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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2016 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2016 • Question 10

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph that is a retract of \(\Sigma \). Prove that: \[ \operatorname {rank}(H_1(X))\le g. \]

Raw LaTeX

Let \(\Sigma\) be a closed oriented surface of genus \(g\) and suppose that
\(X\subset\Sigma\) is a graph that is a retract of \(\Sigma\). Prove that:
\[
  \rank(H_1(X))\le g.
\]

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups