Stable homotopy, 1-dimensional NCCW complexes, and Property (H)

TL;DR

This paper proves that homomorphisms between certain one-dimensional NCCW complexes are stably homotopic if they share the same KK-class, and establishes that all such complexes have Property (H).

Contribution

It demonstrates that homomorphisms with the same KK-class are stably homotopic and confirms Property (H) for all one-dimensional NCCW complexes.

Findings

Homomorphisms with the same KK-class are stably homotopic.

All one-dimensional NCCW complexes possess Property (H).

Establishes a link between KK-theory and homotopy properties.

Abstract

In this paper, we show that the homomorphisms between two unital one-dimensional NCCW complexes with the same KK-class are stably homotopic, i.e., with adding on a common homomorphism (with finite dimensional image), they are homotopic. As a consequence, any one-dimensional NCCW complex has the Property (H).