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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 9

For \(n\ge 3\) let \(X=S^{n-1}\times S^1\) be the product of an \((n-1)\)-sphere and a circle and let \(Y\) be obtained by attaching an \(n\)-cell to \(X\) where the attaching map is a degree \(k\) from the boundary of the cell to the \((n-1)\)-sphere \(S^{n-1}\times \{p\}\subset X\) via a degree \(k\) map. Calculate the homology of \(Y\).

Raw LaTeX

For \(n\ge3\) let \(X=S^{n-1}\times S^1\) be the product of an \((n-1)\)-sphere and
a circle and let \(Y\) be obtained by attaching an \(n\)-cell to \(X\) where the
attaching map is a degree \(k\) from the boundary of the cell to the \((n-1)\)-sphere
\(S^{n-1}\times\{p\}\subset X\) via a degree \(k\) map. Calculate the homology of \(Y\).

Topic Tags

CW Complexes and Cell Attachments Degree Theory Homology with Coefficients Homotopy and Degree on Manifolds