Net bundles over posets and K-theory

TL;DR

This paper explores the theory of net bundles over posets, establishing connections with homotopy, homology, and K-theory, and extending classical topological concepts to this combinatorial setting.

Contribution

It introduces K-theory for posets and develops analogues of classical theorems, linking homotopy, homology, and cohomology in the context of net bundles.

Findings

Established versions of Hurewicz theorems for posets

Defined K-theory for posets and introduced Chern class analogues

Applied results to locally constant bundles over posets

Abstract

We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fibre bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions of classical Hurewicz theorems. Focusing our attention on Hilbert net bundles, we define the K-theory of a poset and introduce functions over the homotopy groupoid satisfying the same formal properties as Chern classes. As when the given poset is a base for the topology of a space, our results apply to the category of locally constant bundles.