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Good morning,

I have a question concerning the relative homotopy groups of a CW-pair as follows.

Let (X,A) be a CW-pair. What are results known for the relation between $\pi_{\ast}(X,A)$ and $\pi_{\ast}(X/A)$. If in addition, A is (n-1)-connected, do we have the equality $\pi_i(X,A) = \pi_i(X/A)$ for $i<n$? For $i\leq n$?

Any help is appreciated. Thanks in advance.

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I believe what you want is called the "homotopy excision theorem". – Justin Young Jan 25 '12 at 12:44
See Proposition 4.28 in Hatcher for a reference. – user17786 Jan 25 '12 at 14:00
Thank you very much. – Đức Anh Jan 25 '12 at 14:04
up vote 1 down vote accepted

An answer is given by the Relative Hurewicz Theorem, for which a well known form can be stated as follows:

If $(X,A)$ is an $(n-1)$-connected pair, then the pair $(X \cup CA,CA)$ is $(n-1)$-connected and the morphism induced by inclusion

$$\pi_n(X,A) \to \pi_n(X \cup CA,CA) \cong \pi_n(X \cup CA)$$

is given by factoring out the action of $\pi_1(A)$ on $\pi_n(X,A)$.

Note that this implies $X \cup CA$ is $(n-1)$-connected and so the absolute Hurewicz Theorem implies $\pi_n(X \cup CA) \cong H_n(X \cup CA)$; and if $(X,A)$ has the HEP, then the map $X \cup CA \to X/A$ is a homotopy equivalence.

A proof of this form is given in

R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41,

and is shown to be a special case of a higher homotopy van Kampen Theorem, proved without using simplicial approximation or singular homology (but it uses a cubical higher homotopy groupoid!).

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thank you very much – Đức Anh Apr 26 '12 at 19:45

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