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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2008 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2008 • Question 8

Let \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Raw LaTeX

Let \(X\) be a compact topological space and \(\widetilde{X}\) its universal
cover. Show that \(\pi_1(X)\) is finite if and only if \(\widetilde{X}\) is compact.

Topic Tags

Covering Spaces Fundamental Groups Homotopy Groups Point-Set Topology and Separation Theorems