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Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?

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the body of the question doesn't match the title –  user29743 Jun 12 '13 at 15:48
    
@Dominik First you wrote simply connected and then changed the title and your question to involve contractability. –  user38268 Jun 12 '13 at 15:53
    
The sphere $S^n$ for $n \ge 2$ is compact and has trivial fundamental group but is not contractible. –  Sammy Black Jun 12 '13 at 15:53
    
Sorry for the confusion. Now the question is right. –  Dominik Jun 12 '13 at 15:53
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@SammyBlack Well the OP asked for a simply connected manifold at first. –  user38268 Jun 12 '13 at 15:54

1 Answer 1

up vote 5 down vote accepted

No. Closed manifold of dimension $n$ has $H^n(M;\mathbb Z/2)\cong\mathbb Z/2$.

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Could you provide a proof or a reference? –  Dominik Jun 12 '13 at 15:59
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@Dominik see e.g. Wikipedia or section 3.3 of Hatcher's book –  Grigory M Jun 12 '13 at 16:00
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specifically page 236 on this link math.cornell.edu/~hatcher/AT/ATch3.pdf –  citedcorpse Jun 12 '13 at 16:02
    
What you want to look for is Poincare Duality. –  Lee Mosher Jun 22 '13 at 4:10

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