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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 18

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 18

a. Let \(P:M\to N\) be a regular covering map between connected differentiable manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if every deck transformation (covering space automorphism) preserves the orientation of \(M\).

b. Prove that \(\mathbb {R}P^n\) is orientable if and only if \(n\) is odd.

Raw LaTeX

\textbf{a.} Let \(P:M\to N\) be a regular covering map between connected differentiable
manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if
every deck transformation (covering space automorphism) preserves the orientation of
\(M\).

\textbf{b.} Prove that \(\mathbb{R}P^n\) is orientable if and only if \(n\) is odd.

Topic Tags

Covering Spaces Homotopy and Degree on Manifolds Orientability and Volume Forms Real Projective Spaces