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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2004 • Question 17

Question

Math 6510 • Differentiable Manifolds Fall 2004 • Question 17

Let \(S_g\) be the closed, orientable surface of genus \(g\). Show that there exists a map \(f:S_g\to S_h\) with \[ f_*:H_2(S_g)\to H_2(S_h) \] non-trivial if and only if \(g\ge h\).

Raw LaTeX

Let \(S_g\) be the closed, orientable surface of genus \(g\). Show that there exists
a map \(f:S_g\to S_h\) with
\[
  f_*:H_2(S_g)\to H_2(S_h)
\]
non-trivial if and only if \(g\ge h\).

Topic Tags

Fundamental Groups Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups