## Spherical homology II

Here I give the second definition of the spherical homology based on a statement about the Hochschild homology of a BV(f2)-algebra that has an independent interest.

As it was already mentioned in this blog (e.g. here) due to M. Brun, Z. Fiedorowicz, R. Vogt “On the multiplicative structure of topological Hochschild homology” on the Hochschild homology of a 2-algebra (that is an algebra over chains of little discs operad) a structure of homotopy algebra is defined. If our algebra is f2-algebra (that is an algebra over chains of *framed* little discs operad) then a new feature appears.

**Statement: **Given a f2-algebra , let be the homotopy algebra mentioned in the previous paragraph. Then is equipped with a homotopy left(right) action of that enjoy following properties:

1. The map is cyclic, where the cyclic structure on the Hochschild homology is the standard one and on comes from the operad action;

2. The composition of the natural embedding with this left (right) action is the left (right) action of on itself as a homotopy associative algebra.

**Construction** (picture): Left action of chain of the Hochschild complex on is given (up to a scalar) by the chain presented on the picture

where angles run through all possible values. The right action is given by

Note, that two properties uniquely define actions.

**Example:** Consider a BV-algebra that is a (super)commutative polynomial as an algebra and is a derivation, and the corresponding f2-algebra. Let be a generator of , and be the left action. Then image of in acts by , by the first property. By the second property, , where is the Rinehart-Connes-Feigin-Tsygan differential. These formulas define action of on .

**Definition** (second)**:** The *spherical homology* of a f2-algebra is , where is left (right) module over given by the statement.

To show that this definition coincide with the one given in the previous post it is enough to produce a map from one functor to another and show that on a free object this is a quasiisomorphism. What about free objects it will be clear from the next post. Let us construct a map from the second definition to the first one. At the beginning given a chain of the Hochschild complex produce a chain in by the picture

where radius of the ring is fixed, take all possible values, N is for the North Pole and the dash line is the Greenwich Meridian. Now take the standard complex for that is with the obvious differential. Given an element let us send it to a chain in as follows. Put and to the North and South poles:

and put rings corresponding to in all possible positions along parallels:

(Α, Β, Γ take all possible values).

On the spherical homology group acts. It follows that the subgroup (rotation around the Earth’s axis) acts trivially, at least on the level of rational chains. The equivariant homology under this action is , where is the cyclic homology and . As the action is trivial, . The rigidity property of the cyclic homology follows the rigidity of the spherical homology.

Besides , it is interesting to consider . This complex may be used for example to prove rigidity of the spherical homology.

**Example** (see e.g. section 3 of E. Getzler, “Manin pairs and topological field theory”): let be a Lie bialgebra. Then the Chevalley complex is a BV algebra with the differential operator of the second order given by the Lie cobracket (it is not **exactly** true, because the differential does not commute with in general, but it seems could be fixed by a kind of twist). It seems that the Hochschild homology of the corresponding f2-algebra equals (as an algebra) to the cohomology of functions on the quantum group as a -module. The meaning of the spherical homology is not clear. But seems to be a thing that is Koszul dual to the Drinfeld double.

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