Timeline for Classification of mapping tori
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Feb 27, 2017 at 18:43 | comment | added | Ben Wieland | @DylanThurston Actually, I was wrong. In high dimensions even if $Wh_2$ vanishes, as for free abelian groups, there is an additional obstruction for pretty much any space that is not simply connected. For the nicest spaces around, $S^n\times S^1$ and $(S^1)^n$, the group of homeomorphisms pseudo-isotopic to the identity but not isotopic is the $\mathbb Z/2$ vector space on the set of nontrivial conjugacy classes of elements in the fundamental group (or maybe the subspace invariant under inversion?). Same for diffeomorphisms. | |
| Oct 13, 2016 at 2:43 | comment | added | Ben Wieland | @DylanThurston Right, in 2d pseudoisotopy implies homotopy implies isotopy. In both low and high dimensions (but not smooth 4), the answer is that the difference between isotopy and pseudoisotopy is a higher Whitehead group, which should vanish for aspherical spaces (Borel conjecture). Waldhausen proved the pseudoisotopy=isotopy for Haken 3-manifolds, suggesting that the K-theory gadget vanishes. Hence his seemingly sharp turn into K-theory. (I imagine that for some 3d lens space pseudoisotopy is coarser than isotopy.) | |
| Oct 10, 2016 at 5:31 | comment | added | Dylan Thurston | I'm not sure about the high-dimensional situation, but in two dimensions it's worth noting that pseudoisotopy and isotopy are the same (by a non-obvious theorem). | |
| Apr 22, 2010 at 21:54 | history | answered | Ben Wieland | CC BY-SA 2.5 |