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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2012 • Question 14

Question

Math 6510 • Differentiable Manifolds Fall 2012 • Question 14

A map \(f:S^n\to S^n\) is said to be even if \(f(-x)=f(x)\) for every \(x\in S^n\). Show that even maps have even degree, and in fact that the degree must be 0 when \(n\) is even.

Raw LaTeX

A map \(f:S^n\to S^n\) is said to be even if \(f(-x)=f(x)\) for every
\(x\in S^n\). Show that even maps have even degree, and in fact that the degree must
be 0 when \(n\) is even.

Topic Tags

Degree Theory Fixed Point Theorems and Borsuk-Ulam Homotopy and Degree on Manifolds