Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\chi=1$?

share|improve this question

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.