Timeline for the space of continuous maps between 3-manifolds

6 events

when toggle format	what		by	license	comment
Jul 15, 2015 at 7:20	comment	added	David Roberts♦		@QiaochuYuan yes, I looked it up. And, I forgot to say above, clearly $[M,K(Z,3)] \simeq H^3(M,Z)$!
Jul 15, 2015 at 5:47	comment	added	Qiaochu Yuan		@David: this is a corollary of the Dold-Thom theorem; more generally, the infinite symmetric product of $S^n$ is $B^n \mathbb{Z}$.
Jul 15, 2015 at 5:17	comment	added	David Roberts♦		@QiaochuYuan I didn't know that was a K(Z,3) :-) I was also trying to think of a geometric (smooth!) construction, rather than a topological/homotopical one.
Jul 15, 2015 at 2:14	comment	added	Qiaochu Yuan		@David: what's wrong with the infinite symmetric product of $S^3$?
Jul 15, 2015 at 1:52	comment	added	David Roberts♦		One can see that there is an isomorphism since if $K(Z,3)$ is the appropriate Postnikov stage of $S^3$, whence there is a map $tr_3\colon S^3 \to K(Z,3)$ inducing isomorphisms on homotopy groups in dimension 3 and below, since $M$ is 3-dimensional $tr_{3\ast}\colon[M,S^3] \to [M,K(Z,3)]$ is an isomorphism. I'm trying to think of a geometric model (in case that's useful) for $tr_3$, involving $SU(2)$ and $BPU(H)$, but it's eluding me for now.
Jul 15, 2015 at 1:16	history	answered	Igor Rivin	CC BY-SA 3.0