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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 12

Let \(S_g\) denote the closed1 1I.e. compact and without boundary. connected orientable surface of genus \(g\). Suppose that \(f:S_m\to S_n\) is a continuous map of degree \(\pm 1\).

  • Show that \(f_\#:\pi _1(S_m)\to \pi _1(S_n)\) is onto.
  • Conclude that \(m\ge n\).

Raw LaTeX

Let \(S_g\) denote the closed\footnote{I.e. compact and without boundary.}
connected orientable surface of genus \(g\). Suppose that
\(f:S_m\to S_n\) is a continuous map of degree \(\pm 1\).

\begin{itemize}
\item Show that \(f_\#:\pi_1(S_m)\to\pi_1(S_n)\) is onto.
\item Conclude that \(m\ge n\).
\end{itemize}

Topic Tags

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