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:
- 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.
- 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?
- 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?
Best Answer
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.