Let $A$ be an abstract simplicial complex, and let $\partial$ be the top-level boundary operator. Suppose we want to know whether $\ker(\partial)$ contains any terms (besides 0 itself) built by nonnegative linear combinations (i.e. coefficients in $\mathbb{R}_{\ge 0}$) of the top-level simplices. Is this information reflected in any way in the homology groups of $A$? If not, are there any alternate homology-like theories that can make this sort of distinction?
Can homology record the existence of a hole built from nonnegative combinations of simplicies?
1
$\begingroup$
homology-cohomology
simplicial-complex
-
0Is this really a question on homological algebra? I think you are asking if the zero eigenspace of a matrix intersects the non-negative cone outside zero. – 2017-02-14