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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2011 • Question 10

Question

Math 6510 • Differentiable Manifolds Spring 2011 • Question 10

Let \(M\) be a closed orientable connected 4-manifold with \(H^1(M)=H^3(M)=0\) and \(H^2(M)=H^4(M)=\mathbb {Z}\). What are the possible cup product structures on \(H^*(M)\)? State all theorems you are using. All cohomology groups are with integral coefficients.

Raw LaTeX

Let \(M\) be a closed orientable connected 4-manifold with
\(H^1(M)=H^3(M)=0\) and \(H^2(M)=H^4(M)=\Z\). What are the possible cup product
structures on \(H^*(M)\)? State all theorems you are using. All cohomology groups
are with integral coefficients.

Topic Tags

Cohomology and Cup Products Homology with Coefficients Homotopy and Degree on Manifolds Orientability and Volume Forms