this paper by Loday for a reference.