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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2017 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 2017 • Question 12

Suppose \(M\) is a closed, orientable, simply connected, smooth manifold of dimension 3. Let \(S_3\) be the symmetric group on 3 letters, and suppose \(S_3\) acts on \(M\) freely, by orientation preserving diffeomorphisms. Let \(N=S_3\backslash M\), and find \(H_n(N;\mathbb {Z})\) and \(H^n(N;\mathbb {Z})\) for all \(n\).

Raw LaTeX

Suppose \(M\) is a closed, orientable, simply connected, smooth manifold of
dimension 3. Let \(S_3\) be the symmetric group on 3 letters, and suppose \(S_3\) acts
on \(M\) freely, by orientation preserving diffeomorphisms. Let \(N=S_3\backslash M\),
and find \(H_n(N;\Z)\) and \(H^n(N;\Z)\) for all \(n\).

Topic Tags

Diffeomorphisms and Isotopy Methods Fundamental Groups Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds