On Topological Complexity of Gorenstein spaces

TL;DR

This paper introduces an algebraic approach to topological complexity for Gorenstein spaces using Sullivan's rational homotopy theory, comparing new invariants with classical ones and emphasizing Adams-Hilton models in specific cases.

Contribution

It develops an $ ext{Ext}$-based algebraic framework for higher topological complexity of Gorenstein spaces and compares it with traditional invariants, highlighting advantages of Adams-Hilton models.

Findings

$ ext{Ext}$-based invariants provide new insights into Gorenstein spaces.

Comparison shows conditions where algebraic invariants differ from classical topological complexity.

Adams-Hilton models are particularly beneficial in odd characteristic fields for certain spaces.

Abstract

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q},C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then make comparisons between these invariants and their respective ordinary ones for Gorenstein spaces. We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.