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mieda
United States85 Posts
 ) If you just look at the max spectra for the points and complex manifold topology on varieties over C it really isn't true at all. For example, take any complex torus, which has nontrivial fundamental groups and is a complex curve (in fact an elliptic curve). Which explains why topologically Zariski space isn't so interesting from topological point of view. This is why we use etale fundamental groups instead. Edit: Probably should assume that X is irreducible also ;p  | ||
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incnone
17 Posts
+ Show Spoiler + The mSpec-Zariski topology on the torus T has it that a set is open iff it is cofinite (or empty). So if X is a topological space, then f: X -> T is continuous iff the preimage of every point is closed. So suppose I have a continuous map f from [0,1] into T with f(0) = f(1). I'm going to extend it to a homotopy F : [0,1] x [0,1] -> T with F( *, 1) = pt. Choose any injection G of [0,1] x (0, 1) into T, and define F to be equal to G on [0,1] x (0, 1), and to be some point p on [0, 1] x {1}. Then the preimage of any point q of T is the union of some closed subset f^{-1}(q) of [0,1] x {0} along with possibly a single point of [0,1] x (0,1), and also (if q = p) the line [0,1] x {1} and is therefore certainly closed. So F is continuous and f is nulhomotopic. | ||
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mieda
United States85 Posts
Edit: OP is Edited to clarify this. | ||
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spoolinoveryou
United States503 Posts
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Terranlisk
Singapore1404 Posts
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mieda
United States85 Posts
On September 10 2010 21:39 MyHeroNoob wrote: Is this something they asked you at your interview too? No, it's more something you might try on the side as you read SGA 1 . The question is just for trivia interest, since no one considers traditional fundamental groups on algebraic varieties anyway (especially for positive characteristic case). | ||
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mieda
United States85 Posts
The solution is quite simple / silly in a way: Suppose S^n -> X is any continuous map. Send the interior, D^{n+1} (without the boundary), to the generic point of X. This is continuous and extends the map S^n -> X to D^{n+1} -> X and the latter clearly homotopic to a constant map. | ||
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Muirhead
United States557 Posts
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mieda
United States85 Posts
On September 11 2010 07:16 Muirhead wrote: Where do you go to school mieda? I think the fact that Zariski topology isn't Hausdorff is enough to show you shouldn't be applying methods better suited to CW complexes Sure, as I keep repeating, no one considers these fundamental groups on these spaces anyway. That's why it's a "trivia" :p , just to see what happens when one does :p | ||
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