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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2009 • Question 14

Question

Math 6510 • Differentiable Manifolds Fall 2009 • Question 14

Let \(M\) be a closed connected 5-manifold such that \(\pi _1(M)=\mathbb {Z}/7\mathbb {Z}\) and \(H_2(M;\mathbb {Z})=\mathbb {Z}^2\). Compute \(H_k(M;\mathbb {Z})\), \(H^k(M;\mathbb {Z})\) and \(H_k(M;\mathbb {Z}/7\mathbb {Z})\) for all \(k\).

Raw LaTeX

Let \(M\) be a closed connected 5-manifold such that
\(\pi_1(M)=\Z/7\Z\) and \(H_2(M;\Z)=\Z^2\). Compute \(H_k(M;\Z)\), \(H^k(M;\Z)\) and
\(H_k(M;\Z/7\Z)\) for all \(k\).

Topic Tags

Cohomology and Cup Products Fundamental Groups Homology with Coefficients Homotopy Groups