CW complex is contractible if union of contractible subcomplexes with contractible intersection

Question

Exercise 0.23 from Algebraic Topology by Hatcher reads:

Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible.

I have an intuitive idea that should lead to a proof, but I do not know how to justify some steps. I want to argue as follows:

1. The projection $A \cup B \rightarrow \dfrac{A \cup B}{A \cap B}$ is a homotopy equivalence. This follows because $A \cap B$ is a contractible subcomplex.
2. After $A \cap B$ is collapsed to a point, we are left with the wedge sum of $\dfrac{A}{A \cap B}$ and $\dfrac{B}{A \cap B}$. This is intuitively clear, but how can we prove it?
3. I now want to claim that both spaces involved in the wedge sum are contractible, showing that the original space is contractible. How can we prove the former claim?

Answer 1

Obviously $A/A\cap B$ and $B/A\cap B$ cover $A\cup B/A\cap B$. Their intersection is a point. Indeed, take any point $p$ in their intersection. Take pre-image of $p$ under the quotient $A\cup B\to A\cup B/A\cap B$. It will be a subset of the intersection of pre-images, i.e. a subset of the $A\cup B$. Thus $p$ is the unique point in the intersection.

Since $A\cap B$ is contractible, the quotient $A/A\cap B$ is homotopy equivalent to $A$ itself. But $A$ is contractible. So you are done.

Answer 2

It should be pointed out that this is a special case of a general gluing theorem for homotopy equivalences. For more references see this mathoverflow question:

https://mathoverflow.net/questions/96071/general-gluing-theorem-for-adjunction-spaces