If $Z$ is a strong deformation retract of $Y$ and $Y$ is a strong deformation retract of $X$, then $Z$ is a strong deformation retract of $X$.

Question

If $Z$ is a strong deformation retract of $Y$ and $Y$ is a strong deformation retract of $X$, then $Z$ is a strong deformation retract of $X$.

I can see that this boils down to finding a function $F_3: X \times [0,1] \rightarrow Z$ such that $$F_3(x,0) = 1_X, F_3(x,1) = r_z(x) \text{ retract into $Z$ }, F_3(z,t) = z, z \in Z$$ where $F_1$ and $F_2$ are the strong deformation retracts of $YZ$ and $XY$, respectively.

But I'm having trouble thinking about how to find a relative homotopy here.

Anyone have any ideas?

Answer 1

You have $F_1:Y\times [0,1]\rightarrow Y$ and $F_2:X\times [0,1]\rightarrow X$. Define $F_3:X\times [0,1]\rightarrow X$ by $F_3(x,t)=F_2(x,2t), t\leq 1/2$, $F_3(x,t)=F_1(F_2(x,1),2t-1), t\geq 1/2$.