Talk:Homotopy groups of spheres
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Historical surprise?
I searched a while for a source backing the claim this article makes that it "came as a great surprise historically" that the homotopy groups of spheres are not always trivial for i > n and I wasn't able to find a source. I find this concerning because there are some unreliable online sources that make this claim (e.g. Google AI Overview, Wolfram MathWorld), and I think they probably get it from Wikipedia. Can anyone find a source for this claim, or is it a myth, or maybe is it true but unsourceable? Mathwriter2718 (talk) 03:11, 2 January 2025 (UTC)
- It was not known whether a higher-dimensional sphere can have a non-trivial map to a lower-dimensional sphere until Heinz Hopf's 1931 discovery of the Hopf fibration H : S3 —> S2, which is indeed an example of such a non-trivial map. ~2025-34696-49 (talk) 19:28, 28 November 2025 (UTC)
- The fact that it wasn't known until 1931 doesn't by itself imply that it was seen as a "great surprise" by the community when it was discovered. Mathwriter2718 (talk) 04:07, 29 November 2025 (UTC)
False statement
This sentence appears in the section History:
"Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper."
No. Hurewicz may have invented the higher homotopy groups πn(X,x0), homotopy classes of continuous maps of the n-sphere Sn for n > 1 into a space X that fix a basepoint.
But it was Henri Poincaré who discovered the fundamental group π1(X,x0) in his 1895 paper "Analysis situs" — 40 years earlier than the Hurewicz paper.
~2025-34696-49 (talk) 19:17, 28 November 2025 (UTC)
- This is fixed by adding the single word "higher" in front of the phrase "homotopy groups". Just edited the page. Mathwriter2718 (talk) 04:18, 29 November 2025 (UTC)
Another false statement
This sentence currently appears in the second paragraph:
"An 'addition' operation defined on these equivalence classes makes the set of equivalence classes into an abelian group."
This is not necessarily the case for the fundamental group π1, which is very often non-abelian. (For example, the fundamental group of a figure-eight is the free group on two generators.) ~2025-34696-49 (talk) 19:42, 28 November 2025 (UTC)
- That sentence is about
- which is an abelian group for all n, i. Mathwriter2718 (talk) 04:12, 29 November 2025 (UTC)
