Nikita Markarian’s mathblog

Spherical homology I

Posted in Mathematics by nikitamarkarian on November 10, 2009

Here I  give a definition of spherical homology, a functor from the category of BV-algebras to the category of complexes, that is analogous to the Hochschild homology of associative algebras.

Notation: algebra over the operad of chains of  framed little discs := f2-algebra.

Let B be an algebra over the BV operad. By  P. Severa, “Formality of the chain operad of framed little disks”, the operad of rational chains of little framed discs is formal, so B is (homotopically) an algebra over the latter operad as well, that is a f2-algebra.

Definition (first): Let C_*(fS^2(k)) be the complex of rational chains of the space of framed discs on a 2-sphere. Define spherical homology HS(B) of  B considered as a f2-algebra as follows. As a first approximation it is a complex \oplus_k (C_*(fS^2(k))\otimes B^{\otimes k}) modulo relations

spheres2

where C is an element of the operad of rational chains of little framed discs and x,...,u\in B (compare with the previous post). To be more precise one should consider a realization of the operad of framed little discs as a real blow-up of the moduli space of stable curves \underline{\mathcal M}, see J. Giansiracusa, P. Salvatore “Cyclic formality of the framed 2-discs operad and genus zero stable curves”, section 2.2. Then the complex is defined by the same picture, but loop on the sphere shrinks and presents one (nodal – on the left hand side) point.

There is a well known statement about Hochschild homology of associative algebras (see e.g.  J-L. Loday, “Cyclic homology”, proposition 1.3.3): inner derivations act trivially on the Hochschild homology. That is for v\in A the map ad(v) on the Hochschild complex given by ad(v)(a_0\otimes\cdots \otimes a_n)=\sum a_0\cdots\otimes [a_i, v]\otimes\cdots \otimes a_n is homologically trivial. The homotopy is h(v) \colon a_0\otimes\cdots \otimes a_n\mapsto \sum a_0\otimes\cdots\otimes v\otimes\cdots \otimes a_n. Another proof is based on the observation that the cohomological Hochschild complex  (in other words, the deformation complex) acts on the homological Hochschild complex, and an inner derivation is a coboundary in this complex.

The analogous statement for BV(f2)-algebras is about invariance of the spherical homology under inner derivations. By inner derivation for a BV-algebra I mean \{v, \cdot\}, where \{a,b \}=\Delta(ab)-\Delta (a)b -a\Delta(b) is the Gerstenhaber bracket. For a f2-algebra inner derivation is \{v,\cdot\}, where \{,\} is the operation that produces the Gerstenhaber bracket on the homology. It seems that this notions coincide under the quasiisomorphism of Tamarkin-Severa. An inner derivation i of degree 1 gives an infinitesimal deformation of BV(f2)-algebra by d'=d+\varepsilon i. Thus one may think about the mentioned invariance as about rigidity of spherical homology.

Analog of the homotopy h(v) above is the map from the complex calculating the spherical homology to itself such that a chain from C_n(fS^2(k)) goes to C_{n+2}(fS^2(k+1)), as v sits in the additional hole and this hole runs around all possible positions. One have to choose framing in it somehow. If \Delta v=0 this framing is not important and this map retracts infinitesimal deformation d'=d+\varepsilon \{v, \cdot\}.

Questions:

1. Have spherical homology of the BV-algebra introduced in the seminal paper of M. Chas and D. Sullivan some geometric meaning?

2. What about homology in a module over a BV-algebra? And more, about “torsion products” of modules?

3. One may consider higher genera as well. Does it give something new? It seems that the case of torus already appeared here under the name of 2-Hochschild homology. Note that the later functor depends only on the Gerstenhaber structure of a BV-algebra.

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