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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2008 • Question 14

Question

Math 6510 • Differentiable Manifolds Spring 2008 • Question 14

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph which 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 which 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