Let $\Sigma$ be the doubled torus (a compact oriented) surface of genus 2) and let $T$ be the torus. Suppose $f: \Sigma \rightarrow T$. Prove that $f$ is not a local homeomorphism.

Attempt at solution: Suppose $f$ was a local homeomorphism. Then by this exercise A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces it would be a covering map. Covering maps induce an injection from the fundamental group of the covering to that of the base. However, for the doubled torus, the fundamental group is given by $\langle a,b,c,d\mid aba^{-1}b^{-1}cdc^{-1}d^{-1}\rangle$ while that of the torus is $\langle a,b\mid aba^{-1}b^{-1}\rangle$, hence no such injection can exist.

  • $\begingroup$ the number of holes (genus of surface) is topological invariant, as such any continuous map from 2-torus to 1-torus would necessarily swallow one of the holes of the 2-torus $\endgroup$ – Nikos M. Aug 15 '14 at 21:05
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    $\begingroup$ @NikosM., what do you mean by "topological invariant"? I generally hear this term used for things that are invariant under homeomorphism, but not necessarily under all continuous maps. For example, the map from any topological space to the one-point space is always continuous, but it doesn't preserve the number of connected components, which is a topological invariant. $\endgroup$ – Vectornaut Aug 15 '14 at 21:12
  • $\begingroup$ @Vectornautm yes exactly invariant under homeomorphisms which i think is what is meant in the question by continuous map $\endgroup$ – Nikos M. Aug 15 '14 at 21:14
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    $\begingroup$ @NikosM. A local homeomorphism is quite different from a homeomorphism, and many topological invariants, like the number of connected components and the Euler characteristic, are not invariant under local homeomorphism. $\endgroup$ – Vectornaut Aug 15 '14 at 21:20
  • $\begingroup$ @Vectornaut, thanx, i missed that part! $\endgroup$ – Nikos M. Aug 16 '14 at 0:38

If you have the machinery of the Euler characteristic available, you can use the fact that if $f$ were an $n$-fold covering map, the Euler characteristic of $\Sigma$ would have to be $n$ times the Euler characteristic of $T$.


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