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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2019 • Question 10

Question

Math 6510 • Differentiable Manifolds Spring 2019 • Question 10

Let \(X\) be a connected compact orientable manifold of dimension \(n\). Explain carefully how Poincaré duality implies the existence of a bilinear pairing \[ H^k(X;\mathbb {Z})\times H^{n-k}(X;\mathbb {Z})\to \mathbb {Z} \] that modulo torsion becomes a perfect (i.e., non-singular) pairing. Deduce from this the structure of the cohomology ring \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\).

Raw LaTeX

Let \(X\) be a connected compact orientable manifold of dimension \(n\).
Explain carefully how Poincar\'e duality implies the existence of a bilinear
pairing
\[
  H^k(X;\Z)\times H^{n-k}(X;\Z)\to\Z
\]
that modulo torsion becomes a perfect (i.e., non-singular) pairing. Deduce from
this the structure of the cohomology ring \(H^*(\CP^n;\Z)\).

Topic Tags

Cohomology and Cup Products Orientability and Volume Forms Poincare Duality Smooth Manifolds and Embedded Submanifolds