91Ó°ÊÓ

Show that the complement of a finite set of points in \(\mathbb{R}^{n}\) is simply-connected if \(n \geq 3\)

Construct a simply-connected covering space of the space \(X \subset \mathbb{R}^{3}\) that is the union of a sphere and a diameter. Do the same when \(X\) is the union of a sphere and a circle intersecting it in two points.

Let \(X \subset \mathbb{R}^{2}\) be a finite graph that is the union of the edges of a convex polygon and a finite number of line segments having endpoints on these edges. (a) Show that \(\pi_{1}(X)\) is free with a basis consisting of loops formed by the boundaries of the bounded complementary regions of \(X,\) joined to a basepoint by paths in \(X .\) (b) Show this is true for all choices of paths to the basepoint.

Show that for a space \(X,\) the following three conditions are equivalent: (a) Every map \(S^{1} \rightarrow X\) is homotopic to a constant map, with image a point. (b) Every map \(S^{1} \rightarrow X\) extends to a map \(D^{2} \rightarrow X\) (c) \(\boldsymbol{\pi}_{1}\left(X, x_{0}\right)=0\) for all \(x_{0} \in X\)

Let \(X\) be the wedge sum of \(n\) circles, with its natural graph structure, and let \(\tilde{X} \rightarrow X\) be a covering space with \(Y \subset \tilde{X}\) a finite connected subgraph. Show there is a finite graph \(Z \supset Y\) having the same vertices as \(Y,\) such that the projection \(Y \rightarrow X\) extends to a covering space \(Z \rightarrow X\).

Let \(Y\) be path-connected, locally path-connected, and simply-connected, and let \(G_{1}\) and \(G_{2}\) be subgroups of Homeo(Y) defining covering space actions on \(Y .\) Show that the orbit spaces \(Y / G_{1}\) and \(Y / G_{2}\) are homeomorphic iff \(G_{1}\) and \(G_{2}\) are conjugate subgroups of Homeo(Y).