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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 8

A. Compute the DeRham cohomology (or if you prefer, the singular homology) of \((S^2\times S^2)\#\mathbb {C}P^2\), where, \(M\#N\) denotes the connected sum of the manifolds \(M\) and \(N\).

B. Let \(p\) be a positive integer. Compute the singular homology with coefficients in \(\mathbb {Z}_p\) of real projective 3-space (\(\mathbb {Z}_p\) being the cyclic group of order \(p\)). That is, find \(H_k(\mathbb {R}P^3,\mathbb {Z}_p)\) for \(k\ge 0\).

Raw LaTeX

\textbf{A.} Compute the DeRham cohomology (or if you prefer, the singular homology) of
\((S^2\times S^2)\#\mathbb{C}P^2\), where, \(M\#N\) denotes the connected sum of the
manifolds \(M\) and \(N\).

\textbf{B.} Let \(p\) be a positive integer. Compute the singular homology with
coefficients in \(\mathbb{Z}_p\) of real projective 3-space (\(\mathbb{Z}_p\) being the
cyclic group of order \(p\)). That is, find
\(H_k(\mathbb{R}P^3,\mathbb{Z}_p)\) for \(k\ge0\).

Topic Tags

Cohomology and Cup Products Homology with Coefficients Real Projective Spaces de Rham Cohomology