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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 17

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 17

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle in the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus. Compute \(\pi _1(X)\) and describe the universal cover of \(X\).

Raw LaTeX

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a
M\"obius band via a homeomorphism from the boundary circle in the M\"obius band to
the circle \(S^1\times\{x_0\}\) in the torus. Compute \(\pi_1(X)\) and describe the
universal cover of \(X\).

Topic Tags

CW Complexes and Cell Attachments Covering Spaces Fundamental Groups Homotopy Groups