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This question is related to my another question:Homotopy class of map from torus to sphere?

I want to construct all different homotopy class of map from $T^n \rightarrow S^n$. From above question, the homotopy class is isomorphic to $m\mathbb{Z}\cong \mathbb{Z}$ with $m$ the lowest non-zero degree of map.

My question:

  1. What's the $m$ for the case $T^n\rightarrow S^n$?

  2. How to explicitly construct the map with lowest non-zero degree of map? Then I can combine the $k$ covers of $S^n$ to $S^n$ to get all different homotopy class of $T^n$ to $S^n$.

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  • $\begingroup$ A closed, connected, orientable manifold always admits a degree one map to a sphere of the same dimension. See this question for example. $\endgroup$ – Michael Albanese Nov 7 '18 at 12:21

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