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Stream: theory: algebraic topology

Topic: Crossed module cohomology of torus

Alonso Perez-Lona (Feb 05 2023 at 22:52):

Is it known how to compute cohomology groups of torus valued in a crossed module? I am particularly interested in computing H^2(T^2,(H->G)) where (H->G) is a crossed module, but I am not really sure how to proceed. I am aware that there exist certain homomorphisms with usual (simplicial) cohomology if (H->G) described an injective or surjective map, but a. I do not think it applies to degree-2, and b. I would like to consider general crossed modules.

David Michael Roberts (Feb 06 2023 at 00:03):

You could possibly look at the exact sequences one can get for crossed-module-valued cohomology. I presume here you mean the torus as a space, and not as a group in its own right, and hence the analogue of group cohomology (see eg https://arxiv.org/abs/0902.0161).

David Michael Roberts (Feb 06 2023 at 00:07):

If you consider $t\colon H\to G$, then there's the ses $(ker(t) \to 1) \to (H\to G) \to (im(t)\to G)$ of crossed modules, where the rightmost one is weakly equivalent to $1\to coker(t)$. Then you can calculate the cohomology valued in the left and right crossed modules.

David Michael Roberts (Feb 06 2023 at 00:10):

This is because you can consider the fibre sequence of classifying spaces corresponding to the ses of crossed modules, and so mapping out of the torus can be written in terms of mapping from circle to the loop space, and everything is really up to homotopy, so it's really calculating $\pi_1's$ of a bunch of free loop spaces of classifying spaces.

John Baez (Feb 06 2023 at 00:37):

It sounds like you're confirming something I suspected, @David Michael Roberts, namely that for this particular sort of cohomology we might as well just figure it out for the circle and then do some sort of "tensor product" trick to figure it out for the torus.

David Michael Roberts (Feb 06 2023 at 00:58):

@John Baez the cohomology of the circle (as a space) valued in $H\to G$ is $\pi_0(L B(G\to H))$, where $L$ means free loop space, and $B$ here means the geometric realisation of the nerve of the one-object 2-groupoid associated to the monoidal groupoid associated to the crossed module (i.e. the classifying space in the sense that you and Danny wrote about). I was being a little careful about getting to the torus directly from this, but since @Alonso Perez-Lona mentioned being able to calculate special cases, reducing it to those, and thence to more standard calculations is definitely do-able.

David Michael Roberts (Feb 06 2023 at 01:05):

Now I go looking, I think one can answer a lot of the question as I reframed it using this paper of Ronnie Brown https://arxiv.org/abs/1003.5617 — he calculates $\Pi_2(L B(G\to H))$! Theorem 1.1 gives a description of $\pi_0(B L(G\to H))$, and also a way to extract its $\pi_1$, which is closely related to what one needs for the original question (which needs the $[S^1,-]$ of the free loop space of the classifying space, if one is jumping straight to the end result, rather than via the l.e.s. of cohomology groups associated to the s.e.s. of crossed modules)

Alonso Perez-Lona (Feb 06 2023 at 01:10):

David Michael Roberts said:

Now I go looking, I think one can answer a lot of the question as I reframed it using this paper of Ronnie Brown https://arxiv.org/abs/1003.5617 — he calculates $\Pi_2(L B(G\to H))$! Theorem 1.1 gives a description of $\pi_0(B L(G\to H))$, and also a way to extract its $\pi_1$, which is closely related to what one needs for the original question (which needs the $[S^1,-]$ of the free loop space of the classifying space, if one is jumping straight to the end result, rather than via the l.e.s. of cohomology groups associated to the s.e.s. of crossed modules)

Looks good, thanks!

Alonso Perez-Lona (Feb 07 2023 at 15:25):

David Michael Roberts said:

Now I go looking, I think one can answer a lot of the question as I reframed it using this paper of Ronnie Brown https://arxiv.org/abs/1003.5617 — he calculates $\Pi_2(L B(G\to H))$! Theorem 1.1 gives a description of $\pi_0(B L(G\to H))$, and also a way to extract its $\pi_1$, which is closely related to what one needs for the original question (which needs the $[S^1,-]$ of the free loop space of the classifying space, if one is jumping straight to the end result, rather than via the l.e.s. of cohomology groups associated to the s.e.s. of crossed modules)

to compute $H^2(T^2,H\to G)$ one needs $[S^1\times S^1,B^2(H\to G)]=[S^1,LB^2(H\to G)]$, how is this related to the $\pi_1(LB(H\to G)$ you mentioned?

John Baez (Feb 07 2023 at 19:55):

That equation doesn't look right to me, but @David Michael Roberts understands this stuff better.

David Michael Roberts (Feb 08 2023 at 00:22):

Hi @Alonso Perez-Lona you can't apply B twice, without knowing the crossed module corresponds to a braided 2-group. What definition of cohomology are you using? The indexing might be off by 1 in the notation.

Alonso Perez-Lona (Feb 08 2023 at 01:09):

David Michael Roberts said:

Hi Alonso Perez-Lona you can't apply B twice, without knowing the crossed module corresponds to a braided 2-group. What definition of cohomology are you using? The indexing might be off by 1 in the notation.

Hi @David Michael Roberts , I double-checked and indeed, what I really need to compute is $H^{-1}(T^2,H\to G)$ and $H^0(T^2,H\to G)$. I am aware how these reduce to usual cohomology groups in the special cases where $H\to G$ is either injective or surjective, but I would be interested in more general cases where the map is neither injective nor surjective.

David Michael Roberts (Feb 08 2023 at 01:41):

@Alonso Perez-Lona now I'm really confused! What's your reference for these cohomology groups? What does the negative index on the cohomology group mean?

Alonso Perez-Lona (Feb 08 2023 at 01:47):

David Michael Roberts said:

Alonso Perez-Lona now I'm really confused! What's your reference for these cohomology groups? What does the negative index on the cohomology group mean?

@David Michael Roberts I am looking at this paper, the results I mentioned for inj/surj are in Examples 2.2. At the top of p.4 they give a characterization of $H^0(E,H\to G)$, but I was wondering if there is a more straightforward way to compute it.

David Michael Roberts (Feb 08 2023 at 02:36):

@Alonso Perez-Lona Ah, ok. The "cohomology" $H^{-1}(X,H\to G)$ is really just a fancy way of thinking about $[X,\Omega B(H\to G))$. Unfortunately, while the paper of Breen that the paper cites is a better source to use, it's also less easily available (and in French ymmv).

David Michael Roberts (Feb 08 2023 at 02:47):

Giraud's book, also cited, is old and at that point he didn't really know what he was aiming for. The role of functoriality in the coefficients wasn't really understood. Better is the thesis of Debremaeker https://arxiv.org/abs/1702.02128

Alonso Perez-Lona (Feb 08 2023 at 03:21):

@David Michael Roberts thank you very much for your help!

David Michael Roberts (Feb 08 2023 at 03:33):

@Alonso Perez-Lona No problems! Ultimately, the "long" exact sequence is how you calculate these things. Getting that l.e.s. is the most important contribution of Debremaeker's work, which comes from the realisation that crossed modules (aka 2-groups) are the correct coefficient objects. These days we can get a l.e.s. from a s.e.s. of crossed modules and butterflies (à la Noohi), which is a little more flexible, but I don't think you need this.