Timeline for Models for P map in EHP sequence
Current License: CC BY-SA 2.5
10 events
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| Feb 13, 2011 at 4:05 | history | edited | John Klein | CC BY-SA 2.5 |
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| Feb 12, 2011 at 23:53 | history | edited | John Klein | CC BY-SA 2.5 |
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| Feb 12, 2011 at 6:39 | vote | accept | Dev Sinha | ||
| Feb 11, 2011 at 20:05 | comment | added | John Klein | I found a reference for $P$ (probably there are more definitive ones): projecteuclid.org/DPubS/Repository/1.0/… | |
| Feb 11, 2011 at 19:56 | history | edited | John Klein | CC BY-SA 2.5 |
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| Feb 11, 2011 at 12:12 | comment | added | John Klein | Method 3 elaborated: take the Samelson product of $G = \Omega \Sigma Y$, to get a map $G \wedge G \to G$. Precompose it with the evident map $Y \wedge Y \to G \wedge G$ to get a map $Y \wedge Y \to \Omega \Sigma Y$. Now take the adjoint. | |
| Feb 11, 2011 at 12:00 | comment | added | John Klein | Method 2: Ganea defines the homotopy fiber of the inclusion $i: \Sigma Y \vee \Sigma Y \to \Sigma Y \times \Sigma Y$ and shows that it can be identified with $\Sigma (\Omega \Sigma Y) \wedge (\Omega \Sigma Y)$. Now use the evident map $ u: \Sigma Y \wedge Y \to \Sigma (\Omega \Sigma Y) \wedge (\Omega \Sigma Y)$ to obtain $P$ as the composite of $u$ followed by the usual map from a homotopy fiberof $i$ into the domain of $i$. | |
| Feb 11, 2011 at 11:45 | comment | added | John Klein | The map $P: \Sigma Y \wedge Y\to \Sigma Y$ is a composition of the map $f:\Sigma Y \wedge Y \to \Sigma Y \vee \Sigma Y$ with the fold map $\Sigma Y \vee \Sigma Y \to \Sigma Y$. Here $f$ has homotopy cofiber identified with $\Sigma Y \times \Sigma Y$. Alternatively, $P$ can be defined explicitly as in my next comment. | |
| Feb 11, 2011 at 5:15 | comment | added | Dev Sinha | Hi John - Thanks. I'm not aware of the generalized Whitehead product map you are referring to. I understand Whitehead products only through the most elementary definitions - the standard one using the attaching map for a torus (which is just in homotopy), and the space level commutator map $G \times G \to G$ which defines the Samelson product and coincides with the Whitehead product (up to sign) when $G = \Omega X$. If you could elaborate on $P$ and perhaps give a reference I'd appreciate it. | |
| Feb 10, 2011 at 23:18 | history | answered | John Klein | CC BY-SA 2.5 |