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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2006 • Question 14

Question

Math 6510 • Differentiable Manifolds Winter 2006 • Question 14

Let \(X_d\) be the CW-complex obtained by attaching a disk with a degree \(d\) map to the circle \(S^1\times \{x_0\}\) on the torus \(S^1\times S^1\) (and \(x_0\in S^1\) is a basepoint). Calculate the homology and cohomology of \(X_d\) with \(\mathbb {Z}\), \(\mathbb {R}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

Raw LaTeX

Let \(X_d\) be the CW-complex obtained by attaching a disk with a degree \(d\) map to
the circle \(S^1\times\{x_0\}\) on the torus \(S^1\times S^1\) (and \(x_0\in S^1\) is a
basepoint). Calculate the homology and cohomology of \(X_d\) with \(\mathbb{Z}\),
\(\mathbb{R}\) and \(\mathbb{Z}_p\) coefficients where \(p\) is prime.

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Homology with Coefficients Surface Topology and Surface Groups