From: rusin@math.niu.edu (Dave Rusin) Subject: Re: This week in the mathematics arXiv (28 Aug - 1 Sep) Date: 9 Sep 2000 15:00:04 -0500 Newsgroups: sci.math.research Summary: [missing] In article <8p3u1j$pcu$1@manifold.math.ucdavis.edu>, Greg Kuperberg wrote: >One article that caught my attention is "The >cohomology of the Sylow 2-subgroup of the Higman-Sims group", by >Adem, Carlson, Karagueuzian, and Milgram [math.AT/0008230]. If you >are wondering whatever happened to the study of finite simple groups, >one answer is that the algebraic topologists picked them up in order to >compute their cohomology. In topologists' terms, they want the cohomology >of a classifying space BG = K(G,1) of a finite group G, meaning a space >with fundamental group G whose universal cover is contractible. I don't >know why they want this information, but it's certainly cool. Greg suggested that I post some comments I made to him in response. "Why?" is often a difficult question for a mathematician to answer; "Because it's cool" is enough for me, but one could also justify the research on other grounds :-) . One perspective is that if it is ever of interest to compute the cohomology of a space, one must ask what ingredients are to be used in the computation. Well, cohomology only depends on homotopy type and in that category, a space is "determined by" its homotopy groups, so we ought to ask how the individual homotopy groups affect cohomology. At that point the cleanest way to begin is to say, OK, if you know a space has first homotopy group = G, say, and all other homotopy groups are trivial, what's its cohomology? That's what H^*(BG) is. So you can view these computations as an attempt to begin to figure out how cohomology depends on the basic data about a space. Alternatively you can take the group theorists' perspective, that the cohomology of BG tells us something about the group G. Just what information about G is reflected in the cohomology is not yet completely clear, although what information _has_ been found in the cohomology has proven useful to clarify certain group-theoretic constructions, such as the general description of how groups can be built from normal subgroups and their quotient groups. >Lesson #1 is that in studying the cohomology with coefficients in Z/p^n, >you really want the cohomology of the Sylow p-subgroup G_p of G and not >G itself. [...] This is why math.AT/0008230 is devoted to the Sylow >2-subgroup of Higman-Sims and not to Higman-Sims itself. For reasons >that I understand less well, the prime 2 is more interesting than other >primes in this game. The cohomology of G_p is in general a more complicated object than that of G itself; indeed, as Greg explained, the cohomology of G_p has to be at least as complicated as that of any group containing G_p as a Sylow subgroup. If indeed it is appropriate to study the cohomology of these Sylow p-subgroups, the case p=2 is of course a reasonable place to begin, but it also tends to draw special attention for a few reasons. Not the least of these is the fact that +1=-1 mod 2, so that those of us prone to sign errors are safe :-). It's also true that there can be families of p-groups (one for each prime p) having descriptions which are equally simple for all primes p, and yet whose mod-p cohomology rings require a description whose complexity grows with p. But these details mask the general pattern of cohomology rings, which is the same for all primes p. If G is a finite group and k is a field of characteristic p, then H^*(BG, k) is (almost) a commutative Noetherian graded ring -- just the sort of thing one encounters in algebraic geometry. When G is abelian, the ring is essentially a polynomial ring over k; more generally, the prime ideals in the cohomology ring correspond to the abelian subgroups of G. In this sort of analysis, p=2 is no better than any other prime. But I've been merrily avoiding precision and quite a few necessary qualifying conditions; if detailed computations are required, they tend to be significantly messier at odd primes, and so papers preferring p=2 are common. It should also be noted that group theory itself draws attention to p=2. If one begins with the premise that finite simple groups are worthy of study, then it will be the Sylow 2-subgroups which will require the most attention, since they are usually much larger than the other Sylow p-subgroups. Also the groups of order 2^n are catalogued for small n, and starting with n=3 are distinctly different from the groups of order p^n. As is well known, 2 certainly is an odd prime. dave ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: The Monster and the Leech Date: 4 Oct 2000 22:46:04 GMT Newsgroups: sci.physics.research In article <8rfkji$43u$1@nnrp1.deja.com>, wrote: >In article <8rdkhe$ag1$1@Urvile.MSUS.EDU>, > baez@math.galaxy.edu (John Baez) wrote: >> Extensions of a group G by an abelian group A are >> classified by the 2nd cohomology group H^2(G,A). (By the way, there's a chance that I really meant "central extensions" here - I find this point incredibly hard to remember for some idiotic reason.) >Sorry for the stupid question, but is group cohomology defined? :) Sure! You think I'm trying to mess with your head? Well, you're right, I am... but only by legitimate means. You can read about group cohomology in any book on homological algebra, but the basic idea is pitifully simple. Suppose I have any group G. Then I can cook up a space X whose fundamental group is G and has trivial higher homotopy groups. This space is actually unique up to homotopy equivalence. It's basically a sneaky way of encoding the group G in a space! Now if you give me an abelian group A, I can take the homology groups of X with coefficients in A, and call them the homology groups of G: H_n(G,A) := H_n(X,A) Ditto for cohomology. The space X is usually called the "Eilenberg-MacLane space" of G and denoted K(G,1). I've talked about these a bunch on This Week's Finds, and explained how you explicitly construct them. They play a basic role in algebraic topology. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Group cohomology for physicists Date: 6 Oct 2000 00:51:38 GMT Newsgroups: sci.physics.research In article <8riq91$prd$1@nnrp1.deja.com>, wrote: >In article <8rgbvc$16b$1@Urvile.MSUS.EDU>, > baez@galaxy.ucr.edu (John Baez) wrote: >> You can read about group cohomology in any book on homological >> algebra, but the basic idea is pitifully simple. Suppose I have any >> group G. Then I can cook up a space X whose fundamental group is G >> and has trivial higher homotopy groups. >Sounds a lot like a resolution. I doubt there's any relation, though... Yes, there's a relation! I'm really just describing the usual trick for getting a free resolution of G as a Z-module; however, I'm describing it using in the language of topology, rather than the language of algebra. That's because personally, I found homological algebra more comprehensible after I translated it into topology. I never understood the deep significance of group cohomology until I realized one could break it down into two independent steps: 1) To understand a group G, it's good to turn it into a space K(G,1): the space having fundamental group is G and vanishing higher homotopy groups. 2) To understand a space X, it's good to calculate its cohomology groups. If we do both steps we get the cohomology of the group G. I have a lot of intuition for each step separately, which gives me intuition for group cohomology. At least on a good day. I haven't been thinking about it much since James Dolan went to Australia, so I can tell now that my intuition is slipping. >> This space is actually unique up to homotopy equivalence. >Well, obviously. The homotopy groups define the space up to homotopy >invariance. No they don't!!! If life were so simple, the homotopy theorists could all go home now and take a nap. In addition to the homotopy groups you need "Postnikov data", which says how the homotopy groups talk to each other. However, when there is only one nonvanishing homotopy group, the Postnikov data are trivial, so the homotopy groups suffice. >> Now if you give me an abelian group A, I can take the homology groups >> of X with coefficients in A, and call them the homology groups of G: >> >> H_n(G,A) := H_n(X,A) >> >> Ditto for cohomology. >> >> The space X is usually called the "Eilenberg-MacLane space" of >> G and denoted K(G,1). > >Ah, that's what it is! Yeah, now I can see why, when you said it. The >loop space of each Eilenberg-MacLane space is the previous one. Hmm, I >heard the claim all of those homology groups vanish except for the >first one for finitely generated Abelian groups, or something of the >sort. Why would that be true? It's not. Consider K(Z/2,1). This is RP^infinity, which has nonvanishing homology groups in all dimensions. ============================================================================== From: squark@my-deja.com Subject: Re: Group cohomology for physicists Date: 9 Oct 2000 23:28:34 GMT Newsgroups: sci.physics.research In article <8rivpv$g6l$1@mortar.ucr.edu>, baez@galaxy.ucr.edu (John Baez) wrote: > In article <8riq91$prd$1@nnrp1.deja.com>, wrote: > > Well, obviously. The homotopy groups define the space up to homotopy > > invariance. > > No they don't!!! If life were so simple, the homotopy theorists could > all go home now and take a nap. Oops... Yeah, now I talked to a person I know, and realized I was thinking of the Whitehead theorem which requires an actual map between the spaces establishing an isomorphism between the homotopy groups. > In addition to the homotopy groups you need "Postnikov data", which > says how the homotopy groups talk to each other. What's the Postnikov data? Sounds like something cool. :) > > Hmm, I heard the claim all of those homology groups vanish except > > for the first one for finitely generated Abelian groups, or > > something of the sort. Why would that be true? > > It's not. Consider K(Z/2,1). This is RP^infinity, which has > nonvanishing homology groups in all dimensions. Hmm, right. I wonder what was that all about, then. Well, whatever... Best regards, squark. P.S. Almost forgot: ain't there a purely group theoretical definition too? I think I read something about it once. Best regards, squark. Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Algebraic topology for physicists Date: 10 Oct 2000 17:52:56 GMT Newsgroups: sci.physics.research In article <8rn3hc$2ar$1@nnrp1.deja.com>, wrote: >Oops... Yeah, now I talked to a person I know, and realized I was >thinking of the Whitehead theorem which requires an actual map between >the spaces establishing an isomorphism between the homotopy groups. Right, it's sorta sneaky: a map between spaces that induces an isomorphism on homotopy groups must be a homotopy equivalence, but it ain't true that spaces with the same homotopy groups are homotopy equivalent. (Here by "spaces" I mean reasonably nice spaces, like CW complexes.) >> In addition to the homotopy groups you need "Postnikov data", which >> says how the homotopy groups talk to each other. > >What's the Postnikov data? Sounds like something cool. :) It's VERY cool. In the game called "Postnikov towers", we describe any connected space X as follows. First we work out the homotopy groups pi_n(X). Then for each n we cook up a space K(pi_n(X),n), called an Eilenberg-MacLane space, which has vanishing homotopy groups except for the nth one, which equals pi_n(X). Then the trick is to put these spaces together to form a space homotopy equivalent to X. We could just take the product of all of them, and get a space with the same homotopy groups as X. But that would not be right, because it would be forgetting how the homotopy groups of X interact. So what we do instead is start with X_1 = K(pi_1(X),1). Then we form a space X_2 which is the total space of a bundle whose base is X_1 and whose fiber is K(pi_2(X),2). Then we form a space X_3 which is the total space of a bundle whose base is X_2 and whose fiber is K(pi_3(X),3). Then we form a space X_4 which is the total space of a bundle whose base is X_3 and whose fiber is K(pi_4(X),4). ... and we go on forever, and when we're done we have a space that's homotopy equivalent to X! If all these bundles were trivial, X would just be a big product of Eilenberg-MacLane spaces --- a very dull sort of space where the homotopy groups don't talk to each other at all. But usually these bundles are not trivial. So to describe these bundles we need extra information: POSTNIKOV DATA! It's like a layer cake: the Eilenberg-MacLane spaces are the layers, but we also need to say how each layer is stuck on top of the preceding layers. Bundle upon bundle upon bundle... an infinitely tall layer cake. So: together with the homotopy groups, the Postnikov data says what the space X is really like, up to homotopy equivalence. >Almost forgot: ain't there a purely group theoretical definition too? I >think I read something about it once. Of group cohomology, you mean? Sure! You said what it was, too: you take your group G, you form a free resolution, you hom that resolution into A, then you take the cohomology of the resulting cochain complex to get the cohomology groups H^n(G,A). This yoga can be found in any book on homological algebra - just go to your local bookstore. This kind of stuff leads to some good ways to actually *compute* the cohomology of groups. In fact, these days you can get computer programs that do this for you - just ask at your local computer software store. (Yeah, right.) The only problem with this purely algebraic approach to group cohomology is that, taken by itself, it doesn't offer enough intuition about the deep inner meaning of it all. Throwing some topology in the mix helps. For example, it lets you actually visualize what's going on.