Timeline for When is a classifying space a topological manifold?
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13 events
| when toggle format | what | by | license | comment | |
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| Mar 8, 2012 at 21:17 | comment | added | Misha | Henry, since Davis' examples, Wall's conjecture reads as: Every $PD(n)$ group $G$ admitting a finite $K(G,1)$ is the fundamental group of a closed aspherical $n$-manifold. There are ample reasons to believe that this conjecture holds for $n=3$ (e.g., via connection to Cannon's conjecture) and ample reasons to disbelieve for $n\ge 5$ (e.g., existence of "exotic" manifolds of Bryant-Ferry-Mio-Weinberger). | |
| Mar 8, 2012 at 21:07 | vote | accept | berl13 | ||
| Mar 8, 2012 at 19:20 | answer | added | Misha | timeline score: 11 | |
| Mar 8, 2012 at 19:19 | comment | added | berl13 | Thank you very much for your answer. I believe I understand now how it works. | |
| Mar 8, 2012 at 17:34 | comment | added | HJRW | Misha - by 'Wall's conjecture', do you mean the conjecture that every PD_n group is the fundamental group of a closed, orientable n-manifold? I thought there were non-finitely presentable counter-examples to this... | |
| Mar 8, 2012 at 16:35 | comment | added | Misha | Eilenberg-Ganea theorem states that if $G$ had cohomological dimension $n$ then geometric dimension $gd(G)$ of $G$ is at most $n+1$. Furthermore, unless $n=2$, they proved that $gd(G)=n$. Hence, the conjecture. | |
| Mar 8, 2012 at 15:55 | comment | added | berl13 | Thanks for your answer. I do not insist on the manifold being closed. What is the Eilenberg-Ganea theorem? I know only the conjecture which was proven to be false for groups of cohomological dimension larger than 2. | |
| Mar 8, 2012 at 15:47 | comment | added | Misha | It all depends on your definition of a topological manifold. If you use the textbook definition, then the result is: $G$ admits a manifold $K(G,1)$ iff $G$ is countable and of finite cohomological dimension. Proof is a combination of a theorems by Eilenberg-Ganea, Whitehead (Theorem 13 from "Combinatorial Homotopy-I"), and Whitney's embedding theorem for locally finite CW complexes. However, if you insist on your manifold being closed, then you are facing Wall's conjecture that most topologists do not believe in. | |
| Mar 8, 2012 at 15:17 | comment | added | Xiaolei Wu | The following page provides some useful information: map.mpim-bonn.mpg.de/Aspherical_manifolds | |
| Mar 8, 2012 at 14:03 | comment | added | berl13 | I have edited the question in asking only for homotopy equivalence. | |
| Mar 8, 2012 at 14:02 | history | edited | berl13 | CC BY-SA 3.0 |
added 54 characters in body
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| Mar 8, 2012 at 12:47 | answer | added | Ulrich Pennig | timeline score: 3 | |
| Mar 8, 2012 at 12:20 | history | asked | berl13 | CC BY-SA 3.0 |