ON TOPOLOGICAL COMPLEXITY OF GORENSTEIN SPACES

Year 2024, Early Access, 1 - 16

Smail Benzakı , Youssef Ramı

https://doi.org/10.15672/hujms.1310709

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 compare 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.

Keywords

Higher topological complexity, Eilenberg-Moore functor, Sullivan algebra, Gorenstein spaces

References

• [1] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30, 305-330, 1955.