91Ó°ÊÓ

Show that a finite product of covering maps is a covering map: If \(p_{i}: \widetilde{X}_{i} \rightarrow X_{i}\) are covering maps for \(i=1, \ldots, n\), then so is the map $$ p_{1} \times \cdots \times p_{n}: \bar{X}_{1} \times \cdots \times \bar{X}_{n} \rightarrow X_{1} \times \cdots \times X_{n} $$

Suppose \(p: \widetilde{X} \rightarrow X\) is a covering map and \(X\) is a compact manifold. Show that \(\bar{X}\) is compact if and only if \(p\) is a finite-sheeted covering.

Let \(S\) be the following subset of \(\mathbb{C}^{2}\) : $$ S=\left\\{(z, w): w^{2}=z, w \neq 0\right\\} $$ (It is the graph of the two-valued complex square root "function" described in Chapter 1, with the origin removed.) Show that the projection \(\pi_{1}: \mathbb{C}^{2} \rightarrow \mathbb{C}\) onto the first coordinate restricts to a twosheeted covering map \(p: S \rightarrow \mathbb{C} \backslash\\{0\\}\).

Let \(\widetilde{M}, M\), and \(N\) be connected manifolds of dimension \(n\) and suppose \(p: \widetilde{M} \rightarrow M\) is a \(k\)-sheeted covering map. Show that there exists a \(k-\) sheeted covering of \(M \\# N\) by the connected sum of \(M\) with \(k\) copies of \(N\). [Hint: Choose the ball to be cut out of \(M\) to lie inside an evenly covered neighborhood.]

Show that a proper local homeomorphism between connected, path connected, and locally compact Hausdorff spaces is a covering map.

Show that every even map \(f: \mathbb{S}^{1} \rightarrow \mathbb{S}^{1}\) has even degree.

Suppose \(X\) is a compact polyhedron and \(p: \tilde{X} \rightarrow X\) is a covering map. (a) Show that \(X\) and \(\tilde{X}\) admit triangulations such that \(p\) is induced by a simplicial map. [Hint: Use barycentric subdivision.] (b) Suppose \(\mathcal{K}, \widetilde{\mathcal{K}}\) are finite complexes such that \(|\mathcal{K}|=X,|\widetilde{\mathcal{K}}|=\vec{X}\), and \(p\) is induced by a simplicial map from \(\widetilde{\mathcal{K}}\) to \(\mathcal{K}\). If \(p\) is an \(n\)-sheeted covering, show that \(\chi(\widetilde{\mathcal{K}})=n \chi(\mathcal{K})\).