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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2020 • Question 7

Question

Math 6510 • Differentiable Manifolds Summer 2020 • Question 7

State the homotopy lifting property of covering maps and use it to prove that the covering map \(S^n\to \mathbb {R}\mathrm {P}^n\) from the \(n\)-sphere to the real projective \(n\)-space is not nullhomotopic for any \(n\ge 1\).

Raw LaTeX

State the homotopy lifting property of covering maps and use it to prove
that the covering map \(S^n\to\RP^n\) from the \(n\)-sphere to the real projective
\(n\)-space is not nullhomotopic for any \(n\ge1\).

Topic Tags

Covering Spaces Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Real Projective Spaces