Timeline for Basic questions on the homotopy category
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 26, 2015 at 1:38 | comment | added | Tom Goodwillie | To explain Mather's quoted comment: If a map $A\to B$ has a kernel $K$ then $\pi_jK\cong ker(\pi_jA\to \pi_jB)$. When $A=K(G,m)$ and $\pi_mB=0$ then this implies $K\sim A$; $A$ is the kernel. But that means that every map $X\to A$ gives trivial $X\to B$, for every $X$. And that means $A\to B$ is trivial. | |
| Feb 3, 2013 at 8:52 | comment | added | Adam Epstein | Well, a lot can happen in 29 years. | |
| Feb 3, 2013 at 2:09 | comment | added | Tom Goodwillie | I'm not sure I knew this when I taught you in Math 272. | |
| Feb 2, 2013 at 9:14 | comment | added | Adam Epstein | True, true. I don't think you got that far when you taught ne in Math 272. | |
| Feb 2, 2013 at 3:32 | comment | added | Tom Goodwillie | Well, the first example of an essential map between E-M spaces of different "dimensions" is the inclusion $\mathbb RP^\infty\to \mathbb CP^\infty$. This has no kernel. I guess neither does the inclusion $\mathbb RP^2\to \mathbb CP^2$. | |
| Feb 2, 2013 at 0:05 | comment | added | Adam Epstein | Thanks, David. Still, is that (failure of) coimit example so much easier than the limit example with Eilenberg-MacLane spaces? I'm no algebraic topologist either, so I wouldn't know one way or the other. Maybe it's the verification that's easier? I'm not particularly comfortable with any of these spaces - I think I'd feel better about an example, say a map with no kernel, arising in the context of compact manifolds, finite simplicial complexes, etc. Can this be arranged? | |
| Feb 1, 2013 at 23:41 | comment | added | David White | Yes, I do happen to know of a simpler example and posted it right on this thread moments after you posted this answer. It's in the book by Jeffrey Strom, and I linked to a google books preview of the relevant pages | |
| Feb 1, 2013 at 23:28 | history | answered | Adam Epstein | CC BY-SA 3.0 |