# Nikita Markarian’s mathblog

## S-homology

Posted in Mathematics by nikitamarkarian on March 18, 2010

$S$-homology is an obvious generalization of the spherical homology. I already mentioned it here. In the present post I would like to summarise their properties.

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.

UPD: Following M. Kapranov, I will use term manifoldic homology for the higher dimensional generalization of $S$-homology.

Let $S$ be an oriented compact 2-manifold possibly with a border. Let $C_*(fS(k))$ be the complex of rational chains of the space of framed discs and on $S$ and semi-discs on the border. $S$-homology $HS_*$ of a dg f2-algebra $A$ (that is an algebra over rational chains of framed little discs operad) is the product of $C_*(fS(k))$ on $A$ over the operad of chains of framed little discs, that is $\oplus_k (C_*(fS(k))\otimes A^{\otimes k})$ modulo relations like in this definition.

Consider some examples.

1. $S$ is a disc. $HS_*(A)=A$ (to be more precise, the underlying complex). It is not suprising, because $S$-homology is homotopical invariant with respect to $S$.

2. $S$ is a annulus.  $HS_*(A)=HH_*(A)$, the Hochschild homology. Picture

proves that $HH_*(A)$ is a (homotopy) algebra (this fact is true for any 2-algebra, structure of f2-algebra is not necessary). $S$-homology of of a surface with $n$ holes is a module over $HH_*(A)^{\otimes n}$. Glueing of surfaces gives maps between derived tensor products over $HH_*(A)$.

Picture

proves that $A$ is  a (homotopy) module over $HH_*(A)$, see also this statement (in contrast with the previous picture, here $A$ is necessarily a f2-algebra).

3. $S$ is a torus. $HS_*(A)=HH_*(HH_*(A))$, more exactly the total homology of the corresponding bicomplex. Note that this complex is acted by automorphisms of the torus, that is $SL_2(\mathbb{Z})$, what is not so obviuos if one define it as the double Hochschild complex.

4. $S$ is a sphere. $HS_*(A)$ is the spherical homolgy that was discussed here, here and here.

5. $S$ is a pair of pants or surface of higher genus. The meaning of the corresponding $S$-homology is not clear.

### 8 Responses

1. Damien Calaque said, on January 5, 2011 at 11:22 am

Dear Nikita,

it seems to me that what you are doing is closely related to
– higher Hochschild homology (Pirashvili)
– topological chiral homology (Lurie)
– factorization homology (Costello-Gwilliam)
There is a very recent preprint of Ginot-Tradler-Zeinalian (see http://people.math.jussieu.fr/~ginot/papers/HighHoc.pdf) that explain how they are related (they are basically the same in the case of Commutative algebras). i guess that your S-homology should coincide with those when they intersect.

Damien

• nikitamarkarian said, on January 6, 2011 at 4:27 pm

Dear Damien,

thank you for your comment, nice to hear from you.
Thank you for the reference on Costello-Gwilliam, I did not know it. It seems that this is close to things I am interested in, but there are too many physics there. The Lurie’s work in dimension two must be also close. But I don’t understand, if he needs only little discs action, not framed discs? What about Ginot-Pirashvili’s things, it seems that the case of commutative algebras is much more simpler. For example, it works for any simplicial sets, not only for manifolds.

I start studying S-homology only for S=2-sphere to work with Hycomm-algebras. Then I discovered that this is a particular case of a more general definition. I do not know how to make use of these more general things. I would be very grateful for any suggestion.

2. Damien said, on January 24, 2011 at 8:30 pm

Hi Nikita,

sorry for answering so late. I agree that Pirashvili’s higher Hochschild homology does work for any simplicial set. But this is not a so serious issue since
– Costello-Gwilliam approach works for any topological space
– I guess that Lurie’s approach does work more generally for CW-complexes (not only manifolds).

About framing, one might be missleaded by the word. Lurie does not need framed little disks, but he needs the tangent space of the manifold he considers to be equipped with a framing (he needs some nontrivial additional data on the manifold, and the construction actually depends on this data).

At the moment I have no relevant suggestion to make. I still don’t understand how your definition is related to the other ones I mentionned. In particular I don’t understand where framed little disks are necessary in your approach.

• nikitamarkarian said, on January 27, 2011 at 9:13 am

Hi Damien,

– Costello-Gwilliam approach works for any topological space
– I guess that Lurie’s approach does work more generally for CW-complexes (not only manifolds).

I don’t know about the Costello-Gwilliam approach. But in the Lurie’s one I guess that the thing must be a manifold. Indeed the idea is we take the configuration space of the thing and consider it in a sense as a right module over the little discs operad. But it works only if the thing locally looks like a disc.

About framing, one might be missleaded by the word. Lurie does not need framed little disks, but he needs the tangent space of the manifold he considers to be equipped with a framing (he needs some nontrivial additional data on the manifold, and the construction actually depends on this data).

Exactly, he needs that the manifold must be parallelizable (by the way, if it works for any CW-complex what does it mean then?). To avoid this restriction one have to consider framed discs.

In fact, I have no particular approach, I solved a practical problem: to construct a functor from (homotopy) BV algebras to complexes with prescribed properties, something like it must be invariant under “inner” deformations. By the formality theorem one can replace BV with framed 2-discs. Firstly I tried to construct such a functor from an algebra over the operad of non-framed discs and found S-homology when S=torus. It seems that there no other such functors on algebras over non-framed discs. The explanation is that torus is the only parallelizable oriented compact surface. Then I constructed S-homology when S=sphere and was quite happy, because it was exactly the functor I was looking for. And then I recognized that sphere may be replaced by any oriented compact surface. And I would like to understand how to make use of all these functors.

The moral is that in dimension 2 the non-framed life is much more boring that the framed one. I guess that the circle action produce reach structures that appears in sciences about quantum cohomology and Witten conjecture, I am going to discuss it sometime.

Please, write me again if some questions remain.

• Damien said, on February 24, 2011 at 9:14 pm

Dear Nikita,

I have a few more comments and questions. I was quite optimistic when saying that Lurie’s approach works for CW-complexes. My guess was (following what happens for higher Hochschild homology and S-homology) that Lurie’s topological chiral homology should somehow be an homotopy invariant of the manifold he considers. Then I vaguely remembered a statement like “any CW complex has the homotopy type of a topological manifold”… but now I have some doubts.

By the way, it is indeed very nice that using framed little disk then you seems to be able to avoid the use of a trivialization of the tangent bundle in defining some avatar of topological chiral homology.

Then I have a few questions :
1. what are algebras over the framed little disk operad (I mean, not going to chain level, but staying at the topological level) ?
2. dealing with QFTs with boundary conditions then one needs to use an avatar of the swiss-cheese operad. I have an idea of what “framed swiss-cheese” is… but it obviously does not contain framed little disks. Say it in another way, Lurie’s stuff seems to work for manifolds with corners (see his paper on extended QFTs and the cobordism hypothesis). I see that you have a construction for surfaces with boundary, but I have the strange feeling that it misses some information from the boundary (I might be wrong).
3. You should have an (easier) analog in dimension 1: namely, for any curve $C$ and any $E1$-algebra $A$, associate something like $H_C(A)$. For $C$ the affine line you should get $A$ itself and for $C=S^1$ you should get Hochschild homology of $A$.
4. Do you have cochain version of this, and associated Deligne’s type conjecture ?

Best

Damien

• nikitamarkarian said, on March 2, 2011 at 10:54 am

Dear Damien,

1. what are algebras over the framed little disk operad (I mean, not going to chain level, but staying at the topological level) ?

The recognition principle for an algebras over framed discs states that this is the same as spaces with a circle action. This is well written in the Salvatore-Wahl’s paper. By the way, every space may be equipped with the trivial circle action, thus every algebra over little discs may be canonically lifted to an algebra over framed little discs. This is not true on the level of complexes! Perhaps, it works for complexes with augmentation.

2. dealing with QFTs with boundary conditions then one needs to use an avatar of the swiss-cheese operad. I have an idea of what “framed swiss-cheese” is… but it obviously does not contain framed little disks. Say it in another way, Lurie’s stuff seems to work for manifolds with corners (see his paper on extended QFTs and the cobordism hypothesis). I see that you have a construction for surfaces with boundary, but I have the strange feeling that it misses some information from the boundary (I might be wrong).

Yes, your are right, some work must be done to work with a border. In dimension two this is much easier, because the border is always a circle and I guess everything is clear. Manifolds with corners is much more delicate subject. In dimension two this is essentially the same as homotopy cyclic 2-category. I can’t imagine what’s going on in higher dimensions.

3. You should have an (easier) analog in dimension 1: namely, for any curve $C$ and any $E1$-algebra $A$, associate something like $H_C(A)$. For $C$ the affine line you should get $A$ itself and for $C=S^1$ you should get Hochschild homology of $A$.

Yes, I met such construction somewhere. Essentially paper of T. Goodwillie “Cyclic homology, derivations, and the free loop space” is about it.

4. Do you have cochain version of this, and associated Deligne’s type conjecture ?

No. I doubt that it could be, for example Hochschild cohomology is not a functor.

Best,
Nikita.

3. Grégory said, on February 25, 2011 at 3:02 pm

Dear Nikita and Damien,

the chiral homology as mentioned in the previous comment is really an invariant of a (stably) framed manifold (i.e. the choice of a framing of Mx D^n where D^n is a disk). It does depend on the framing and not just of the homotopy type of the manifold. However the more commutative is your E_n-algebra, the more homotopical invariant it becomes.
More precisely, if you have two framing on a manifold M of dimension n (which induce equivalent framing on Mx D) and an E_{n+1}-algebra A, then the chiral homology of M with value in A is the same whatever the framing you choose. However, there may be E_n algebra for which they are different. That explains why, if you take E_\infty algebras, you get a theory that applies to all manifolds and doesn’t care about choices of stable framing; inducing a theory that makes sense for any CW-complex (and thus simplicial set).

In Lurie’s TFT paper (and DAG VI too I think), there are some paragraph in which it is explained that you can look to topological chiral homology for oriented manifolds if you look to E_n-algebras which are homotopy fixed point with respect to the natural SO(n) action on the framed little disk operad. I think that framed little disk algebras are naturally examples of such. Further, such things can be computed by using (homotopy) coends in between a coalgebra over the framed little disk operad and an algebra over it. You can find some notes about that (well in the parallelizable case) here at the end of these notes (which grew up from discussions with D. Ayala):
http://www.math.ku.dk/english/research/conferences/strings/strings.gregory.pdf/

This looks pretty close from your approach (I only found time to have a quick look to your posts so far so I may have missed many things) and I’m pretty sure what you got is a very nice description of this version of topological chiral homology.

Best,

Grégory Ginot

• nikitamarkarian said, on March 2, 2011 at 10:56 am

Dear Grégory,

thank you very much for a substantial comment. I will read the note with great interest.

Best,
Nikita.