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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 15

Prove that \(\mathbb {C}P^3\) and \(S^2\times S^4\) have cell complex structures with isomorphic cellular chain complexes, and therefore \(H^i(\mathbb {C}P^3;\mathbb {Z})\cong H^i(S^2\times S^4;\mathbb {Z})\) for all \(i\). Also prove that cohomology rings \(H^*(\mathbb {C}P^3;\mathbb {Z})\) and \(H^*(S^2\times S^4;\mathbb {Z})\) are not isomorphic. Are \(\mathbb {C}P^3\) and \(S^2\times S^4\) homotopy equivalent? You are allowed to use the standard cell complex structure of spheres and complex projective spaces.

Raw LaTeX

Prove that \(\mathbb{C}P^3\) and \(S^2\times S^4\) have cell complex structures
with isomorphic cellular chain complexes, and therefore
\(H^i(\mathbb{C}P^3;\Z)\cong H^i(S^2\times S^4;\Z)\) for all \(i\). Also prove that
cohomology rings \(H^*(\mathbb{C}P^3;\Z)\) and \(H^*(S^2\times S^4;\Z)\) are not
isomorphic. Are \(\mathbb{C}P^3\) and \(S^2\times S^4\) homotopy equivalent? You are
allowed to use the standard cell complex structure of spheres and complex
projective spaces.

Topic Tags

CW Complexes and Cell Attachments Cellular Homology Chain Complexes and Chain Homotopy Homotopy Equivalence, Retracts, and Contractibility