Obstruction Cocycles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:37:41Z http://mathoverflow.net/feeds/question/38497 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38497/obstruction-cocycles Obstruction Cocycles Juan OS 2010-09-12T18:13:32Z 2010-09-13T06:27:32Z <p>Hey everyone, I was reading about obstruction theory, here's a bit of a summary. Take a cellular space $X$ and a fibre bundle $p:E \to X$ with fiber $F$; consider the problem of extending a section $s$, defined on the $(n-1)$-skeleton over to the $n$-skeleton. We work cell by cell pulling back the bundle via the characteristic map and the section via the restriction of the Char. Map to the boundary of our $n$-cell, since the cell is contractible, the bundle is isomorphic to $D^n \times F$ so the section defines a map from $S^{n-1} \to D^n \times F$ i.e. an element of $\pi_{n-1}(D^n \times F) \cong \pi_{n-1}(F)$. Define the obstruction cochain as the element in $C^n(X^n,\pi_{n-1}(F))$ taking each $n$-cell to the element in the $(n-1)$-homotopy group constructed before.</p> <p>Here's what's bothering me, in Steenrod's book (The topology of Fibre Bundles) he proves that this cochain is actually a cocycle in a really weird way, it looks to me as if he makes no distinction between the homological boundary and the topological boundary of a cell. Roughly he writes the following composition: $C_{q+1}(X) \stackrel{\partial_*}\to Z_q(X) \stackrel{hurewicz}\to \pi_q(X^q) \stackrel{f=(p_2\circ \sigma)_*}\to \pi_q(F)$</p> <p>And claims that this composition is the value of the obstruction cochain in an $n+1$ cell, how might one verify this?</p> <p>Not being happy with this proof i went and looked at the one Kirk and Davis' book (Lecture notes on algebraic topology) and found it too complex (I know, I dont like anything sorry).</p> <p>What I was wondering is if there was a way to prove this affirmation (the obstruction cochain is a cocycle) directly, i.e. denoting the cochain by $\Theta$ doing something like:</p> <p>$\delta \Theta (e) = \Theta( \partial e) = \Theta (\sum [w_i;e]w_i) = \sum [w_i;e]\Theta(w_i) = \dots = 0$ (where $e$ is a $(n+1)$-cell, $w_i$ is a $n$-cell and $[w_i;e]$ is their incidence number, so that the third term is $\Theta$ evaluated on the cellular (homological) boundary).</p> <p>Any help on the subject or a good refference is very very much appreciated! Thanks and have a great week.</p> http://mathoverflow.net/questions/38497/obstruction-cocycles/38505#38505 Answer by Evan Jenkins for Obstruction Cocycles Evan Jenkins 2010-09-12T19:05:57Z 2010-09-12T19:05:57Z <p>You should have a look at the paper given in the answer to <a href="http://mathoverflow.net/questions/31147/obstruction-theory-for-non-simple-spaces" rel="nofollow">my earlier question on obstruction theory</a>. It gives a very nice and direct proof that the obstruction cochain is a cocycle that also works in the case of non-simple spaces (a setting that most modern treatments gloss over). It's written in simplicial rather than cellular language, but I imagine the techniques could be carried over with a bit of effort.</p> http://mathoverflow.net/questions/38497/obstruction-cocycles/38510#38510 Answer by Somnath Basu for Obstruction Cocycles Somnath Basu 2010-09-12T20:29:36Z 2010-09-12T20:29:36Z <p>You may find <a href="http://www.math.sunysb.edu/~tony/stiefel/cocycles.pdf" rel="nofollow">this</a> and <a href="http://www.math.sunysb.edu/~tony/stiefel/stiefel.pdf" rel="nofollow">this</a> by Tony Phillips useful. </p> http://mathoverflow.net/questions/38497/obstruction-cocycles/38513#38513 Answer by Paul for Obstruction Cocycles Paul 2010-09-12T21:38:05Z 2010-09-12T21:38:05Z <p>How are you defining the cellular chain complex and $[w_i;e]$? The usual way is to <em>define</em> $C_n(X):=H_n(X^n,X^{n-1})$ and the differential as the composite $H_n(X^n,X^{n-1})\to H_{n-1}(X^{n-1})\to H_{n-1}(X^{n-1},X^{n-2})$, where the first map is the connecting homomorphism for the pair. Steenrod's observation is then straightforward, and follows from the long exact sequence of the appropriate pair. <em>After the fact</em>, you can show that $C_n(X)$ is free on the $n$ cells and that $[w_i;e]$ is the degree of the composite of the attaching map $S^{n-1}\to X^{n-1}$ for $e$ composed with the projection $X^{n-1}\to X^{n-1}/X^{n-2}=\vee_i S^{n-1}\to w_i/\partial w_i\cong S^{n-1}$. That typically requires reconciling different notions of degree, e.g. homological and differential topological and perhaps this is where you are having difficulties.</p>