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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 17

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 17

Let \(M\) be a Möbius strip and \(D\) a disk. Let \(f:\partial D\to \partial M\) have degree \(n\). Set \(X_n=M\coprod D/{\sim }\) where \(x\sim y\) if \(f(x)=y\). Calculate \(\pi _1(X_k)\) and construct all of its connected covering spaces.

Raw LaTeX

Let \(M\) be a M\"obius strip and \(D\) a disk. Let \(f:\partial D\to\partial M\) have
degree \(n\). Set \(X_n=M\coprod D/{\sim}\) where \(x\sim y\) if \(f(x)=y\). Calculate
\(\pi_1(X_k)\) and construct all of its connected covering spaces.

Topic Tags

Covering Spaces Fundamental Groups Homotopy Groups Homotopy and Degree on Manifolds