As a reference, we permit the use of homotopy theorems established in Topology and Geometry [Bredon1993] up till The Whitehead theorem. Exercise 1.Solution.Exercise 1. Let $n\geq 2$ and let $X$ be a homology $n$-sphere. If $X$ is a simply-connected CW-complex, show that $X\simeq \S^n$. Solution. Since $X$ is simply-connected, by inverse Hurewicz, there is an…
We use some of the results established in Topology and Geometry [Bredon1993] implicitly. In particular, the fact that the join of two spheres results in another sphere with degree $n+m+1$ (which can also be proven independently using CW-structure) and some results about Stiefel manifolds. Exercise 1.Solution.Exercise 1. Show that $X$ is Hausdorff if and only…