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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 11

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 11

Apply the Lefschetz fixed point theorem to show that every map \(f:\mathbb {C}\mathrm {P}^n\longrightarrow \mathbb {C}\mathrm {P}^n\) has a fixed point when \(n\) is even (state the Lefschetz fixed point theorem and any fact about the ring structure of \(H^*(\mathbb {C}\mathrm {P}^n)\) you are using). Construct a fixed point free map \(f:\mathbb {C}\mathrm {P}^n\longrightarrow \mathbb {C}\mathrm {P}^n\) when \(n\) is odd.

Raw LaTeX

Apply the Lefschetz fixed point theorem to show that every map
\(f:\CP^n\longrightarrow\CP^n\) has a fixed point when \(n\) is even (state the
Lefschetz fixed point theorem and any fact about the ring structure of
\(H^*(\CP^n)\) you are using). Construct a fixed point free map
\(f:\CP^n\longrightarrow\CP^n\) when \(n\) is odd.

Topic Tags

Cohomology and Cup Products Euler Characteristic and Lefschetz Fixed Point Theorem Fixed Point Theorems and Borsuk-Ulam Homology with Coefficients