Yoshinaga - Magnitude homotopy type

Abstract

Magnitude homology group was introduced by Hepworth-Willerton and Leinster-Shulman. It was mentioned that the magnitude homology group can be considered as the reduced homology of certain simplicial set (also by Bottinelli-Kaiser for general metric spaces). Inspired by Asao-Izumihara’s work for finite graphs, we introduce a pair of simplicial complexes whose quotient is isomorphic to the above mentioned simplicial set. The construction is functorial and enables us to apply discrete Morse theory on magnitude homology groups. This is joint work with Y. Tajima.

Motivation

We want to study magnitude homology combinatorically. The idea is to transform a poset into on ordered complex. This allows us to apply morse theory.

Notations:

• $S$ is a simplicial complex

• $M\subseteq \{(\sigma ,\tau )\in S\times S:\sigma \subset \tau ,|\tau \backslash \sigma |=1\}$

$M$ is called an acycling matching iff

• each $\sigma \in S$ apperrs at most one
• There is no cyle ${\sigma }_1⊢{\tau }_1\succ {\sigma }_2⊢{\tau }_2\succ ...+{\tau }_{}\succ {\sigma }_1$. Where ${\sigma }_i⊢{\tau }_i⟺({\sigma }_i,{\tau }_i)\in M$ and ${\tau }_i\succ {\sigma }_{i+1}⟺{\sigma }_{i+1}\subseteq {\tau }_i,|{\tau }_i\backslash {\sigma }_{i+1}|=1$

$C:=\{\sigma \in S:\sigma \text{ does not appear in }M\}$ is called critical cells.

Fundamental theorem of DMT

$|S|\cong \text{ a cw cpx constructed from critical cells}$

Setting

Let $(X,d)$ be a quasimetric space. (no smmetry)

We will constrcut a poset which is suitable for $MH$.

Causal order (poential on space-time)

Take a look at events in space time. One can incluence an other or they are not related. This give a natural poset struture on space time events.

We say $(x_1,t_1)<(x_2,t_2)\in X\times \mathbb{R}$ if the second point lies in thecausality cone of the first as dictated by the metric von $X$. This makes $X\times \mathbb{R}$ a poset. Copare Minkovski timelike and light like.

Def: Causal interval. Let $a,b\in X$. $l\ge 0$. ${\text{Cau}}^l(X,:a,b)=[(a,0),(b,l)]=\{x,t)|(a,0)\le (x,t)\le (b,l)\}$ i.e. The Intersection a forward and backward lightcone.

Defintion: $X$ is a causal space iff there are two orders $<,\ll$s.t. $x_1\ll x_2⟹x_1<x_2$

Mangitude homotopy type

$\Delta Ca{u}^l(a,b)\supset {\Delta }^{\prime }:=\{((x_0,t_{0)}<...<(x_n,t_n)):d(x_0,x_1,...,x_n)<l\}$ Definition: $M^l(a,b)=|\Delta Ca{u}^l(a,b)|/|{\Delta }^{\prime }|$. Quotient: ${\Delta }^{\prime }$ ist contracted to one point.

We have $M{H}_n^l(X):={⨁}_{a,b\in X}M{H}_n^l(X:a,b)$ Theorem:

1. ${\tilde{H}}_n(M^l(X:,a,b))\cong M{H}_n^l(X:a,b)$
2. $M^l(X:a,b)$ is a double suspension of $|\Delta (Ca{u}^l(a,b)\backslash \{(a,0),(b,l)\}/{\Delta }^{\prime }\cap ...\}|$