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In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the Q-vector space Ext₂^ (X;Q) (Q, C^ (X;Q) ) with a graded commutative algebra structure. This leads us to introduce the Ext-version of higher (resp. module, homology) topological complexity of X₀, the rationalization of X (resp. of X over 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.
Benzakı et al. (Sun,) studied this question.