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The Blakers-Massey theorem in the homotopy theory of pointed topological spaces is concerned with algebraically describing the first obstruction to excision for relative homotopy groups. There is also a weaker version just describing vanishing conditions which should be called the Blakers-Massey connectivity theorem.
This obstruction is measured by triad homotopy groups $\pi_m(X:A,B)$ for a pointed space $X$ with two subspaces $A,B$ each containing the base point. Here the group structure is defined for $m \geq 3$ and is abelian for $m \geq 4$. There is an exact sequence
where $\epsilon$ is the excision map. The main result of Blakers and Massey is as follows:
Suppose the triad X $=(X;A,B)$ is such that: (i) the interiors of $A,B$ cover $X$; (ii) that $A,B$ and $C=A \cap B$ are connected; (iii) that $C$ is simply connected; (iv) and that $(A,C)$ is $(m-1)$-connected and $(B,C)$ is $(n-1)$-connected, $m,n \geq 3$. Then X$=(X;A,B)$ is $(m+n-2)$-connected and if $C$ is simply connected then the morphism given by the generalised Whitehead product
is an isomorphism.
(Blakers-Massey 51, tomDieck 08, theorem 6.4.1).
A more intrinsic statement in the language of homotopy theory of the connectivity part of the theorem is that for $f_1$ and $f_2$ two maps out of the same domain which are $n_1$-connective and $n_2$-connective, respectively, then the canonical map from that domain into the homotopy pullback of their homotopy pushout
is $(n_1 + n_2 - 1)$-connective.
For the special case that $Y_1 \simeq Y_2 \simeq \ast$ are point contractible, the Blakers-Massey theorem reduces to the Freudenthal suspension theorem.
Since the tensor product is zero if one of its factors is zero, this result also gives criteria for the excision morphism $\epsilon$ to be an isomorphism in a certain range of dimensions. For this reason the excision consequences of that sequence are also called the excision theorem of Blakers and Massey and have been given quite separate proofs for example in (Hatcher), and in (tom Dieck). The first non zero triad homotopy group is also called the critical group. Note that in algebraic topology one wants algebraic results, not just connectivity results.
A natural question is what happens if the conditions that $m,n \geq 3$ and $C$ simply connected are weakened. For example in the case $m=n=2$ we have the additional structure that the morphisms $\pi_2(A,C) \to \pi_1(C), \pi_2(B,C) \to \pi_1(C)$ are crossed modules, and so the required relative homotopy groups are in general nonabelian. If $m \geq 3 ,n \geq 3$ then $\pi_m(A,C), \pi_n(B,C)$ are still $\pi_1(C)$-modules.
The extension to the non simply connected case was given by Brown and Loday; one simply replaces the usual tensor product by the nonabelian tensor product of groups which act on each other and on themselves by conjugation. This result is a special case of a Seifert-van Kampen Theorem for $n$-cubes of spaces. Notice that the assumption (i) of the theorem is reminiscent of such a type of theorem. The useful fact is that one gets such a theorem for a certain kind of structured space which allows for the development of algebraic structures which have structures in a range of dimensions.
Thus one of the intuitions is that the Blakers-Massey Theorem, and hence also the FST, is of the Seifert-van Kampen type, since we are assuming that $X$ is the union of the interiors of $A,B$.
The Blakers-Massey connectivity theorem in the form of remark 1 holds in every (∞,1)-topos of (∞,1)-sheaves.
This is shown in (Rezk 10, prop. 8.16) with reference to (∞,1)-sites. An intrinsic proof in homotopy type theory is announced in (HoTTBook, theorem 8.10.2, Lumsdaine-Finster-Licata 13). The fully formal computer-checked version of this proof appears as HoTT-Agda code in (Favonia 14).
This translates to an internal language proof of Blakers-Massey valid in all (∞,1)-toposes (including elementary (∞,1)-toposes). Unwinding of the fully formal HoTT proof to ordinary mathematical language is, for the special case of the Freudenthal suspension theorem, in (Rezk 14).
There are higher analogs of the BM-theorem with (pushout) squares replaced by higher dimensional cubes. The higher BM-theorem (Goodwillie 91) says equivalently that the identity (∞,1)-functor on ∞Grpd is a 1-analytic (∞,1)-functor. See (Munson-Volic 15, section 6).
The original connectivity statement of the theorem is due to
Review includes
Stanley Kochmann, theorem 3.2.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Alan Hatcher, theorem 4.23 Algebraic Topology
Tammo tom Dieck, theorem 6.4.1 Algebraic Topology, EMS Textbooks in Mathematics, (2008) (pdf)
The algebraic statement and proof is in
The Blakers-Massey’s Connectivity Theorem was shown to be a consequence of Farjoun’s “cellular inequalities”
is Theorem 1.B of
This would constitute a purely homotopy-theoretic proof.
The generalisation of the algebraic statement is Theorem 4.3 in:
which relies essentially on the paper
for the van Kampen Theorem and for the nonabelian tensor product of groups. Here is a link to a bibliography of 137 items on the nonabelian tensor product.
Further applications are explained in
R. Brown, Triadic Van Kampen theorems and Hurewicz theorems, _ Algebraic Topology, Proc. Int. Conf. March 1988_, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57. pdf
The following paper applies the methods of the above two Brown-Loday papers to the well known problem of $n$-ad connectivity and to determination of the critical group, see Theorem 3.8 of:
The methods work because of their equivalence between cat$^n$-groups and crossed $n$-cubes of groups. This can be explained by saying that we need two kinds of algebraic categories for calculations with $(n+1)$-types: broad categories for conjecturing and proving theorems, and narrow algebraic categories for calculations and relations with classical ideas. In this case the broad category is that of cat$^n$-groups, and the narrow category is that of crossed $n$-cubes of groups, which are related geometrically to the homotopy groups of $r$-ads and to generalised Whitehead products. The tricky equivalence between the two kinds of categories is one of the engines behind the results, since it enables the use of whichever category is most convenient at any given time. Note also these two categories model weak, pointed, homotopy $(n+1)$-types, as shown by Loday in his paper
Further background to these ideas is in
Discussion of Blakers-Massey for ring spectra/E-∞ rings and other algebras over operads is in
The higher cubical version of Blakers-Massey is due to
a textbook account is in
A proof of Blakers-Massey connectivity in general ∞-stack (∞,1)-toposes is in prop. 8.16 of
A general version of the connectivity theorem in homotopy type theory (and thus in (infinity,1)-topos theory) was found by
A fully computer-checked version of this proof in HoTT-Agda was produced in
the statement appeared also as
and an announcement was given in
A writeup finally appeared as
Another unwinding to ordinary mathematical language of the above code was meanwhile given
prompted by online discussion here.
Further developments along these lines are in
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, A Generalized Blakers-Massey Theorem (arXiv:1703.09050)
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, Goodwillie’s Calculus of Functors and Higher Topos Theory (arXiv:1703.09632)
Let (X, A, x) be an n-connected inverse system of CW-pairs such that the restriction (A, x) is m-connected. We prove that there exists an isomorphic inverse system (Y, B, y) having n-connected terms such that the terms of the restriction (B, y) are m-connected. This result is then applied in proving analogues of Hurewicz and Blakers-Massey theorems for homotopy pro-groups and shape groups.