Proving that lower homotopy groups of spheres are trivial: does there exist a simpler proof than showing the full cellular approximation theorem?

As the title says, I’d like to know if there is a simple proof for the triviality of the nth homotopy group of Sk if n<k without talking about CW-complexes and the cellular approximation theorem.

Does anyone know any way to do this? Maybe some way to show that every map from Sn to Sk is homotopic to a map which is not surjective?

EDIT: I am also interested if someone knows a way to skip “some” of the details of the cellular approximation theorem, it doesn’t have to be a completely different way. I used the triviality of these homotopy groups in a proof and would like to show why these groups are trivial without taking too much time.

submitted by /u/Melevisione
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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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