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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 7

Show that the total space of the tangent bundle \(T(S^2)\) of the 2-sphere is not diffeomorphic to \(S^2\times \mathbb {R}^2\). Hint: One strategy is to find a 2-sphere in \(T(S^2)\) whose self-intersection number is nonzero and argue that there are no such 2-spheres in \(S^2\times \mathbb {R}^2\).

Raw LaTeX

Show that the total space of the tangent bundle \(T(S^2)\) of the 2-sphere is not
diffeomorphic to \(S^2\times\mathbb{R}^2\). Hint: One strategy is to find a 2-sphere in
\(T(S^2)\) whose self-intersection number is nonzero and argue that there are no such
2-spheres in \(S^2\times\mathbb{R}^2\).

Topic Tags

Tangent Bundles and Vector Bundles Transversality and Intersection Theory