Take the torus $T=S_1 \times S_1$. Choose two points $x,y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental group of $X$. b) Prove that $X$ is not homeomorphic to a surface.
Quotient space of the torus with two points identified
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general-topology
algebraic-topology
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0Do you know how to compute the fundamental group of a cell complex? (if yes, then use it:) For b): the glued point has a connected neighourhood which becomes disconnected when you remove the point. – 2011-05-03
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0Thanks, it's not from a homework, but from a past exam and I was just training. I posted mostly because I wanted to see how to write such a proof with van Kampen's theorem (just to figure out the amount of details required). – 2011-05-03
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0@user8268: Thanks, I think X is homeomorphic to the wedge sum of a torus and a sphere, I would however like to see a formal argument, if possible :-). – 2011-05-03
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2Consider the torus with an added segment (outside it) that goes from x to y. If you contract this segment, you get your space but if you contract an arc within the torus that goes from x to y, you get the wedge sum of a torus and a circle. So your X is homotopy equivalent to (torus v circle) and the Seifert-van Kampen theorem gives you easily the fundamental group. – 2011-05-04
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0@Jacob: can you please explain more clearly why the torus without two points (or without $n$ points in general) is a bouquet of circles? Thanks. – 2011-05-04
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0@Beatty: please do not use answers to make comments. I have merged your accounts, so you should be able to comment on your own questions now, at least. – 2011-05-04