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I know that a sphere and a torus are not homeomorphic. But exists a continuous function from the torus to the sphere? or to be not homeomorphic implies that they are imposible (continuous function)

Joe
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1 Answers1

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Given two nonempty topological spaces, there is always a continuous map from one to the other; simply map the first space to a point in the second space. So the question remains, are there any interesting maps, and of course that depends on what you find interesting.

From the sphere $S^2$ to the torus $T^2$ I'd say no. The second homotopy group of the torus $\pi_2(T^2)$ is zero. So every continuous map from $S^2$ to $T^2$ is homotopic to a constant map.

From $T^2$ to $S^2$ I'd say yes. There's a double cover from $T^2$ to $S^2$ (ramified at $4$ points; think elliptic curves). This induces a nontrivial map between the second homology groups $H_2(T^2,\Bbb Z)$ and $H_2(S^2,\Bbb Z)$ (both are isomorphic to $\Bbb Z$) so isn't homotopically trivial.

Angina Seng
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  • What is the map on homology of a branched cover? Is it the degree? – Elle Najt Apr 23 '17 at 06:47
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    There is also a map $T^2 \to S^2$ which induces isomorphism on $H_2$: Think of $T^2$ as the square with sides identified, then the map given by collapsing all four sides to a point is such an example. –  Apr 23 '17 at 06:59