Timeline for Reference request: Whitehead product and the Borel construction
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2016 at 21:10 | comment | added | Ronnie Brown | Theorem 2.4 of my article math.AT/0109091 relates the Generalised Whithead product, from dimensions (2,2) to 3, to the algebra of crossed squares, due to Guin-Walery and Loday. But I have not tried to relate this to the Borel construction as asked! | |
| Feb 23, 2016 at 21:03 | comment | added | Ronnie Brown | Have you tried J.F. Adams, "Algebraic topology: a student's guide"? | |
| Feb 23, 2016 at 4:16 | comment | added | Dylan Wilson | But I think the short answer is: there don't seem to be any good references... | |
| Feb 23, 2016 at 4:15 | comment | added | Dylan Wilson | I'm still mystified but maybe the following references will be useful: (1) G Whiteheads "generalization of the hopf invt" section 3, (2) caftan 1958/59 expose 5 section 2 (especially the statement of a lemma due to Samelson identifying the sign that appears when you use the adjoint isomorphism), (4) the only relatively recent references I found are Baues algebraic homotopy sec 15 and B. Gray's paper 'on generalized whitehead products' | |
| Feb 23, 2016 at 1:07 | comment | added | Charles Rezk | Also, what is a good reference for the Whitehead product? I thought G. Whitehead's book would be good, but I can't figure out where his basepoint is! (I realize it is not the same Whitehead.) | |
| Feb 23, 2016 at 1:06 | comment | added | Charles Rezk | Of course, this means I have to figure out things like the convention for path composition (i.e., product on $\Omega X$): is $\gamma*\delta$ in "temporal order" (put $\gamma$ on $[0,1/2]$) or "composition order" (put $\gamma$ on $[1/2,1]$, cause we are thinking of it as a morphism $\gamma(0)\to \gamma(1)$, so the output of $\delta$ is the input of $\gamma$)? | |
| Feb 23, 2016 at 1:04 | comment | added | Charles Rezk | @DylanWilson I agree with your line of thinking. The point, I guess, is that the only real sign issues are lurking in the "adjoint" relation $S^p\to G$ vs. $S^{p+1}\to BG$, which can be fixed relating the path fibration $P(BG)\to BG$ to the fibration $EG\to BG$. ... | |
| Feb 21, 2016 at 23:06 | comment | added | Dylan Wilson | If we think of $G$ as acting on $\Omega X$ then it looks like there's a map $G \star \Omega X \longrightarrow X$ which also induces a pairing on homotopy groups like the one you described. This seems related to both of your constructions since (1) the Whitehead product comes from the canonical pairing $\Omega X \star \Omega X \rightarrow X$ for any space, and (2) there should be some kind of naturality relating this construction to fiber sequences, e.g. path-loops and homotopy orbits. Not sure if this is useful... | |
| Feb 21, 2016 at 20:12 | history | edited | Charles Rezk |
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| Feb 21, 2016 at 20:01 | history | asked | Charles Rezk | CC BY-SA 3.0 |