The Up Topology for Mirrored Topological Posets

TL;DR

This paper extends McCord's theorem to topological posets, showing that the order complex and the poset with the Up topology are weak homotopy equivalent, enabling transfer of homotopy information in topological combinatorics.

Contribution

It proves an analog of McCord's theorem for a broad class of topological posets, linking their order complex to the Up topology in terms of weak homotopy equivalence.

Findings

The natural map from the order complex to the topological poset with Up topology is a weak homotopy equivalence.

This result applies to the Grassmann poset, allowing transfer of homotopy type information.

The theorem generalizes McCord's classical result to a wider class of topological posets.

Abstract

For a discrete poset $\mathcal X$, McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$, from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, \u{Z}ivaljevi\'{c} defined the notion of order complex for a topological poset. For a large class of such topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. An example is the Grassmann poset of proper non-zero linear subspaces of $\R^{n+1}$. Here, Vassiliev had computed the homotopy type of the order complex. Our theorem allows us to transfer that information (up to weak homotopy type) to the Grassmann poset itself with the Up topology.