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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 15

a. Let \(S^3\) be the 3-sphere, and let \(T^2=S^1\times S^1\) be the 2-torus. Compute the singular homology of \(S^3\times T^2\) using its CW-complex structure.

b. Recompute the singular homology of \(S^3\times T^2\) using the Eilenberg-Zilber and Künneth Theorems.

Raw LaTeX

\textbf{a.} Let \(S^3\) be the 3-sphere, and let \(T^2=S^1\times S^1\) be the 2-torus.
Compute the singular homology of \(S^3\times T^2\) using its CW-complex structure.

\textbf{b.} Recompute the singular homology of \(S^3\times T^2\) using the
Eilenberg-Zilber and K\"unneth Theorems.

Topic Tags

Cellular Homology Homology with Coefficients Relative Homology and Long Exact Sequences Surface Topology and Surface Groups