Homotopy equivalence of Double mapping cylinder

Question

I shall quote from "Algebraic topology;Tammo Dieck".

Suppose we have inclusions $f_\pm:X_0\subset X_\pm$ and $j_\pm:X_\pm\subset X$ such that $X=X_-\cup X_+$. (If it is necessary, we can assume that the interiors $X^\circ_\pm$ covers $X$ and $X_0=X_1\cap X_2$, then the space $X$ is a pushout in $\textsf{Top}$.)

Let $Z(f_-,f_+)$ be the double mapping cylinder and $N(X_-,X_+)=X_-\times\{0\}\cup X_0\times I\cup X_+\times\{1\}$ be the subspace of $X\times I$. We have a canonical bijective map $\alpha:Z(f_-,f_+)\to N(X_-,X_+)$.

And there is a claim that $\alpha$ is a homotopy equivalence. There is a proof here: (Proposition 4.2.4)

Let $\gamma:I\to I$ be defined by $\gamma(t)=0$ for $t\leq\frac{1}{3}$, $\gamma(t)=1$ for $t\geq\frac{2}{3}$ and $\gamma(t)=3t-1$ for $\frac{1}{3}\leq t\leq\frac{2}{3}$. We define $\beta:N(X_-,X_+)\to Z(f_-,f_+)$ as $1_{X_0}\times\gamma$ on $X_0\times I$ and the identity otherwise. Homotopies $\alpha\beta\simeq1$ and $\beta\alpha\simeq1$ are induced by a linear homotopy in the $I$coordinate.

Q: Is there necessity to define $\gamma$ as above?

I substituted $\gamma$ with another function $\gamma^\prime(t)=t$, namely the identity function on $[0,1]$, and thought there was no problem on the continuity of $\beta$. Of course, I believe I am wrong and there is some reason to define $\gamma$ piecewisely. And if there is an elaboration of the continuity of $\beta$ above, it will be very thankful.

Answer 1

The problem is that $\alpha$ is in general not a homeomorphism. Working with your $\gamma'$ would give $\beta = \alpha^{-1}$ which in general is not continuous.

The benefit of tom Dieck's map $\gamma$ is this. Let $Z_0 = X_-\times\{0\}\cup X_0\times [0,1/3]$, $Z_1 = X_0\times [1/3,2/3]$, $Z_2 = X_0\times [2/3,1]\cup X_+\times\{1\}$. These are three closed subspaces of $N(X_-,X_+)$ whose union is $N(X_-,X_+)$. To show that $\beta$ is continuous it suffices to show that the restrictions $\beta_i = \beta \mid_{Z_i}$ are continuous.

Let $e_\pm : X_\pm \to Z(f_-,f_+)$ be the canonical embeddings and $r_0 : Z_0 \to X_-$ be the restriction of the projection $X_- \times I \to X_-$. Then $\beta_0 = j_- \circ r_0$ which is continuous. Similarly $\beta_2$ is continuous.

Let $\pi : X_0 \times I \to Z(f_-,f_+)$ be the map occuring in the pushout diagram defining $Z(f_-,f_+)$. Then $\beta_1 = \pi \circ (id_{X_0} \times \gamma \mid_{[1/3,2/3]} )$ which is continuous.

Remark: We can of course split up $N(X_-,X_+)$ into the three subspaces $Z'_0 = X_-\times\{0\}$, $Z'_1 = X_0\times I$, $Z'_2 = X_+\times\{1\}$. However, $Z'_1$ is in general not closed, thus piecewise definition of a map on the $Z'_i$ will in general not guarantee continuity on $N(X_-,X_+)$.