An algorithm to determine LS-category and Ginsburg invariant of any rationally elliptic space

TL;DR

This paper introduces three algorithms leveraging Gorenstein algebra structures to compute the LS-category and Ginsburg invariant of rationally elliptic spaces, providing effective computational tools for these topological invariants.

Contribution

It presents novel algorithms for calculating the LS-category and Ginsburg invariant of rationally elliptic spaces using algebraic and spectral sequence methods.

Findings

Algorithms successfully compute the rational LS-category.

Algorithms determine the Ginsburg invariant using spectral sequences.

Provides a systematic approach for these topological invariants.

Abstract

Let $X$ be a rationally elliptic space. Utilizing the Gorenstein algebra structure of $X$, we present three algorithms that together induce a generating class of $Ext^N_{(\Lambda V,d)}(\mathbb{Q},(\Lambda V,d))$ with $N$ being the formal dimension of $X$. From these algorithms, we derive an algorithm to compute the rational Lusternik-Schnirelmann category $cat_0(X)$. Furthermore, by applying a spectral sequence argument based on the {\it Eilenberg-Moore spectral sequence}, we compute the rational Ginsburg invariant $l_0(X)$ introduced by M. Ginsburg in \cite{Gin}.