I'm having trouble understanding Serre Spectral sequence given in Hatcher. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf

In the beginning of the section, (Page 526 of the book, or rather, Pg 9 of the pdf file) he writes "Since $(B, B^p)$ is $p$-connected, the HLP implies that $(X, X_p)$ is also $p$-connected..."

I'm unable to show this simple(?) statement - maybe I'm overthinking it, or I've missed something.

What I've tried :

We need to show that the relative homotopy groups are 0. So, let $f \in [D^n, S^{n-1}; X, X_{p}]$. (where $n < p$) Then $\phi = \pi \circ f \in [D^n, S^{n-1}; B, B^{p}]$. Since $(B, B^p)$ is $p$-connected, $\phi$ is homotopic to a constant map, say $e_{b_0}$. ($e_{b_0}$ is the constant map that sends everything to $b_0 \in B$.) Let the homotopy be $F$. Then, by the HLP, we get a map $G:D^n \times I \rightarrow X$ such that $G\circ i = f$ (or $G(x,0) = f(x) \forall x \in D^n$) and $\pi \circ G = F$, i.e. $\pi \circ G(x,1) = F(x,1) = e_{b_0}$.

(Basically I've used the fact that fibration => Serre's fibration)

Now I want to show that $G$ is the required homotopy between $f$ and the constant map, but I'm not able to see why.

Also, in Theorem 5.3, how is $X$ filtered if $B$ is not a CW complex?

  • $\begingroup$ Hint: The restriction $X_p\rightarrow B_p$ is a pullback. What do you know about the fibres of the maps in a pullback square? $\endgroup$ – Tyrone May 8 at 16:55
  • $\begingroup$ I haven't checked theorem 5.3, but the standard way to get a filtration in this case is to take a CW approximation. $\endgroup$ – Tyrone May 8 at 16:56

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