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.
Categories: math.AT