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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 13

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 13

Let \(S^4\) be the 4-dimensional sphere and \(F\subset S^4\) be a smoothly embedded surface of genus \(g\) (this surface is orientable, compact and has no boundary). Compute the homology groups \[ H_k(S^4-F,\mathbb {R}) \] for \(k=2,3\). (You may assume that \(F\) is contained in an open subset \(U\) of \(\mathbb {R}^4\), which is homeomorphic to \(F\times \mathbb {R}^2\) so that \(F\subset U\) corresponds to \(F\times \{0\}\).)

Raw LaTeX

Let \(S^4\) be the 4-dimensional sphere and \(F\subset S^4\) be a smoothly
embedded surface of genus \(g\) (this surface is orientable, compact and has no
boundary). Compute the homology groups
\[
  H_k(S^4-F,\mathbb{R})
\]
for \(k=2,3\). (You may assume that \(F\) is contained in an open subset \(U\) of
\(\mathbb{R}^4\), which is homeomorphic to \(F\times\mathbb{R}^2\) so that \(F\subset U\)
corresponds to \(F\times\{0\}\).)

Topic Tags

Homology with Coefficients Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups