Filtered order complexes and magnitude homology of finite graded posets

Abstract

In this paper, we study the family of subcomplexes of the order complexes of finite graded posets, defined via its rank function. We address three main topics. \upshape{(1)} We describe the general topological properties of these subcomplexes in relation to magnitude homology of graded posets. \upshape{(2)} For posets whose order complexes are simplicial subdivisions of closed manifolds, we show that the homology groups of these subcomplexes agree with that of the undelying manifold except for the top dimension, where it is a nontrivial free abelian group. \upshape{(3)} For shellable graded posets, we prove that each of the subcomplexes are also shellable. Moreover, in the case of geometric semilattices, we show that each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres of the same dimension.