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If you accept that every map from a contractible space is trivial, notice that{ $S^n -x$} is iso. to $\mathbb R^{n-1}$. Then $f$ can be expressed/"factored" as a composition of maps, one map with domain $\mathbb R^n$. For $\pi_1$ use the same "factorization" of f and functoriality properties of $\pi_1$. EDIT Note we use simplicial approximation , as ...


One particular consequence of simplicial approximation of maps $f:S^m\to S^n$ with $m<n$ is that any such map is homotopic to one which is not surjective: indeed, a simplicial map maps simplice to simplices of no greater dimension, and it follows than any point in the interior of an $n$-dimensional simplex of the codomain is not in the image. Now it is ...


This is a useful construction in homotopy theory and is probably easiest explained as the augmented simplex category.


The claim is a special case of Example 23.8 of [Shulman, Homotopy limits and colimits and enriched homotopy theory].


Self-answer: This is the clique complex.

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