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For any given topological space $X$ does $\chi(X)>2 \Rightarrow $ more than 1 connected component?

If not, when does it. And if true, can someone point me towards a proof. Thanks

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up vote 3 down vote accepted

Certainly not. For example, $\chi(\mathbb CP^n)=n+1$, but of course $\mathbb CP^n$ is connected.

It is true for surfaces, though: Euler char of any connected surface $\le2$ (this follows, for example, from the classification of surfaces).

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Thanks. What about for simplicial complexes in general? Are there bounds? –  npfedwards Nov 6 '12 at 10:46
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$\mathbb S^2\vee\ldots\vee \mathbb S^2$, the wedge of $n$ spheres, has Euler characteristic $n+1$. –  user17786 Nov 6 '12 at 14:02
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