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  • Exercises #6 – Climbing Algebraic Topology – #4

    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…
  • Exercises #6 – Climbing Algebraic Topology – #3

    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…