Hatcher Exercise 0.16
Theorem: is contractible.
Proof: Give the cell-complex structure described in Hatcher where the spheres and hemispheres of each dimension are subcomplexes – that is, regarding each as obtained from by adding two -cells which are the components of .
For an arbitrary in , it must be the case that for some . Furthermore, is precisely the boundary of , which is another sub-complex of . So each lies inside a closed disc.
For each , let be the deformation retract of to the represented by half of its boundary, where we take to be a point. Of course, each could retract onto one of two copies of . Distinguish a basepoint (one of the two subcomplexes), and simply define each to retract onto the containing .
Let be a homotopy defined as follows: For a point such that we have in particular for some , let:
The continuity of is worth discussing, as it is not immediately obvious: A map on a CW-complex is continuous if and only if its restriction to every -skeleton is continuous (appendix of Hatcher). For any , note that is ‘motionless’ for the interval , after which it merely performs a finite sequence of continuous homotopies. So the restriction to any skeleton is continuous and is continuous.
It’s clear that , and . Then, by construction, is a homotopy from the identity map on and the constant map to and thus, is contractible.