0
$\begingroup$

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that:

Let $\pi:X\to B$ be a fibration with $B$ being a CW complex. Considering $\Phi_{\alpha}:D_{\alpha}^p\to B^p$ being the characteristic map for the $p$-th cell of the CW complex $B$ and defining $\tilde{D_{\alpha}^p}=\Phi_{\alpha}^*(X_p)$ we have a map $\tilde{\Phi}:\sqcup_{\alpha}(\tilde{D_{\alpha}^p},\tilde{S_{\alpha}^{p-1}})\to (X_p,X_{p-1})$, where $ \tilde{S_{\alpha}^{p-1}}$ is the part of $\tilde{D_{\alpha}^p}$ over $S_{\alpha}^{p-1}$. Which will induce the map on homology.

It's written that as there is a neighbourhood $N$ of $B^{p-1}$ in $B^p$, which deformation retracts onto $B^{p-1}$, by the homotopy lifting property of $\pi$, we have $X_{p-1}\hookrightarrow \pi^{-1}(N)$ is a homotopy equivalence. Now it's written that using excision,this implies $\tilde{\Phi}$ induces the isomorphism $\tilde{\Phi_*}$.

I couldn't understand the last line, how excision is used. Please give some hint.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.