Timeline for Are there pairs of highly connected finite CW-complexes with the same homotopy groups?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 11, 2020 at 0:55 | comment | added | Connor Malin | Shouldn't the unit sphere bundle of $TS^{2k-1}$ give a wealth of easy examples. It has a section, but is not trivial (for k large) by the Hopf invariant 1 problem. It's even the case by the Thom isomorphism (I believe) that this sphere bundle has the same homology as $S^{2k-1} \times S^{2k-2}$. | |
| Apr 8, 2013 at 6:48 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
added latex to answer
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| Jan 24, 2011 at 14:09 | comment | added | John Klein | I do realize that you are using Bott + $J$ to construct many examples for $(n,k)$ arbitrarily large. What I'm merely trying to point about is that once you know that $\pi_{k-1}^{\text{st}}(S^0)$ is non-trivial, you can cook up an example for some $n$ for that $k$. | |
| Jan 24, 2011 at 13:25 | comment | added | John Klein | Allen: you can avoid the J-homomorphism and Bott periodicity. All you need is a fibration $S^{k-1}\to E\to S^n$ with section given by clutching a essential map $S^{k-1} \to G_n \to F_n$ ($G_n$ = unbased homotopy automorphisms of $S^{n-1}$, $F_n$ = based homotopy automorphisms of $S^n$). Then the attaching map for the Thom complex is given by the Hopf construction of the adjoint to the clutching map $S^{k-1}\times S^{n-1} \to S^{n-1}$ and this is non-trivial because of essentialness of the clutching map. We only require non-triviality homotopy groups of spheres for this argument + Freudenthal. | |
| Nov 13, 2009 at 20:15 | vote | accept | Charles Rezk | ||
| Nov 13, 2009 at 20:00 | history | answered | Allen Hatcher | CC BY-SA 2.5 |