Timeline for Easiest proof of computability of homotopy groups of spheres
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| Feb 11, 2022 at 17:41 | comment | added | Timothy Chow | Here's another interesting reference I found: Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension. But again, I don't think it, or the papers it cites, yields an "easier" proof that homotopy groups of spheres are computable. | |
| Feb 11, 2022 at 16:37 | comment | added | Timothy Chow | @Gro-Tsen I think Joe Shipman is saying that at bottom, these "new" proofs are essentially the same as the original proof. I don't know the subject well, so that may be an accurate assessment. I think the primary value of the modern approach is that it takes a systematic approach to computability questions. So for example, while spectral sequences are definitely a "computational tool" in an informal sense, effective homology insists on computability in a formal sense. I think this is clear conceptual advance, but it may not make specific computability proofs "easier." | |
| Feb 11, 2022 at 14:03 | comment | added | Gro-Tsen | @JoeShipman I don't know anything about the subject (I'm reading this to learn), but in the question you seemed to say that you wanted a simple proof even if it is inefficient, now you seem to be saying that you don't like this approach because it's inherently superexponential. Is this because its description is necessarily complex, or do you care about efficiency after all? | |
| Feb 11, 2022 at 12:49 | comment | added | Joe Shipman | It look like the only proofs that homotopy groups of spheres are effectively computable are still the same proofs that work for ALL simply connected spaces of finite type via Whitehead or Postnikov towers. Sergeraert and others have given us better tools for managing the process, but you must still iteratively and laboriously kill off homotopy groups one dimension at a time by building higher dimensional complexes, an inherently superexponential process (at least 2^(2^n), at most an exponential tower of height n). | |
| Feb 11, 2022 at 12:40 | comment | added | Joe Shipman | I have reviewed all the references.My conclusion: the answer to my original question is “no”. | |
| Feb 11, 2022 at 0:11 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added another reference
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| Feb 11, 2022 at 0:00 | history | answered | Timothy Chow | CC BY-SA 4.0 |