Timeline for Can the various proofs of the connectivity in the Blakers-Massey theorem be algebraicised?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 14, 2016 at 21:32 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
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| May 29, 2013 at 15:45 | comment | added | Ronnie Brown | @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. | |
| May 29, 2013 at 6:50 | comment | added | John Klein | @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. | |
| May 28, 2013 at 14:02 | comment | added | Ronnie Brown | @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. | |
| May 28, 2013 at 7:50 | comment | added | John Klein | @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. | |
| May 27, 2013 at 22:01 | history | asked | Ronnie Brown | CC BY-SA 3.0 |