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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2010 • Question 13

Question

Math 6510 • Differentiable Manifolds Spring 2010 • Question 13

Let \(S^3=\{(z,w)\in \mathbb {C}^2\mid |z|^2+|w|^2=1\}\) and \(S^1\subset S^3\) is given by \(w=0\). Let \(X\) be the quotient space of \(S^3\) obtained by collapsing \(S^1\) to a point. Use any method to compute \(H_k(X;\mathbb {Z})\) for all \(k\in \mathbb {Z}\).

Raw LaTeX

Let \(S^3=\{(z,w)\in\C^2\mid |z|^2+|w|^2=1\}\) and \(S^1\subset S^3\) is given
by \(w=0\). Let \(X\) be the quotient space of \(S^3\) obtained by collapsing \(S^1\) to
a point. Use any method to compute \(H_k(X;\Z)\) for all \(k\in\Z\).

Topic Tags

Fundamental Groups Homology with Coefficients