Timeline for What is the intuition for higher homotopy groups not vanishing?

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May 6, 2021 at 21:51	comment	added	Ryan Budney		This framed cobordism perspective highlights the fact that the only ingredient that is interesting in cobordism of co-dimension one submanifolds is separation. So the homotopy groups of $S^1$ are different from the homotopy groups of $S^k$ with $k>1$ in much the same way that $S^0$ is different from $S^k$ for $k>0$, i.e. $S^0$ is not connected.
Jul 26, 2019 at 22:19	history	edited	Dev Sinha	CC BY-SA 4.0	added 1 character in body
Jul 26, 2019 at 17:07	history	edited	Dev Sinha	CC BY-SA 4.0	added 1 character in body
Jul 26, 2019 at 17:07	comment	added	Dev Sinha		Sorry to be brief: if we consider $\pi_n(S^k)$ then the submanifolds in question are codimension $k$ submanifolds of $\mathbb{R}^n$. (The ambient manifold can be taken to be $\mathbb{R}^n$ rather than $S^n$ because the map is based so the preimage of a non-basepoint will be disjoint from the basepoint of $S^n$.)
Jul 25, 2019 at 22:44	comment	added	D. Zack Garza		This is probably well-known, but could you describe what the ambient manifold is here when you say "submanifolds of Euclidean space"? Is this something like R^\infty with the direct limit topology?
Jul 24, 2019 at 5:00	history	answered	Dev Sinha	CC BY-SA 4.0