用户: Eee/Milnor&Stasheff

1Obstruction

Prepare: Stepwise Extension of A Cross Section.

 Reference: Basic Algebraic Topology and its Application by Mahima Ranjan Adhikari.

Definition 1.1. A path-connected topological space is said to $n$-simple if there is point $x_0\in X$ such that ${\pi }_1(X,x_0)$ acts trivially on ${\pi }_n\ast X,x_0)$ in the sense that each element of ${\pi }_1(X,x_0)$ acts on ${\pi }_n(X,x_0)$ as the identity.

 This subsection considers the problems of constructing a cross section of a fiber bundle. Throughout this subsection it is assumed that for the fiber bundle $\xi :F\subset E\to X$, the base space $X$ is a finite CW-complex. Suppose that the fiber space $F$ is path-connected and $n$-simple.

 We assume that we have a partially defined cross section $f:X^n\to E$, the problem is to extend it over $X^{n+1}$. In such a problem, obstruction may appear. Indeed, if $\sigma$ is an $(n+1)$-cell of $X$, the cross section $f{|}_{\partial \sigma }$ might describe a nontrivial element in ${\pi }_n(F)$ and in this case $f$ will not have a continuous extension over $\sigma$. Consider the $${\psi }_f:\{(n+1)\text{-cells}\sigma \text{of}X\}\to {\pi }_n(F),\sigma \mapsto [f{|}_{\partial \sigma }].$$

 As by hypothesis, $F$ is $n$-simple, and $f{|}_{\partial \sigma }$ is a topological $n$-sphere, the function ${\psi }_f$, is well defined and ${\psi }_f$ sends an $(n+1)$-cell $\sigma$ to an element of ${\pi }_n(F)$ which is determined by $f{|}_{\partial \sigma }$ through some random trivialization $\psi {|}_{\sigma }\cong \sigma \times F$. Then ${\psi }_f$ can be extended by linearity and ${\psi }_f$ can be regarded as a ${\pi }_n(F)$-valued cochain, abbreviated $c^{n+1}(f)\in C^{n+1}(X;{\pi }_n(F))$.

 Again for every $(n+2)$-cell $\sigma$, we have $\delta {\psi }_f(\sigma )={\psi }_f(\partial \sigma )=[f{|}_{\partial \partial \sigma }]=0$. This shows that $c^{n+1}(f)$ is a cocycle. Hence its cohomology class $[c^{n+1}(f)]$ is an element of $H^{n+1}(X;{\pi }_n(F))$, the element is usually abbreviated ${\gamma }^{n+1}(f)$.

Remark 1.2. Because $c^{n+1}(f)$ being the zero indicator that all of these elements of $H^{n+1}(X;{\pi }_n(F))$ vanish, it asserts that the given partially cross section can be extended to $X^{n+1}$ using the homotopy between $f{|}_{\partial \sigma }$ and the constant map.

 If we start with a different partially defined cross section $g$ that agrees with $f{|}_{X^{n-1}}$, then the resulting cocycle $c^{n+1}(g)$ would differ from $c^{n+1}(f)$ by a coboundary. This asserts that there is a well-defined element of the cohomology group $H^{n+1}(X;{\pi }_n(F))$ such that if a partially defined cross section on $X^{n+1}$ exsits that agrees with the given choice on $X^{n-1}$, then the cohomology class ${\gamma }^{n+1}(f)$ must be trivial. Its converse is also true as homotopy section is in the sense $pf\sim 1_d$.

Definition 1.3. Given an $(n+1)$-cell $\sigma$ of $X$, a function $c(f,\sigma )$ is defined by the rule $c(f,\sigma )={\psi }_f(\sigma )=[f{|}_{\partial \sigma }]\in {\pi }_n(F)$.

Definition 1.4. The function ${\psi }_f$ of $\sigma$ given ${\psi }_f(\sigma )=[f{|}_{\partial \sigma }]=c(f,\sigma )\in {\pi }_n(F)$ is called the obstruction cocycle of $f$ and is sometimes denoted by $c(f)$.

 The above discussion with corresponding notations leads to the following important results.

Theorem 1.5. A cross section over $X^n$ can be extended over $X^{n+1}$ iff $c(f,\sigma )$ vanishes for each $(n+1)$-cell $\sigma$ of $X$.

Proof.

Proof. Let $X_{\sigma }$ be a base point of a given $(n+1)$-cell $\sigma$ of $X$ and $F_{\sigma }$ be the fiber over $x_{\sigma }$. Choose an orientation of $\sigma$. Consider $\sigma$ the oriented cell and $\partial \sigma$ oriented boundary. If $f$ is a cross section $f:X^n\to E$, define $c(f,\sigma )={\psi }_f(\sigma )=[f{|}_{\partial \sigma }]\in {\pi }_n(F)$. As $F$ is $n$-simple, the theorem follows.

$\square$

Corollary 1.6. Given a fiber bundle $p:E\to X$ whose fiber $F$ is a path-connected $n$-simple space, then the element ${\gamma }^{n+1}(f)\in H^{n+1}(X;{\pi }_n(F))$ vanishes iff there are cross sections of $p:E\to X$ defined over the $n$-skeleton of $X$ that extend over the $(n+1)$-skeleton.

Back to Milnor&Stasheff

 Now suppose that the base space $B$ is a CW-complex.

Definition 1.7. We say $B$ is $k$-connected if ${\pi }_i(X)$ is triviall for $i\le k$.

 Steenrod shows that the fiber $V_k(F)$ is $(n-k-1)$-connected, so it is easy to construct a cross-section of $V_k(\xi )$ over the $(n-k)$-skeleton of $B$.

 There exists a cross-section over the $(n-k+1)$-skeleton of $B$ if and only if a certain well defined primary obstruction class in $$H^{n-k+1}(B;{\pi }_{n-k}(V_k(F)))$$is zero.

 Setting $j=n-k+1$, we will use the notation $${\mathfrak{o}}_j(\xi )\in H^j(B;{\pi }_{j-1}(V_{n-j+1}(F)))$$for this primary obstruction class. If $j$ is evern, and less than $n$, then Steenrod shows that the coefficient group ${\pi }_{j-1}(V_{n-j+1}(F))$ is cyclic of order $2$, hence canonically isomorphic to ${\mathbb{Z}}_2$. If $j$ is odd, or $j=n$, the group ${\pi }_{j-1}(V_{n-j+1}(F))$ is infinite cyclic. However it is not canonically isomorphic to $\mathbb{Z}$ (In general twisted).

 In any case, there is certainly a unique non-trivial homomorphism $h$ from ${\pi }_{j-1}(V_{n-j+1}(F))$ to ${\mathbb{Z}}_2$. Hence we can reduce the coefficients modulo $2$, obtaining an induced cohomology class $h_{\ast }{\mathfrak{o}}_j(\xi )\in H^j(B;{\mathbb{Z}}_2)$.

Theorem 1.8. This reduction modulo $2$ of the obstruction class ${\mathfrak{o}}_j(\xi )$ is equal to the Stiefel-Whitney class $w_j(\xi )$.