nLab Blakers-Massey theorem

Idea

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.

Statement

Traditional

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

$$\cdots \to \pi_{n+1}(X;A,B) \to \pi_n (A, A \cap B) \to^{\epsilon} \pi_n( X,B) \to \pi_n(X;A,B) \to \cdots$$

where $\epsilon$ is the excision map. The main result of Blakers and Massey is as follows:

Theorem

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

$$\pi_{m}(A,C) \otimes \pi_{n}(B,C) \to \pi_{m+n-1}(X;A,B)$$

is an isomorphism.

(Blakers-Massey 51, tomDieck 08, theorem 6.4.1).

Remark

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

$$\array{ X &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{f_2}} & \searrow \\ Y_2 && Y_2 \underset{Y_2 \underset{X}{\coprod} Y_1}{\times} Y_1 }$$

is $(n_1 + n_2 - 1)$-connective.

Remark

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.

Remark

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.

Remark

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 $A,B$.

In higher topos theory

Proposition

The Blakers-Massey connectivity theorem in the form of remark 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).

Higher cubical BM-theorems and analytic $\infty$-functors

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).

References

Classical

The original connectivity statement:

• Albert Blakers, William Massey, The homotopy groups of a triad I , Annals of Mathematics 53: 161–204, (1951) (jstor:1969346)

The algebraic statement and proof:

• Albert Blakers, William Massey, The homotopy groups of a triad. {III}, Ann. of Math. (2), 58: (1953) 409–417 (jstor:1969744)

Textbook accounts:

• Tammo tom Dieck, Klaus Heiner Kamps, Dieter Puppe, Section 15 of: Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (doi:10.1007/BFb0059721)

• 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 Blakers-Massey’s Connectivity Theorem was shown to be a consequence of Farjoun’s “cellular inequalities”

• Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localizationm No. 1621-1622. Springer, 1996]

is Theorem 1.B of

• Wojciech Chachólski, A generalization of the triad theorem of Blakers-Massey Topology 36.6 (1997): 1381-1400

This would constitute a purely homotopy-theoretic proof.

The generalisation of the algebraic statement is Theorem 4.3 in:

• R. Brown and Jean-Louis Loday, Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces, Proc. London Math. Soc.

  (3) 54 (1987) 176-192. pdf

which relies essentially on the paper

• R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of

  spaces, Topology 26 (1987) 311-334,

for the van Kampen Theorem and for the nonabelian tensor product of groups. Here is a link to a bibliography of 170 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:

• 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 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

• Jean-Louis Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179-202.

Further background to these ideas is in

• R. Brown“A philosophy of modelling and computing homotopy types” Presentation to CT2015, Aveiro, Portugal, June 14-19. pdf and in “Modelling and computing homotopy types: I” Indag.Math. 29 (2018) 459-482 pdf

Discussion of Blakers-Massey connectivity for ring spectra/E-∞ rings and other algebras over operads is in

• Michael Ching, John Harper, Higher homotopy excision and Blakers-Massey theorems for structured ring spectra (arXiv:1402.4775)

The higher cubical version of Blakers-Massey connectivity is due to

• Tom Goodwillie, Calculus II: Analytic functors, K-Theory 01/1991; 5(4):295-332. DOI: 10.1007/BF00535644

a textbook account is in

• Brian Munson, Ismar Volic, Cubical homotopy theory, Cambridge University Press, 2015 pdf

In $\infty$-topos theory and homotopy type theory

A proof of Blakers-Massey connectivity in general ∞-stack (∞,1)-toposes is in prop. 8.16 of

• Charles Rezk, Toposes and homotopy toposes (2010) (pdf)

A general version of the connectivity theorem in homotopy type theory (and thus in (infinity,1)-topos theory) was found by

• Peter LeFanu Lumsdaine, Eric Finster, Dan Licata (to appear)

A fully computer-checked version of this proof in HoTT-Agda was produced in

• Favonia, BlakersMassey.agda

the statement appeared also as

• Univalent Foundations Project, theorem 8.10.2 of Homotopy Type Theory – Univalent Foundations of Mathematics

and an announcement was given in

• Peter LeFanu Lumsdaine, The Blakers-Massey theorem in homotopy type theory talk at Conference on Type Theory, Homotopy Theory and Univalent Foundations (2013) (talk abstracts pdf).

A writeup finally appeared as:

• Kuen-Bang Hou (Favonia), Eric Finster, Dan Licata, Peter LeFanu Lumsdaine, A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory, LICS ‘16 (2016) 565–574 [arXiv.1605.03227, doi:10.1145/2933575.2934545]

Another unwinding to ordinary mathematical language of the above code was meanwhile given in

• Charles Rezk, Proof of the Blakers-Massey theorem, 2014 pdf.

prompted by online discussion at

• Urs Schreiber, Explaining the point of HoTT on FOM, Google+ thread 2014-10-22 (archived version)

  (scroll down a fair bit through the list of replies there to see the exchange between Charles Rezk and Favonia).

Further developments along these lines are in

• Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, A Generalized Blakers-Massey Theorem, Journal of Topology 13 4 (2020) 1521-1553 $[$arXiv:1703.09050, doi:10.1112/topo.12163$]$

  (generalizing to the case where the (n-connected, n-truncated) factorization system may be replaced by more general modalities)

• Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, Goodwillie’s Calculus of Functors and Higher Topos Theory (arXiv:1703.09632)

In shape theory

• Šime Ungar, $n$-Connectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371-396 pdf

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.