Subject: What are and what might be `higher order Blakers-Massey theorems?
From: "Ronnie Brown"
Date: Sun, 23 Jul 2006 15:20:01 +0100
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{\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
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