TITLE: A specific question about Proposition 4.36 in Hatcher
QUESTION [1 upvotes]: Let $D^{n}$ denote the closed unit $n$-dimensional ball.
Let $c : D^{n} \rightarrow D^{n} \vee D^{n}$ denote the map collapsing the equatorial plane $D^{n}$ to a point. Let $q_{1}, q_{2} : D^{n} \vee D^{n} \rightarrow D^{n}$ denote the quotient maps onto the first and second summand respectively.
Hatcher claims in the proof of proposition 4.36 that $q_{1}c$ and $q_{2}c$  are homotopic to the identity through maps $(D^{n},\partial D^{n}) \rightarrow (D^{n},\partial D^{n})$. I kinda see why this is reasonable to expect, but I have no idea how to prove this at any level of rigour.
Please can someone help me write down this homotopy explicitly, or help me see why such a homotopy can be written down. As mentioned in the title, this comes from the proof of Proposition 4.36 in Hatcher, where he proves that the Hurewicz map is a group homomorphism.

REPLY [0 votes]: The idea is that we have identifications $D^n/D_{\leq t} \cong D^n$ for every $t$ that can be chosen to "continuously depend on t", where the subscript denotes what part of the disk we are looking at and $-1 \leq t <1$.
Continuous in what sense? Well the composition you wrote down is the map $D^n \rightarrow D^n/D^n_{\leq 0} \cong D^n$ and we have a map $D^n \times I \rightarrow D^n$ which sends $D^n \times \{t\}$ to $D^n$ via the map $D^n \rightarrow D^n/D_{\leq -t} \cong D^n$. So this homotopy starts at the composition you ask about in your question and ends at $D^n \rightarrow D^n/D_{\leq -1} \cong D^n$, but this is just the identity. Since each of these maps preserves the boundaries, we have what we are after.
So if you want something concrete to check, you should check that this homotopy is continuous.
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