Nikita Markarian’s mathblog

Spherical homology II

Posted in Mathematics by nikitamarkarian on November 13, 2009

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 B, let HH(B) be the homotopy algebra mentioned in the previous paragraph. Then B is equipped with a homotopy left(right) action of HH(B) that enjoy following properties:

1. The map HH(B)\otimes B\to B is cyclic, where the cyclic structure on the Hochschild homology is the standard one and on B comes from the operad action;

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

Construction (picture): Left action of chain a\otimes b\otimes c\otimes \cdots\otimes d of the Hochschild complex on x\in B is given (up to a scalar) by the chain presented on the picture

left_action

where  angles \alpha, \beta, \gamma,... run through all possible values. The right action is given by

right_action

Note, that two properties uniquely define actions.

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

Definition (second): The spherical homology of a f2-algebra B is HS(B):= \mathop{Tor}_{HH(B)}(B_l, B_r), where B_{l(r)} is left (right) module over HH(B) 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 a\otimes b\otimes c\otimes\cdots\otimes d of the Hochschild complex produce a chain in \oplus_k (C_*(fS^2(k))\otimes B^{\otimes k}) by the picture

ring

 

where radius of the ring is fixed, \alpha, \beta, \gamma take all possible values, N is for the North Pole and the dash line is the Greenwich Meridian.  Now take the standard complex for \mathop{Tor}_{HH(B)}(B_l, B_r) that is \oplus (B\otimes HH(B)^{\otimes i}\otimes B) with the obvious differential. Given an element x \in \otimes A\otimes B\otimes C \otimes y let us send it to a chain in \oplus_k (C_*(fS^2(k))\otimes B^{\otimes k}) as follows. Put x and y to the North and South poles:

 

poles

 

and put rings corresponding to A, B,C in all possible positions along parallels:

rings

(Α, Β, Γ take all possible values).

On the spherical homology group SO(3) acts. It follows that the subgroup SO(2) \subset SO(3) (rotation around the Earth’s axis) acts trivially, at least on the level of rational chains. The equivariant homology under this action is \mathop{Tor}_{HC(B)}(B'_l, B'_r), where HC is the cyclic homology and B'=(B, d+u\Delta). As the action is trivial, \mathop{Tor}_{HC(B)}(B'_l, B'_r)=HS(B)\otimes k[u]. The rigidity property of the cyclic homology follows the rigidity of the spherical homology.

Besides \mathop{Tor}, it is interesting to consider \mathop{Ext}^*_{HH(B)}(B_l, B_l). 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 g be a Lie bialgebra. Then the Chevalley complex (S^*g^\vee[1], d_{Lie}) is a BV algebra with the differential operator of the second order \Delta given by the Lie cobracket \Lambda^2 g^\vee \to g^\vee (it is not exactly true, because the differential does not commute with \Delta  in general, but it seems could be fixed by a kind of twist). It seems that the Hochschild homology of the corresponding f2-algebra B(g) equals (as an algebra) to the cohomology of functions on the quantum group as a g-module. The meaning of the spherical homology is not clear. But \mathop{Ext}^*_{HH(B(g))}(B(g)_l, B(g)_l) seems to be a thing that is Koszul dual to the Drinfeld double.

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