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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 13

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 13

Let \(S^n\) denote the unit sphere of \(\mathbb {R}^{n+1}\) and let \(a:S^n\to S^n\) denote the antipodal map \(a(x)=-x\). Show that \(S^n\) admits a nonvanishing tangent vector field if and only if \(a\) is homotopic to the identity. For which values of \(n\) does this happen?

Raw LaTeX

Let \(S^n\) denote the unit sphere of \(\mathbb{R}^{n+1}\) and let
\(a:S^n\to S^n\) denote the antipodal map \(a(x)=-x\). Show that \(S^n\) admits a
nonvanishing tangent vector field if and only if \(a\) is homotopic to the identity.
For which values of \(n\) does this happen?

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Left-Invariant Vector Fields Tangent Bundles and Vector Bundles Vector Fields and Flows