Timeline for Realizing homomorphisms between fundamental groups
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 16, 2014 at 14:15 | comment | added | Denis Nardin | Every differentiable (or even PL) manifold does. Alternatively you could just replace your manifold with an homotopy equivalent CW complex (from corollary A.12 in Hatcher's book) since your question is not really affected by replacing $X$ with an homotopy equivalent space. | |
| May 16, 2014 at 13:53 | comment | added | William of Baskerville | Thanks for useful answers. Hatcher has it in his book as Proposition 1.b9 if $X$ is a connected CW complex, I found it. Does every compact connected manifold admit a structure of a CW complex? | |
| May 14, 2014 at 21:46 | comment | added | Fernando Muro | Let me also recommend Baues's obstruction theory book. | |
| May 14, 2014 at 17:31 | vote | accept | William of Baskerville | ||
| May 14, 2014 at 16:57 | comment | added | Will Chen | It might also be helpful to note that whenever Y has a contractible universal covering space, then any homomorphism $\pi_1(X)\rightarrow\pi_1(Y)$ is induced by a map $X\rightarrow Y$, unique up to homotopy. This is also in Hatcher. | |
| May 14, 2014 at 16:52 | comment | added | Denis Nardin | I forgot: a good reference for this kind of questions is Hatcher's book on algebraic topology, 4.3 - obstruction theory | |
| May 14, 2014 at 16:50 | history | answered | Denis Nardin | CC BY-SA 3.0 |