Talk:List of geometric topology topics

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Mathematics Mid‑priority

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Examples versus Definitions[edit]

Ok, so where does Seifert Fibered Space belong in the three-dimensional list? I would argue that SFS's are important examples, but not a foundational part of three-manifold topology, say in the say way that Moise's theorem, the Loop theorem, or Geometrization are foundational. The three-sphere is perhaps the "most important" example (closely followed by the figure eight knot complement). Best, Sam nead 18:20, 2 January 2007 (UTC)[reply]

Hmmm. I changed my mind. Sam nead 18:31, 2 January 2007 (UTC)[reply]

Looks quite good to me now. My only remaining quibble is that Graph manifolds (in the sense of Waldhausen) are more general than Seifert fibered spaces (in fact, they strictly include the latter class), so shouldn't they be moved up a notch, too? Turgidson 22:07, 2 January 2007 (UTC)[reply]

One more thought. Under the Examples of 3-manifolds heading, I would very much like to add the Poincaré sphere -- after all, it has been the most historically motivating example of them all! Unfortunately, that wikilink goes to Homology spheres, which is something that includes high-dimensional homology spheres, as well. My inclination would be to have separate pages -- one for the general theory of homology spheres, the other for the specific example of Poincaré, which I think merits an article all by itself. I can do some of the work on this when I get the time, if more people think it's a good idea... Turgidson 22:23, 2 January 2007 (UTC)[reply]

The PHS (Poincare Homology Sphere) definitely deserves its own page. See Rolfsen's book for a nice discussion. Cameron Gordon also has a wonderful historical article -- I'll try to find the title, since it doesn't appear to be on-line. I didn't move graph mainfolds up because I don't feel that they are "foundational", I suppose. ps This page is beginning to have large overlap with the three-manifold page. Best, Sam nead 15:10, 3 January 2007 (UTC)[reply]

My personal favorite on the subject is the following paper:

Kirby, R. C.; Scharlemann, M. G., Eight faces of the Poincaré homology 3-sphere. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113--146, Academic Press, New York-London, 1979.

There are 8 different definitions in there, all shown to be equivalent. Very nicely explained. Turgidson 19:06, 3 January 2007 (UTC)[reply]

This page is beginning to have large overlap with the three-manifold page. Yes, indeed. But note that there is in there a category titled Some important classes of 3-manifolds which includes Haken, SFS, and, yes, Graph manifolds (which, by the way, I still think as foundational, but à chacun son goût, as they say). Note though that again the various level at which one should classify these objects get mixed, eg, spherical 3 manifolds and lens spaces on the same level, when of course the latter are a particular case of the former. Maybe there is a better way to organize the taxonomy of (3-) manifolds than the current way, which is not quite up to encyclopedia standards, I think... Turgidson 19:16, 3 January 2007 (UTC)[reply]