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I'm having trouble understanding the last part of Hatcher's proof of Hurewicz' theorem. (It's on page 367, thm. 4.32).

We want to show, that a cellular boundary map: $d:H_{n+1}(X^{n+1},X^n) \rightarrow H_n(X^n,X^{n-1})$ is a map $\oplus_\beta \mathbb{Z} \rightarrow \oplus_\alpha \mathbb{Z}$, where $X$ only has cells in dimensions $n$ and $n+1$ and is given by $(\bigvee_\alpha S_\alpha ^n )\bigcup_\beta e^{n+1}_\beta.$

The argument goes like this:

"$d$ [is a map $\oplus_\beta \mathbb{Z} \rightarrow \oplus_\alpha \mathbb{Z}$] since for each cell $e_\beta ^{n+1}$, the coefficients of $de_\beta ^{n+1}$ are the degrees of the compositions $q_\alpha \phi_\beta$ where $q_\alpha$ collapses all n-cells except $e_\alpha ^n$ to a point, and the isomorphism $\pi_n (S^n) \approx \mathbb{Z}$ in corollary 4.25 us given by degree."

Any help or ideas would be welcome :)

Kristoffer
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  • Sorry, what are $S^n_\alpha$ end $e^n_\beta$? I have some guesses, but.. – Berci Jun 22 '14 at 23:21
  • Hi there! $S^n \alpha$ is an n-sphere and $e^{n+1}\beta $ is a cell of dimension $n+1$, which is homeomorphic to an open n+1-disc. $e^n \alpha$ I guess is the $S^n _\alpha$ without the common point from $(\bigvee\alpha S^n _\alpha)$. – Kristoffer Jun 22 '14 at 23:46
  • The formula $d_n(e_\beta)=\sum_\alpha d_{\beta\alpha}e_\alpha$ where $d_{\beta\alpha}$ is the degree of the composition $q_\alpha\phi_\beta$ is explained in the section about cellular homology starting at page 137. – Stefan Hamcke Jun 23 '14 at 11:18
  • I also have some troubles in identify the boundaru map and the cellular boundary map. – Brooks Jun 06 '17 at 13:31
  • we have to show that d and round maps isomorphisc elements to isomorphisc elements. – Brooks Jun 06 '17 at 13:37

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