Obstruction Cocycles

Question

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.

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) $

And claims that this composition is the value of the obstruction cochain in an $n+1$ cell, how might one verify this?

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).

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:

$\delta \Theta (e) = \Theta( \partial e) = \Theta (\sum [w_i;e]w_i) = \sum [w_i;e]\Theta(w_i) = \dots = 0$ (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).

Any help on the subject or a good refference is very very much appreciated! Thanks and have a great week.

Answer 1

You should have a look at the paper given in the answer to my earlier question on obstruction theory. 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.

Answer 2

How are you defining the cellular chain complex and $[w_i;e]$? The usual way is to define $C_n(X):=H_n(X^n,X^{n-1})$ 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. After the fact, 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.

Answer 3

You may find this and this by Tony Phillips useful.