In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3.
Question: How many different geometries (in the sense of Thurston) do we have in dimension 4 ?
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1 Answer
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The 4-dimensional geometries were classified in the unpublished thesis of Filipkiewicz, which is available here.
answered Nov 8, 2014 at 16:14
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3$\begingroup$ Wonderful, Thank you for the reference! Just for the future reader: the total classification is summarized in pages 129-131. The answer is: There is 19 geometries in dimension 4! $\endgroup$– MaxCommented Nov 8, 2014 at 17:45
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1$\begingroup$ At the bottom of page (ii) it says "there is a countable infinite of inequivalent such geometries." $\endgroup$ Commented Nov 8, 2014 at 18:44
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2$\begingroup$ You are right Peter! I need to find a good formulation, it is like there is 19 geometries, and one of them is parametrized by natural numbers. $\endgroup$– MaxCommented Nov 8, 2014 at 18:55