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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Summer 2022 • Question 6

Question

Math 6520 • Algebraic Topology Summer 2022 • Question 6

Let \(M\) be an orientable \(n\)-manifold that is connected, compact, and without boundary. Let \(f:S^n\to M\) be a continuous map such that the induced map \(f_*:H_n(S^n;\mathbb {Z})\to H_n(M;\mathbb {Z})\) is non-trivial. Calculate \(H^i(M;\mathbb {Q})\) and \(H_i(M;\mathbb {Q})\) for all \(i\).

Raw LaTeX

Let \(M\) be an orientable \(n\)-manifold that is connected, compact, and without boundary. Let \(f:S^n\to M\) be a
continuous map such that the induced map \(f_*:H_n(S^n;\mathbb{Z})\to H_n(M;\mathbb{Z})\) is non-trivial. Calculate \(H^i(M;\mathbb{Q})\) and
\(H_i(M;\mathbb{Q})\) for all \(i\).

Topic Tags

Cohomology and Cup Products Homology with Coefficients