Hatcher's Exercise 0.27 is :
$\mathbf{27.}$ Given a pair $(X,A)$ (this just means that $A$ is a subspace of a space $X$) and a homotopy equivalence $f:A \to B$, show that the natural map $X \to B \cup_f X$ is a homotopy equivalence if $(X,A)$ satisfies the HEP. [Hint: Consider $X\cup M_f$ and use the preceding problem.]
(Here $M_f$ is the mapping cylinder, obtained by identifying $(a,1)$ and $f(a)$ for $a \in A$ in $A \times I \coprod B$)
The preceding problem is :
$\mathbf{26.}$ Use Corollary 0.20 to show that if $(X,A)$ as the HEP, then $X \times I$ deformation retracts onto $X \times {0} ~\cup A \times I $. Deduce from this that Proposition 0.18 holds more generally for any pair $(X_1,A)$ satisfying the HEP.
Corollary 0.20 and Proposition 0.18 are :
$\mathbf{Corollary ~0.20.}$ If $(X,A)$ satisfies the HEP and the inclusion $A \to X$ is homotopy equivalence, then $A$ is a deformation retract of $X$.
$\mathbf{Proposition ~0.18.}$ If $(X_1,A)$ is a CW pair and we have attaching maps $f,g:A \to X_0$ that are homotopic, then $X_0 \cup_fX_1 \simeq X_0 \cup_gX_1$.
Now, my questions are the following:
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Since $f:A \to B $ is a homotopy equivalence, the mapping cylinder $M_f$ deformation retracts onto both $A$ and $B$. Using this, and considering the hint, I showed that $X \cup M_f$ deformation retracts onto both $X$ and $B \cup_fX$. Thus, $X$ and $B \cup_fX$ are homotopy equivalent. But I know that this does not imply that the natural embedding $X \to B \cup_fX$ is a homotopy equivalence. I think that I have to use Exercise 0.26, but I have no idea. How do I have to proceed?
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Does $X \cup M_f$ mean the attaching space $M_f \cup_g X$ where the attaching map $g:A \to M_f$ is given by $g(a)=q(a,0)$? (Here $q:A \times I \coprod B \to M_f$ is the quotient map.)
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I searched google about this question and I found that this question is quite related about "cofibrations" and "pushouts". (I don't know these notions. Hatcher's book is my first AT book.) Are there some references to learn these?
Best Answer
This is a special case (at least in the case $A$ is closed on $X$) of the results discussed in section $7.5$ of Topology and Groupoids, on the homotopy type of adjunction spaces.