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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 10

Let \(X\) be obtained from attaching two tori along a circle in their product structure. In particular each individual torus is \(S^1\times S^1\) and we attach the two tori along circles \(S^1\times \{p\}\) in the first torus and \(S^1\times \{q\}\) in the second torus. Calculate the cohomology and cup product structure on \(X\) with \(\mathbb {Z}\) coefficients. You can use the cup product structure of a torus as a given. (Hint: \(X\) retracts onto each of the individual tori.)

Raw LaTeX

Let \(X\) be obtained from attaching two tori along a circle in their product
structure. In particular each individual torus is \(S^1\times S^1\) and we attach
the two tori along circles \(S^1\times\{p\}\) in the first torus and \(S^1\times\{q\}\)
in the second torus. Calculate the cohomology and cup product structure on \(X\)
with \(\Z\) coefficients. You can use the cup product structure of a torus as a
given. (Hint: \(X\) retracts onto each of the individual tori.)

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Homology with Coefficients Homotopy Equivalence, Retracts, and Contractibility