Timeline for space of homotopy equivalences of $S^1$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 12, 2011 at 22:23 | comment | added | Ian Agol | There may be a way to homotope $HE^+(S^1)$ to "monotonic" maps. For a $\mathbb{Z}$-equivariant map $F:\mathbb{R}\to \mathbb{R}$, define $F'(x)= \max\\{ F(y) | y\leq x\\}$. Then you could straightline homotopy from $F$ to $F'$, maintaining that the map is $\mathbb{Z}$-equivariant. Intuitively, this takes the graph of $F$ and "fills in" the "valleys" to get a monotonic map. This should give a deformation retract to monotonic maps. | |
| May 12, 2011 at 22:12 | comment | added | Ian Agol | Woops, I should have read the question more carefully! This gives a homotopy to SO(2), but obvious not a deformation retract to $Homeo^+(S^1)$. | |
| May 12, 2011 at 21:37 | comment | added | Aaron Magid | Thanks Ian. I believe this is continuous with respect to the topology on HE. You're right that this shows HE retracts to SO(2), but I think Neil is correct to point out I was wrong in my original question since Homeo isn't closed in HE. | |
| May 12, 2011 at 21:33 | vote | accept | Aaron Magid | ||
| May 12, 2011 at 21:33 | |||||
| May 12, 2011 at 20:46 | history | answered | Ian Agol | CC BY-SA 3.0 |