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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2006 • Question 13

Question

Math 6510 • Differentiable Manifolds Fall 2006 • Question 13

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

Raw LaTeX

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