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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 5

A. Let \(f:S^n\to T^n\) be continuous, \(n>1\), where \(S^n\) and \(T^n\) denote the \(n\)-sphere, and the \(n\) dimensional torus respectively. Prove that the degree of \(f\) is zero. (If you prefer, you may assume that \(f\) is smooth).

B. Let \(f:X\to Y\) be a covering map between two connected manifolds. Prove that if \(f\) is homotopic to a constant map then \(X\) is contractible.

Raw LaTeX

\textbf{A.} Let \(f:S^n\to T^n\) be continuous, \(n>1\), where \(S^n\) and \(T^n\) denote the
\(n\)-sphere, and the \(n\) dimensional torus respectively. Prove that the degree of \(f\)
is zero. (If you prefer, you may assume that \(f\) is smooth).

\textbf{B.} Let \(f:X\to Y\) be a covering map between two connected manifolds. Prove
that if \(f\) is homotopic to a constant map then \(X\) is contractible.

Topic Tags

Covering Spaces Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds