Timeline for Failure of smoothing theory for topological 4-manifolds
Current License: CC BY-SA 2.5
3 events
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| Dec 12, 2009 at 20:11 | comment | added | Tim Perutz | Fair point. If I read Dold-Whitney (Annals 1959) correctly, I can create an $SO(4)$ vector bundle with $w_2=0$ and $p_1$ any even integer. Altering this over a 4-ball, via a map $S^3\to SO(4)$ that factors through $SU(2)$, I can adjust its Euler class freely. In this way I can cook up a "tangent bundle" that underlies the Spivak fibration, provided that $o=0$. One then ought to check, using $ks$, whether it underlies the microbundle. | |
| Dec 12, 2009 at 18:13 | comment | added | John Francis | Assuming the $o=ks$ claim (that I, unfortunately, don't know how to show at the moment), this seems to me to give that the plumbing of $E_8\oplus E_8$ has a stable tangent bundle. This might be standard, but how do you go from that to the unstable statement? A priori, it's plausible that a microbundle $\tau_M \oplus \mathrm{R}^k$ could admit a vector bundle structure even if $\tau_M$ doesn't. | |
| Dec 12, 2009 at 5:18 | history | answered | Tim Perutz | CC BY-SA 2.5 |