# Is there a compact contractible manifold?

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
@SammyBlack Well the OP asked for a simply connected manifold at first. –  user38268 Jun 12 '13 at 15:54

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