Subject: What are and what might be `higher order Blakers-Massey theorems? From: "Ronnie Brown" Date: Sun, 23 Jul 2006 15:20:01 +0100 %blakers-massey.tex plain tex file \parskip=1ex \def\br{\hfil \break} \def\em{\it} {\bf What are and what might be `higher order Blakers-Massey theorems'?} Tom Goodwillie in his response of 18 July on `totalisation' mentions the term `higher Blakers-Massey theorems'. I would like to explain some thoughts on this. I think Tom intends to include in that term results from the 1950s of Toda and Barratt-Whitehead on connectivity of n-ads. Now the papers show that those workers were particularly interested also in the algebraic problem of determining the critical group, i.e. the first non vanishing group, in these theorems, to use this group in exact sequences. These theorems can also be seen as `higher order van Kampen theorems' (or GvKT's, for `generalised'). In dimension 1, we are interested in knowing more than that (roughly) the union of simply connected spaces with connected intersection is simply connected. We want the van Kampen Theorem, that the fundamental group of the union is the colimit of the fundamental groups of the parts. This precise, nonabelian colimit information is useful. Also, a colimit of trivial things is trivial, so simple connectivity is a special case. Blakers-Massey (and Barratt-Whitehead) determined in certain cases the critical triad (or n-ad) group as a (sum of) tensor product(s) of relative homotopy groups of the form $\pi_m(A_j,C)$, but only for $m>2$, so that these relative homotopy groups were all abelian. In the triad case, and when m may be 2, the nonabelian tensor product defined in [42,51] is required; this is explained there for the case of $\pi_3(X;A,B)$ and the general triad case is explained in [60] using the results of [49]. The n-ad case needs the notion of `universal crossed n-cube of groups' (a kind of `n-pushout') from [49]; this gives nonabelian information, but there are few explicit calculations apart from the case n=2 (there is one in [ES]). There are lots of results on nonabelian tensor products of groups, see the bibliography\br www.bangor.ac.uk/r.brown/nonabtens.html \br which now has 90 items, some of which are on analogues for structures other than groups, such as Lie algebras. If one wants algebraic, even nonabelian, information at the critical stage, rather than just connectivity information, then it would seem you need an algebra appropriate to the diagrams under consideration. For n-cubes of spaces, Loday's cat$^n$-groups are used in [42,51], and their equivalence with crossed n-cubes of groups is proved and used in [ES]. Evidence in favour of this algebra is the Generalised van Kampen theorem (GvKT) proved in [51], and its consequences, such as in [42,49,51,ES] and elsewhere. Even the bare functors\br (n-cubes of spaces) $\to$ (cat$^n$-groups) $\sim$ (crossed n-cubes of groups)\br contain, in the light of [ES], lots of homotopical interpretation, e.g. on generalised Whitehead products and their laws. [49] gives a general discussion of what is meant by the excision map for n-cubes of spaces, where the usual excision map of pairs is essentially the case n=2. (`Excision' is there regarded as a map of (n-1)-cubes of spaces derived from a given n-cube of spaces.) There should be *lots* more to dig out of this higher order algebraic structure. Cat$^n$-groups, and so crossed n-cubes of groups, model connected weak homotopy (n+1)-types (Loday, Porter, ...), so that in calculating such a crossed n-cube of groups one is calculating a homotopy type, often in the non simply connected case. Grothendieck exclaimed to me in 1986 when he realised that `n-fold groupoids model homotopy n-types': `That is absolutely beautiful!'. There are of course lots of problems, and few calculations except for crossed squares. I do not know if this algebraic information is used in or is relevant to the results of Tom's Calculus series of papers. It would be interesting if this strong algebraic information could be used in that sort of way. Are there possibilities for work in the nonabelian case analogous to that of Barratt-Whitehead and others on the second non vanishing group? Perhaps the GvKT for m-cubes with $m > n$ can yield helpful information on the n-cube case. Are there other possibilities of finding a more algebraic expression for, say, general position, or other subdivision arguments, in order to deduce algebraic results? The aim of the GvKT programme since 1967 has been to explore the possibility that this could lead to new higher dimensional and nonabelian local-to-global methods and results, with applications in and outside of homotopy theory. (numbered references are from my list) 42. (with J.-L. LODAY), ``Excision homotopique en basse dimension'', {\em C.R. Acad. Sci. Paris S\'er.} I 298 (1984) 353-356. 49. (with J.-L. LODAY), ``Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces'', {\em Proc. London Math. Soc.} (3) 54 (1987) 176-192. 51. (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'', {\em Topology} 26 (1987) 311-334. 60. ``Triadic Van Kampen theorems and Hurewicz theorems'', {\em Algebraic Topology, Proc. Int. Conf. March 1988}, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57. [ES] Ellis, G. J. and Steiner, R., {Higher-dimensional crossed modules and the homotopy groups of {$(n+1)$}-ads}, {J. Pure Appl. Algebra}, {46}, {1987}, {117--136}. (The publication of [49], [51], [ES] in the same year was due to delays in getting [51] accepted for publication.) Barratt, M. G.; Whitehead, J. H. C. The first nonvanishing group of an $(n+1)$-ad. Proc. London Math. Soc. (3) 6 (1956), 417--439. Barratt, M. G.; Whitehead, J. H. C. On the second non-vanishing homotopy groups of pairs and triads. Proc. London Math. Soc. (3) 5 (1955), 392--406. (Reviewer: J. Adem) G.J. Ellis, `Crossed squares and combinatorial homotopy', Math. Z. 214 (1993) 93-110. Ronnie Brown www.bangor.ac.uk/r.brown 23 July, 2006 \bye