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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2015 • Question 11

Question

Math 6510 • Differentiable Manifolds Spring 2015 • Question 11

Let \(X\) be a connected surface with nonempty boundary \(\partial X\) and \(p\in X\). Show that inclusion \[ X\backslash \{p\}\hookrightarrow X \] induces an isomorphism in \(H_1\) if and only if \(p\in \partial X\). You may use the fact that \(H_2(X)=0\) without proof.

Raw LaTeX

Let \(X\) be a connected surface with nonempty boundary \(\partial X\) and
\(p\in X\). Show that inclusion
\[
  X\backslash\{p\}\hookrightarrow X
\]
induces an isomorphism in \(H_1\) if and only if \(p\in\partial X\). You may use the
fact that \(H_2(X)=0\) without proof.

Topic Tags

Fundamental Groups Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups