Timeline for Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Current License: CC BY-SA 3.0
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| Apr 26, 2016 at 20:08 | comment | added | HJRW | One last comment. You might also be interested in the discussion under this blog post: ldtopology.wordpress.com/2013/04/23/… . | |
| Apr 26, 2016 at 19:42 | comment | added | HJRW | ... I would certainly regard that as strong motivation. But actually $SL(2,\mathbb{C})$ seems to do quite well: arxiv.org/abs/1510.08493 . | |
| Apr 26, 2016 at 19:41 | comment | added | HJRW | Let's stick to the irreducible case to keep things simple. Then I have in mind the JSJ decomposition, together with the Seifert invariants of the toroidal pieces, the hyperbolic structures of the atoroidal pieces, and the gluing maps. Certainly this is hard to compute in general, but it often works quite well in practice. I agree that this approach is unlikely to, say, find a polynomial-time algorithm to recognize knottedness. Is it conjectured that invariants coming from higher representation varieties might recognize knottedness quickly, say? (cont'd) | |
| Apr 22, 2016 at 16:42 | comment | added | Liviu Nicolaescu | What is the nice complete set of invariants supplied by the geometrization conjecture? What about the classification of knots and links? They are determined by their complements which are $3$-manifolds with boundary. Deciding whether two knots are isotopic is still a very difficult problem. | |
| Apr 22, 2016 at 16:24 | comment | added | HJRW | So the answer is that you might hope to generalize well known existing invariants? It would be nice to know what people then hope to do with those invariants. Since geometrization provides a very nice complete set of invariants for any 3-manifold, the business of finding new invariants for their own sake seems poorly motivated to me. | |
| Apr 20, 2016 at 18:06 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
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| Mar 28, 2016 at 0:56 | comment | added | Liviu Nicolaescu | That's basically the gist of it. You also need to remember that a representation of a fundamental group is a richer object object than a representation of an abstract group since it comes with a locally constant sheaf. | |
| Mar 27, 2016 at 19:06 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Mar 27, 2016 at 17:58 | comment | added | Daniel Moskovich | Thanks! So this answer basically is that representations of 3-manifold groups into any group are interesting, and $SL(n,\mathbb{C})$ is just a class of groups such that (whatever version of) the algebraic variety of representations into it is relatively tractable? | |
| Mar 27, 2016 at 17:51 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |