Subject: n-ad connectivity and critical groups From: "Ronnie Brown" Date: Wed, 26 Jul 2006 13:18:41 +0100 Bill and all, > if this is really true, please write up the details. Why not > just post a sketch here. I'd be shocked if it were true. That is interesting - the Brown-Loday paper was published in Topology almost 20 years ago! We should also mention the proof of triad connectivity in tom Dieck-Kamps-Puppe SLNM volume on `Homotopietheorie'. Jean-Louis and I envisaged first a geometric proof of our main result, analogous to that of the methods of Brown-Higgins, but we could not see how to write it down! So the published proof uses lots of simplicial machinery, which I cannot summarise here. The algebra of cat^n-groups is crucial to the inductive proof. The `details' I had in mind in my email were only those of going between n-ads and n-cubes, which you can all complete. Let's sketch the case n=3. If X is the union of open A,B,C then the inclusions of A,B,C into X define a 3-cube of spaces K whose vertices are X,A,B,C and intersections of A,B,C. This defines the 3-cube K as a pushout 3-cube of various somewhat `degenerate' 3-cubes involving A,B,C, some more degenerate than others. A form of the theorem is: Connectivity: if all these `degenerate' 3-cubes are connected, then so also is the 3-cube K, AND Isomorphism: the fundamental cat3-group of K is the 3-pushout (in cat3-groups) of the cat3-groups of all the various degenerate 3-cubes. The connectivity conclusion seems more general than the Barratt-Whitehead conditions, which, as I stated earlier, involve only relative groups (judging from the review), whereas the above conclusion gives a 4-ad connectivity deduced from r-ad connectivity for all r< 4. The isomorphism part gives COMPLETE information (in principle) on the (in general) NONABELIAN 4-ad group \pi_4 (X; A,B,C), in terms of r-ad groups for r < 4. It can be thought of as a nonabelian tensor product! To dig this out explicitly, you would have to get used to computing with crossed 3-cubes of groups! Lots of work here for young people! There might be some surprising results??!! There is more fun and games to be had by using the cubical jiggery-pokery and excision techniques indicated in my earlier mailings. This would lead for example to notions of `induced and so of free crossed 3-cubes of groups', taking a step further Graham Ellis' work referred to earlier. There are also non trivial connectivity deductions, which may not have been noticed earlier. Let me go back to the start in 1966 or so. I convinced myself in writing my book (now republished and available!!!) that all that I knew of in 1-dimensional homotopy was better expressed using groupoids rather than groups. So could higher homotopy theory be rewritten replacing the word `group' by `groupoid', and if one did that, would the result be more appealing? or yield new results? Or was that an impossible idea? Note the aesthetic consideration. A major step was in 1974, finding with Philip Higgins the homotopy double groupoid of a pair of spaces, and deducing a van Kampen type theorem for Henry Whitehead's crossed module of a pair. (published in Proc LMS in 1978, after early vicissitudes from the then editor, for undisclosed reasons). When I visited Strasbourg in 1981 and explained the work with Philip in all dimensions, Jean-Louis was convinced that analogues worked for cat^n-groups, and we were sure after further work that this would give a useful triadic Hurewicz theorem. Which it did, so confirming an algebraic conjecture of Jean-Louis on the first non vanishing homotopy group of the cofibre of a connected square of spaces. It is interesting that the simplicial methods manage to replace or circumvent possible detailed geometric subdivision and general position arguments. Perhaps a careful examination of the proof would show further possibilities, apart from giving a renewed check. Hope all that is not too shocking! Ronnie `Topology and groupoids' www.bangor.ac.uk/r.brown/topgpds.html Roll up, roll up! 538 pages at only $32 . ____________________________________________________________________________________ Subject: Re: 2 postings From: Tom Goodwillie Date: Wed, 26 Jul 2006 14:27:17 -0400 I've been meaning to weigh in here for a while. I guess it's time. I have several things to say. Besides adding my two cents to the discussion of who said what when, I know that I am also going to end up getting on my soapbox about basepoints and the empty set. Also, I want to suggest that the "incredible diagram chase" that Bill Richter referred to is not as mysterious as I made it appear in the paper. Historically, both these connectivity results for n-cubes and the corresponding computations of critical groups were done first for 1-connected spaces and 2-connected maps. When I reproved the connectivity results, I was able to drop these hypotheses because I was using a direct method which did not involve homology groups. On the other hand, for computations of critical groups things become much more subtle when the maps are not 2-connected. (I guess you need those nonabelian tensor products and cat^n groups and so on, which I never studied. I probably should have studied them. Should study them.) I have made heavy use of these connectivity statements for cubes over the years -- they are the standard tools for proving that a given functor is "analytic" -- but I have never specifically needed the computations of critical groups. Knowing the latter in the abelian case could be said to be the same as knowing the first nontrivial group of the nth derivative of the identity, and therefore one of the many ways of getting the critical group computations would be to use Taylor towers; but this would be overkill. I feel like giving some details, so here we go. Let me start by clarifying what we are talking about, and fixing terminology in the way that I am used to: The old result that I am in the habit of calling the Blakers-Massey theorem, and that is also called the triad connectivity theorem, says that if a commutative square diagram X ---> X_1 | | v v X_2 ---> X_{12} is a pushout up to homotopy and if the maps X-->X_i are highly connected, then the square is highly connected in the sense of being nearly a pullback up to homotopy. Of course, the relationship between squares and triads is that a triad (A;B,C) determines a square with X_{12}=A, X_1=B, X_2=C, X=B\cap C. Every square is weakly equivalent to a square that is cofibrant in the sense that X is (Quillen) cofibrant and the maps X-->X_i and colim( X_1 <--- X ---> X_2 ) --> X_{12} are cofibrations. Or we might as well say that any square is equivalent to the square determined by some CW triad. (Equivalence of squares means a strict map of diagrams that is "pointwise" an equivalence.) By homotopy pushout square I mean that the canonical map hocolim( X_1 <--- X ---> X_2 ) --> X_{12} is a weak equivalence, or equivalently that in a CW triad replacement the map B\cup C --> A is a weak equivalence, or equivalently that the CW triad replacement can be chosen so that A=B\cup C. (Maybe we call it a complete triad in that case.) The original Blakers-Massey triad connectivity statement said something like this: If (A;B,C) is a complete CW triad of simply-connected spaces, and if the homotopy groups of the pairs (A,B) and (A,C) vanish in degrees

X_1 | | v v X_2 ---> X_{12} is a homotopy pushout and the map X-->X_i is k_i-connected, then the canonical map X --> holim ( X_1 --> X_{12} <-- X_2 ) is k-connected where k = k_1 + k_2 - 1 . Notice how have I changed the statement: (1) I made it a square rather than a triad and I wrote k_1 and k_2 for p-1 and q-1. (2) I no longer assume that the spaces are simply connected or that the k_i are positive. (3) I assume connectivity of the maps X-->X_1 and X-->X_2 rather than the maps X_2-->X_{12} and X_1-->X_{12}. (This would have made no difference if it were not for (2), but as it is it is a necessary strengthening of the hypothesis.) (4) I do not explicitly mention triad homotopy groups. They appear here implicitly as follows: the \pi_j of the triad is \pi_{j-1} of the homotopy fiber (or rather of some homotopy fiber) of the map X --> holim ( X_1 --> X_{12} <-- X_2 ) The point about (4) is that I don't know a definition of triad homotopy groups under which their vanishing will tell us that the map X --> holim ( X_1 --> X_{12} <-- X_2 ) induces a surjection of \pi_0 . So now we come to the soapbox talk. SOAPBOX If you have to deal with spaces that might not be path-connected, you have to be careful how you define k-connectedness of a pair, or of a map. When I call a space k-connected, I mean that, for every j with -1\leq j\leq k, every map from S^j to X extends to D^{j+1}. So for example the empty space is not k-connected if k \geq -1. When I say that a pair, or more generally a map X-->Y, is k-connected, I mean that for every point in Y the homotopy fiber (a.k.a. mapping fiber, a.k.a. holim ( * <-- X --> Y ) ) is a (k-1)-connected space. Here is where you have to be careful: If the spaces are path-connected (= 0-connected), then k-connected map means vanishing of \pi_j(pair) for 1\leq j\leq k. But in general k-connected map means vanishing of those (for every base point in X) plus surjectivity of \pi_0. Yes, you can get around this by making an (in my view artificial) definition of relative \pi_0. You can say relative \pi_0(X-->Y) means the obvious quotient of \pi_0(Y). But this doesn't do the job if X is empty; you should instead say \pi_0(X-->Y)=colim(*<--\pi_0(X)-->\pi_0(Y)). And you'll still probably get it wrong when you say "k-connected map means vanishing of relative \pi_j for 0\leq j\leq k", because for j>0 you have to say "for all basepoints in X" while for j=0 you had better not say that (again, what if X is empty?) All right, I know I can get a little nutty about the empty set. But there are reasons. I don't believe that the 0-connectedness of the map X --> holim ( X_1 --> X_{12} <-- X_2 ) for a square can reasonably be expressed as the vanishing of some kind of "triad \pi_{-1}". And, while many homotopy theorists may not have much occasion to think about spaces that are not path-connected, that has not been my fate; in some of my favorite applications of these definitions and results, I want to prove that a space X that is the "first" space in a cube is nonempty by using connectivity information about the cube and about the other spaces. ENDSOAPBOX Anyway, the Blakers-Massey proof of triad connectivity used homology, but there is a more direct argument using general position that yields the result as stated above. I can't remember where I saw it first, but as Bill Richter mentioned some such proof is given in Brayton Gray's book. Terminology: I like to call a square "k-cartesian" if that map X --> holim ( X_1 --> X_{12} <-- X_2 ) is k-connected. I also like to call it "k-cocartesian" if the analogous map hocolim ( X_1 <-- X --> X_2 ) ---> X _{12} is k-connected. So infty-cocartesian means homotopy pushout and infty-cartesian means homotopy pullback. [Both of these notions -- k-cartesian square and k-cocartesian square -- seem to have an equal claim to the name "k-connected square". But even when k=infty they are two different notions (that's the whole point!), so we just don't use that name.] Note that the theorem discussed above says that infty-cocartesian implies (k_1 + k_2 -1)-cartesian if the maps in the square are k_i-connected. There is an easier "dual" result to the effect that infty-cartesian implies (k_1 + k_2 +1)-cocartesian if the maps in the square are k_i-connected. (The proof boils down to the fact that the join of a p-connected and a q-connected space is (p+q+2)-connected. Note that this is valid even when p and/or q is not positive.) Now, briefly, about the generalizations from squares (triads) to n-cubes ((n+1)-ads). Let's call a CW (n+1)-ad (A;A_1,...,A_n) "complete" if A=A_i\cup A_j for all i and j. These (n+1)-ads correspond to n-cubes which are "strongly infty-cocartesian" in the sense that every square face of the cube is infty-cocartesian. Think of these as the ones you get by taking n cofibrations X-->X_i sharing a common domain and making a cube by pushout from there. Let's call an n-cube k-cartesian if the canonical map from the first space to the holim of the others is k-connected, and call it k-cocartesian if the canonical map to the last space from the hocolim of the others is k-connected. (By the way "strongly infty-cocartesian" implies that every >2 dimensional face of the cube, in particular the whole n-cube if n>1, is infty-cocartesian.) Here is the result that I am accustomed to calling the Higher Blakers-Massey Theorem, and that Bill Richter called the "n-ad Connectivity Theorem" in his recent post: HBM: If a strongly infty-cocartesian n-cube is such that for each i from 1 to n the map X-->X_i from the first space to its ith neighbor is k_i-connected, then the cube is k-cartesian where k = k_1 + ... + k_n - n + 1 . My personal history with this result is as follows. When I first became interested in multirelative connectivity questions (in the late 70's, as a student, thinking about spaces of smooth embeddings) and was looking for tool like HBM above, somebody steered me to the Barratt-Whitehead paper "The first non-vanishing group of an (n+1)-ad". B-W stated both connectivity and computation of critical group. It stayed away from \pi_1 (and \pi_0) difficulties in the same way as the original Blakers-Massey result. I suppose it used homology, but I have not looked at the paper in a long time. I had trouble reading that paper. (I have since heard others say that they couldn't figure it out either.) So I reproved the connectivity result myself, in the form stated above as HBM. My method was the same general position method mentioned above, plus induction on n, plus some complicated bookkeeping. Later, when I finally published my "Calculus" papers, I included that proof (in section 2 of Calculus 2). In the meantime I had also learned about yet another proof, due to Ellis and Steiner, using the cat^n stuff, so of course I referenced that. There is also the dual result: HBM*: If a strongly infty-cartesian n-cube is such that for each i the map to the last space from its ith neighbor is k_i-connected, then the cube is k-cocartesian where k = k_1 + ... + k_n + n - 1 . This is also proved in Calculus 2, but the proof does not require induction on n (or complicated bookkeeping). What I meant by Higher Blakers-Massey theorems in my post about cosimplicial spaces was a family of results that all appear in section 2 of my Calculus 2 paper: - The theorem called HBM above. - The dual result called HBM* above. - A more flexible and elaborate variant of HBM, that applies to cubes that are not necessarily strongly infty-cocartesian. - A similar variant of HBM*. Let me say a few words about the latter two. It follows easily from the triad connectivity theorem (i.e. the case n=2 of HBM) that if you have a square in which X-->X_1 is k_1-connected, X-->X_2 is k_2-connected, and the square is k_{12}-cocartesian, then the square is also k-cartesian where k=min(k_1 + k_2 -1, k_{12} - 1). The basic idea is (wlog the square is cofibrant and then) look at the two squares X ---> X_1 = X_1 | | | v v v X_2 --> colim(X_1<-X->X_2) --> X The left one is (k_1 + k_2 -1)-cartesian by triad connectivity; the right one is (k_{12] - 1)-cartesian by the other hypothesis. That's the more flexible version of HBM when n=2. For n=3 the statement is that if you have a 3-cube and you know how cocartesian it is, as well as how cocartesian the three square "faces" X ---> X_i | | v v X_j ---> X_{ij} are, and also how connected the three maps X-->X_i are, then you can write down a lower bound for how cartesian the whole 3-cube is. And so on. This is really an easy corollary of HBM. The complicated "diagram chase" that Richter referred to was just my rather opaque way of writing down the deduction of the corollary from the HBM theorem. But there's this other wrinkle: when I proved HBM I had to simultaneously prove the corollary, because I needed the corollary for n-1 to prove the theorem for n. The point is this: The general-position argument leads from an n-cube to an (n-1)-cube. For example, given a square X ---> X_1 | | v v X_2 ---> X_{12} = colim(X_1<-X->X_2) where X->X_i are high-dimensional CW pairs, it tells you how connected the induced map hf(X-->X_1) ---> hf(X_2-->X_{12}) of homotopy fibers is, which is really what you want to know when you are asking how cartesian the square is. When I generalized this to a strongly infty-cocartesian n-cube, I was led to an (n-1)-cube of homotopy fibers that was not strongly infty-cocartesian, but for which I could read off the sort of information required by the corollary. By the way, my interest in the corollary (the "more flexible and elaborate version") of HBM goes beyond the fact that I found myself needing it to prove HBM; it's really an indispensable tool for dealing with examples. I don't have anything to add to the discussion of calculation of critical group in the general (nonabelian) case. In the abelian case, I mean in the case of a strongly infty-cocartesian n-cube (<--> complete (n+1)-ad) whose numbers k_i are all at least 2, you can easily use HBM for n+1 to reduce the computation of the critical group to a special case where the answer is given by, for example, a small part of the Hilton-Milnor Theorem. This is all related to my Taylor tower idea, if you like, and from one point of view the "critical group" information in the abelian case is the tip of an iceberg called the nth derivative of the identity. Tom Goodwillie ________________________________________________________________ Subject: Re: n-ad connectivity and critical groups From: Bill Richter Date: Wed, 26 Jul 2006 13:32:50 -0500 Ronnie, we had a long private discussion in 1992 when you failed to convince me you had a proof of the n-ad conn thm. This was after the Brown-Loday paper. I even remember you backing off the claim. Maybe I misunderstood you, and maybe you see something new now. But this is an area in which great mathematicians like Barratt & Toda have come to grief. So care is really needed. Thanks for writing something about n = 3, but that's the case that's really well understood, right, the triad conn thm? The n > 3 case is a lot harder. I figured out a real nice Barratt-Ganea theory proof of the EHP sequence for n=3, and that's a big part of n=4, and I could not generalize it. So if you can really do this, you beat me, and you put yourself up there with Mike & Tom, and it would be a really good hook for work: Lots of work here for young people! There might be some surprising results??!! Sure! But folks want to know you have a powerful tool before they spend a lot of time learning how to use it.