My question arose from the proof of proposition 1.26 in Hatchers Algebraic Topology. There he constructs a space $Z$ from a path-connected space $X$ as follows:
- attach a set of 2-cells $e_\alpha$ via maps $\varphi_\alpha : S^1 \rightarrow X$
- choose a basepoint $x_0$ in $X$ and paths $\gamma_\alpha$ from $x_0$ to $\varphi_\alpha(s_0)$ (where $s_0$ is an arbitrary point in $S^1$), then
- for each $\alpha$ take a rectangular strip $S_\alpha = [0,1] \times [0,1]$ and attach one of its edges to $X$ via $\gamma_\alpha$, and the adjacent edge at $\varphi_\alpha(s_0)$ to an arc in $e_\alpha$
- finally glue these strips together along their edges opposite to the ones attached to the $e_\alpha$
There is a picture in the book on page 50.
Now Hatcher claims that
- $Z - X$ is contractible and
- the space obtained from $Z$ by removing one interior point from each $e_\alpha - S_\alpha$ deformation retracts onto $X$
Intuitively this seems quite straightforward, but a detailed proof would get quite messy. That is, unless there is an elegant way of modularizing.
And this is my question: how to prove 1. and 2. elegantly without relying on intuition?
I would also be grateful for references if someone can recommend a book about Algebraic Topology which relies less on intuition.