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This is related to Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?, and other questions, such as the intuition behind the Freudenthal suspension theorem.

The answers and comments to the question give references to nice proofs of the connectivity result. But of course Blakers and Massey proved an algebraic result for the critical (i.e. first non-vanishing) triad group $\pi_n(X;A,B)$ as a tensor product in the case $A \cap B$ simply connected.

The result was generalised to the $m$-ad and non simply connected case by Ellis and Steiner in "Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads" J. Pure Appl. Algebra {46} (1987) 117--136, see Theorem 3.7, using in an essential way the van Kampen theorem for $n$-cubes of spaces proved in the paper by Brown and Loday Topology 26 (1987) 311-334 (available here).

Now the proof of the latter theorem involves sophisticated tools in algebraic topology, of which surely Loday's cat$^n$-groups are essential.

Question 1. Is it possible to algebraicise the connectivity proof or proofs so that they give at least some part of the results on the critical group, including the non simply connected case?

Question 2. Is it possible to give a more geometric proof, based on the techniques pioneered by Puppe, to give a new proof of the theorem Loday and I proved, but instead by a direct verification of the universal property of the proposed colimit?

Note that of course the determination given of the critical group implies by induction the connectivity result. The proof in the paper in Topology goes by induction using the algebraic result in one dimension to prove connectivity in the next dimension.

The point of Question 1 is that if one has clever and nice proofs of a connectivity result, can the arguments be given an algebraic model so that by similar arguments one gets an algebraic result? Is it currently like trying to use arguments as in the proof of the usual Seifert-van Kampen Theorem, but without having a fundamental group or groupoid?

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    $\begingroup$ @Ronnie: For what it's worth, Theorem 3.13 of arxiv.org/pdf/1212.4420.pdf does much more than compute the critical group. In essence, it computes the deviation term of a strongly cocartesian $n$-cube being (homotopy) cartesian in a certain stable range. The idea in the end boils down to the Hilton-Milnor theorem plus games with homotopy colimits. $\endgroup$
    – John Klein
    Commented May 28, 2013 at 7:50
  • $\begingroup$ @John: But your paper does not refer to Ellis-Steiner. Are you claiming that you generalise their results? Do your results imply Corollary 3.2 (which uses the nonabelian tensor product) of the Brown-Loday paper in Topology? That would all be very interesting. $\endgroup$
    – Ronnie Brown
    Commented May 28, 2013 at 14:02
  • $\begingroup$ @Ronnie: Ellis-Steiner do not have a result called "Corollary 3.2," so I'm not sure. Let $X_\bullet$ be a strongly cocartesian $n$-cube (~ $(n+1)$-ad) indexed on a set $J$ having $n$-elements. Let $X_\emptyset$ be the initial vertex of the cube and let $X_J$ be the terminal one. What Bruce and I do is the following: we study the map $X_\emptyset \to \text{holim}_{I\ne \emptyset} X_I$ which we consider as a morphism of spaces over $X_J$. We compute the homotopy cofiber (taken fiberwise over $X_J$) in the stable range. $\endgroup$
    – John Klein
    Commented May 29, 2013 at 6:50
  • $\begingroup$ @John: Sorry if I have confused you. Corollary 3.2 refers to the Brown-Loday Topology paper which determines a $\pi_3(X;A,B)$ as $\pi_2(A,C) \otimes \pi_2(B,C)$. There are many calculations (including computer ones) of the tensor product. This leads to some calculations of $3$-types, and the $n$-cube result leads to some calculations of $n$-types. I presume our $n$-pushouts (in our Proc. LMS paper) are your strongly cartesian $n$-cubes. Can you get the $n$-adic Hurewicz theorem of the latter paper? I don't see the relations between the two methods, and results, I fear. $\endgroup$
    – Ronnie Brown
    Commented May 29, 2013 at 15:45

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