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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2017 • Question 11

Question

Math 6510 • Differentiable Manifolds Fall 2017 • Question 11

Suppose \(f:S^k\to S^k\) is continuous and not surjective. Prove that \(f_*:H_k(S^k;\mathbb {Z})\to H_k(S^k;\mathbb {Z})\) is the zero homomorphism.

Raw LaTeX

Suppose \(f:S^k\to S^k\) is continuous and not surjective. Prove that
\(f_*:H_k(S^k;\Z)\to H_k(S^k;\Z)\) is the zero homomorphism.

Topic Tags

Fundamental Groups Homology with Coefficients Lie Group Actions and Homomorphisms