Does Euler Characteristic > 2 imply not connected

Question

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

Grigory M · Accepted Answer · 2012-11-06 10:43:13Z

up vote 2 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).

answered Nov 6 '12 at 10:43

Grigory M
7,07722150

Thanks. What about for simplicial complexes in general? Are there bounds? – npfedwards Nov 6 '12 at 10:46

3

$\mathbb S^2\vee\ldots\vee \mathbb S^2$, the wedge of $n$ spheres, has Euler characteristic $n+1$. – user17786 Nov 6 '12 at 14:02

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