12
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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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20
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The 4-dimensional geometries were classified in the unpublished thesis of Filipkiewicz, which is available here.

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    $\begingroup$ 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! $\endgroup$ – Max Nov 8 '14 at 17:45
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    $\begingroup$ At the bottom of page (ii) it says "there is a countable infinite of inequivalent such geometries." $\endgroup$ – Peter Samuelson Nov 8 '14 at 18:44
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    $\begingroup$ 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. $\endgroup$ – Max Nov 8 '14 at 18:55

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