What is the origin of and motivation for the notion fake weighted projective spaces? Could you please compare this notion to that of genuine weighted projective spaces, giving significant examples? Is there any good reference on fake weighted projective spaces as the one on fake lens spaces? (Thanks to the "user 15817").
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$\begingroup$ Topologists and algebraic geometers have different notions of a "fake" space. For a topologist, a "fake $X$" means a space homotopy-equivalent to $X$ but not CAT-isomorphic to $X$, where CAT is the DIFF, PL, or TOP category, depending on the context. In algebraic geometry, one typically replaces homotopy-equivalence with equality of Betti numbers. Answer to your question would depend on which of these you are interested in. Lastly, your question about Miles Reid is completely meaningless without context. $\endgroup$– MishaCommented Jun 25, 2013 at 10:33
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2$\begingroup$ You should support your allegations about Reid with direct sources (not "I read that"), and you should also not impute mental states to him (as in whether he does something voluntarily or involuntarily). Stick to facts that you can directly support. $\endgroup$– Todd TrimbleCommented Jun 26, 2013 at 16:54
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$\begingroup$ In reply to an comment via a suggested edit: the reason there are now 'fake lens spaces' is that this is what mathoverflow.net/users/35678/al-amrani suggested as an edit, which while not OP-account seemed to be the same person so that I approved and improved (I think, language wise) the edit. In general, please register an account than you can edit you own things directly. Please check the edit history click the time in the middle for the details, to see that the lens spaces originate from somebody with the same display name. $\endgroup$– user9072Commented Jun 28, 2013 at 19:17
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$\begingroup$ Sorry I take the above back in part. I misunderstood the meaning of a phrase. I make the change which I hope is now the intended meaning. Sorry, for the oversight. $\endgroup$– user9072Commented Jun 28, 2013 at 19:20
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$\begingroup$ @quid: While the edit is a major improvement, it still does not address the key (at least in my mind) question about the question: Does the OP address topological or algebro-geometric notion of "fakeness". The tags used in OP do not help here at all. It is like somebody asking the question "What is $H^2$?" $\endgroup$– MishaCommented Jun 29, 2013 at 2:29
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1 Answer
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Everything you would like to know about Fake lens spaces is on http://www.map.mpim-bonn.mpg.de/Fake_lens_spaces by Tibor Macko.
A Fake lens space is the Quotient of a sphere by a cyclic group acting freely and properly discontinuously. (The latter condition equivalent to the group being finite.)
