Is there a non-contractible Simplicial Complex with Euler Characteristic 1?

It seems apparent that all contractible simplicial complexes have Euler characteristic of $\chi=1$. Are there any non-contractible (path-connected) simplicial complexes with Euler characteristic $\chi=1$?

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1 Answer

Yes, many. For a concrete example, how about an octahedron with an edge running through the middle?

It's worth noting that such a complex must necessarily have dimension at least 2, since a path connected 0-complex is just a point, and a noncontractible path-connected 1-complex is homotopy equivalent to a wedge of circles, which has nonpositive Euler characteristic.

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