Abstract
The invariance of magnitudes for graphs differ by a Whitney twist (with a certain condition) was proved by Leinster, and the problem that “Are their magnitude homology groups isomorphic” is still open. We approach the problem using discrete Morse theory on magnitude homotopy types. In the course of this approach, we had another proof of the invariance of magnitudes under a Whitney twist. This is joint work with Masahiko Yoshinaga (Osaka University).
Introduction
Every graph has a magnitude homotopy type. It is defined as a refinement of the Magnitude homology. The homology can be reconstructed by applying the reduced homology.
Magnitude homotopy type
Define the magnitude homology as alwas.
The magnitude homotopy type