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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 7

A. Prove the Brouwer Fixed Point Theorem using homology theory. That is, if \(f:B^n\to B^n\) is continuous, where \(B^n\) denotes the closed unit ball in \(\mathbb {R}^n\), then \(f\) has a fixed point.

B. Prove the Fundamental Theorem of Algebra using degree theory.

Raw LaTeX

\textbf{A.} Prove the Brouwer Fixed Point Theorem using homology theory. That is, if
\(f:B^n\to B^n\) is continuous, where \(B^n\) denotes the closed unit ball in
\(\mathbb{R}^n\), then \(f\) has a fixed point.

\textbf{B.} Prove the Fundamental Theorem of Algebra using degree theory.

Topic Tags

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