I'm currently reading Kirk's & Davis's Lecture notes in Algebraic Topology. On the page 190 they discuss obstruction for lifting $f:X \rightarrow B $ to $f':X \rightarrow E$ where $E \rightarrow B$ is a fibration. Suppose $f'$ is defined on $X_n$, then for any $n+1$ cell - $e^{n+1}$ - its characteristic map gives us a map $S^n \rightarrow F \hookrightarrow E$ which defines an element of $\pi_{n}(F)$ since we assume that $F$ is $n$ simple. Now the authors note that if$\pi_{1}(B) \neq 0$ then this assignement need not define a $cochain$ since $f$ does not preserve base points. Then they conclude that we can instead define a cocycle in cohomology with local coeaficient system induced by $p$. So we get an element of $Hom_{Z[\pi_1(X)]}(C_{n+1}(\tilde{X}),\pi_{n}(F))$. My question is - how do we define this cochain?
First consider the case that $X=B$, construct the class and then pull it back to $X$ via the map $f$. See section 5.2.1, pg 100 for remarks regarding the interaction between the cell structures of $B$ and its universal cover $\widetilde{B}$. If the cells $e^{n+1}_i$ are a $\mathbb{Z}$-basis for the cellular chain group $C^{n+1}(B)$ then $\tilde{e}^{n+1}_i$ are $\mathbb{Z\pi_1}$ basis for $C^{n+1}(\widetilde{B})$ (see 5.2.1 for notation). Given the lift partial lift $g:B_n\rightarrow E$ with $p\circ g=f|{B_n}$ define $\theta^{n+1}(g)\in C^{n+1}(\widetilde{B};\pi_nF)=Hom_{\mathbb{Z}\pi}(C_{n+1}(\widetilde{B}),\pi_nF)$ on the $\mathbb{Z}\pi$-basis elements by
$\theta^{n+1}(g)[\epsilon\cdot \tilde{e}^{n+1}_i]=\epsilon\cdot [g\circ\varphi_i]$
where $\epsilon\in\mathbb{Z}\pi$ with $\pi=\pi_1B$ acting on $\pi_nF$ through the representation $\rho$ (pg 189), and $\varphi_i:S^n\rightarrow B_n$ is the attaching map for the cell $e^{n+1}_i$ of $B$. Extend $\theta^{n+1}(g)$ to all of $C_{n+1}(\widetilde{B})$ by linearity and you have your cochain.
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$\begingroup$ $[g \circ \varphi_i ] $ will land in $n$-th homotopy groups of fiber over different base points. How do you identify all those groups with one $\pi_n(F)$? $\endgroup$ – leg14able Jul 9 '17 at 15:44
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$\begingroup$ Davis and Kirk use the assumption that $F$ is $n$-simple. In particular $F$ must be connected so $\pi_nF$ does not depend on any particular choice of basepoint. $\endgroup$ – Tyrone Jul 9 '17 at 16:07
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$\begingroup$ Yes, of course. What I meant is the following. $[g \circ \varphi_i] $ defines an elemnt in $p^{-1}(b) \cong F$ but for different base element the point $b$ may be different. How do we identify $\pi_{n(}p^{-1}(b)$ for different b? $\endgroup$ – leg14able Jul 9 '17 at 16:19
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$\begingroup$ Could you please elaborate a little bit? i'll try to be more specific. We take a cell $e^{n-1}$. We have a homotopy $H:\partial e^{n-1} \times I \rightarrow B$ which takes the boundary into a point $b \in B$. This point does not have to be the same for different cells. then a lift of homotopy collapsing a cell to a point gives us an element in $\pi_{n}(p^{-1}(b)$. But since the point may be different for different cells my question is how do we bypass this obstacle? (because from what I understand this is exactly why we want to use local coeffients (to bypass this obstacle)) $\endgroup$ – leg14able Jul 9 '17 at 17:24
