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A m-dumbbell shaped domain is a simply connected domain consisting of m disconnected domain joined by thin tubes. Clearly, because it is simply connected then H^1 is trivial. Moreover is a convex domain but what about the higher order cohomology of this domain? More preciselly, is there an integer d greater than 1 for which H^d is NOT trivial?

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I don't understand where you get convexity from (or what you mean by it). Where is this happening? In $\mathbb{R}^n$? If so, and if it is convex, then it will be contractible and hence have all $H^i=0$ for $i>0$. But, aside from convexity, the 2-sphere (which is two disks joined by a tube) is an example with $H^2\neq 0$. –  HJRW Feb 25 '13 at 14:36
    
What do you mean by thin tube? A neighborhood of an arc? The boundary of a neighborhood of an arc? –  Jim Conant Feb 25 '13 at 14:42
    
The question is ill-posed (lacks definitions and motivation) and any interpretation of it I can figure out are not research-level, I therefore vote to close. –  Benoît Kloeckner Feb 25 '13 at 20:51
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