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It is very well known that $\mathbb{T}^2$ and $\mathbb{S}^2$ are not homeomorphic, and therefore not diffeomorphic. But I was wondering if there is a possibility of a local diffeomorphism $f:\mathbb{T}^2\to\mathbb{S}^2$ between those two manifolds. Is this possible? I know that between spheres of dimension $n>1$ any local diffeomorphism is actually a diffeomorphism. Can anyone help to answer my curiosity ?

Jacaré
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    $S^2$ has only itself as a covering space. – Ted Shifrin Aug 11 '22 at 01:59
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    (As a connected covering space.) – Ted Shifrin Aug 11 '22 at 17:22
  • @TedShifrin And what about a local diffeomorphism $f:\mathbb{S}^2\to\mathbb{T}^2$? – Jacaré Aug 12 '22 at 00:11
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    This, too, is basic topology. The unique simply connected manifold that covers the torus is the plane. You can use basic intersection theory and transversality to see that there is no surjective smooth map from the sphere to the torus. – Ted Shifrin Aug 12 '22 at 01:17
  • Thanks for the answer! – Jacaré Aug 12 '22 at 20:22
  • @TedShifrin: How about a local or global injection from $T^2$ to $S^2$? I'm thinking such map from $T^2 (S^1 \times S^1)$ is an element of $\pi_2(T^2)$, which is trivial by Hurewicz theorem. This means any such map is nullhomotopic. But I dont see how a n injection would be a contradiction, i.e., would be nontrivial homotopically. – MSIS Aug 30 '22 at 21:10
  • You’ve got your algebraic topology backwards. There is a (smooth) map of degree $1$ (hence one of arbitrary degree) from any surface to the sphere — just map an open disk onto the punctured sphere and send everything else to the puncture. – Ted Shifrin Aug 30 '22 at 22:07

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