UPD: Things described here and in some previous posts are particular cases of the topological chiral homology introduced in J. Lurie, “Derived Algebraic Geometry VI: E_k Algebras”. The only difference is that I need framed discs. I thank Damien for this comment.
Let be an oriented compact 2-manifold possibly with a border. Let be the complex of rational chains of the space of framed discs and on and semi-discs on the border. -homology of a dg f2-algebra (that is an algebra over rational chains of framed little discs operad) is the product of on over the operad of chains of framed little discs, that is modulo relations like in this definition.
Consider some examples.
1. is a disc. (to be more precise, the underlying complex). It is not suprising, because -homology is homotopical invariant with respect to .
2. is a annulus. , the Hochschild homology. Picture
proves that is a (homotopy) algebra (this fact is true for any 2-algebra, structure of f2-algebra is not necessary). -homology of of a surface with holes is a module over . Glueing of surfaces gives maps between derived tensor products over .
proves that is a (homotopy) module over , see also this statement (in contrast with the previous picture, here is necessarily a f2-algebra).
3. is a torus. , more exactly the total homology of the corresponding bicomplex. Note that this complex is acted by automorphisms of the torus, that is , what is not so obviuos if one define it as the double Hochschild complex.
5. is a pair of pants or surface of higher genus. The meaning of the corresponding -homology is not clear.