Timeline for Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57) [duplicate]
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| when toggle format | what | by | license | comment | |
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| Jan 9, 2016 at 10:33 | history | closed |
Chris Gerig José Figueroa-O'Farrill András Bátkai Wolfgang Stefan Waldmann |
Duplicate of "Dirty" proof that Eilenberg-MacLane spaces represent cohomology? | |
| Jan 9, 2016 at 5:55 | comment | added | Ryan Budney | Yes, of course there's intuition. It's in the proof! The proof is about as elementary as you could hope for a theorem of this type. Understand the proof and you will understand much more than this theorem. The technique of the proof is very much the "initial spirit" of obstruction theory. A wonderfully productive subject that gave birth to many productive careers. | |
| Jan 8, 2016 at 22:19 | comment | added | Dylan Wilson | You can take, as a model for $K(\mathbb{Z},n)$, the free abelian group on $S^n$ where the basepoint acts as the identity. Then, given an $n$-cocycle $\phi$ you define a map on $X$ by sending the $(n-1)$-skeleton to the basepoint and then sending an $n$-cell $\sigma$ to the sphere by modding out by the boundary, then multiply the result by $\phi(\sigma)$. This gets you to the $n$-skeleton... to extend further you use that $\phi$ was a cocycle. | |
| Jan 8, 2016 at 22:09 | review | Close votes | |||
| Jan 9, 2016 at 10:33 | |||||
| Jan 8, 2016 at 21:46 | comment | added | Chris Gerig | Also here: mathoverflow.net/questions/5518/… | |
| Jan 8, 2016 at 21:37 | comment | added | Chris Gerig | This was answered here: mathoverflow.net/questions/2890/… To repeat: $H^n(K(G,n);G)=Hom(\pi_nK(G,n),G)=Hom(G,G)$ and so there is a distinguished element $u\in H^n(K(G,n);G)$ corresponding to the identity $1:G\to G$. The bijection is given by pull-back, $f\mapsto f^*u$. For your simple example ($G =\mathbb{Z}$, $n = 1$), take $c \in H^1(X)$ to map the 1-skeleton of $X$ to $S^1$, where an edge $e$ will make $c(e)$ loops around $S^1$. | |
| Jan 8, 2016 at 20:28 | review | First posts | |||
| Jan 8, 2016 at 20:49 | |||||
| Jan 8, 2016 at 20:22 | history | asked | Krishna | CC BY-SA 3.0 |