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Suppose $X$ is a topological space and $x_0 \in X$. Let $$ X' = X \cup e^{n+1} $$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the attaching map $\partial e^{n+1} \to X$).

I want to show that the inclusion $X \to X'$ is an $n$-equivalence, i.e. that the induced map $i^\ast: \pi_k(X,x_0) \to \pi_k(X',x_0)$ is bijective for $k<n$ and surjective for $k=n$.

Using cellular approximation I can prove this when $X$ is a CW complex, but I did not succeed in solving the general case.

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  • $\begingroup$ This is true in general, but what kind of attaching map is this? I think this follows from long exact sequence of homotopy groups. $\endgroup$ – Bombyx mori Jan 15 '15 at 8:45
  • $\begingroup$ The attaching map can be any continuous map from $\partial e^{n+1}$ to $X$. $\endgroup$ – user133281 Jan 15 '15 at 8:46
  • $\begingroup$ I think you can work with the pair $(A,B)=(X',X)$. I am not entirely sure if this is a good pair since $X$ is just some topological space. $\endgroup$ – Bombyx mori Jan 15 '15 at 8:49
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I assume $X$ is Hausdorff and connected. (The connectedness is just because I don't want to care about base points. You can adapt the argument to the non-connected case. The assumption that $X$ is Hausdorff is more important. I'm unsure whether your claim holds in the non-Hausdorff case.)

An element in $\pi_i(X',X)$ is represented by a map of pairs $(D^i,S^{i-1})\rightarrow(X',X)$ which is homotopic as a map of pairs to a cellular map of relative CW-complexes by the cellular approximation theorem for relative CW-complexes, so it is zero in $\pi_i(X',X)$ for $0\le i\le n$. Your claim now follows from the LES of relative homotopy groups.

You'll find the relative cellular approximation theorem in the book of Spanier (Section 7.6, Cor. 18).

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