In [Proc. Am. Math. Soc. 144, No. 4, 1829–1840 (2016; Zbl 1336.55008)], Y. Félix et al. described a rational homotopy theoretic model for the Thom space \(\mathrm{Th}(\xi)\) of an oriented vector bundle \(\xi\colon E \to B\) with nilpotent base. The geometric goal there was to specialize to the tangent bundle in order to obtain a (conjectural) Lie model for the configuration space of two points. Here, the authors develop a model without the assumptions of orientability and nilpotency with the ultimate geometric goal of understanding the structure of the Thom space when \(B\) is a smooth complex algebraic variety and then using that to analyze the problem of realizing cohomology classes by submanifolds. For this, the authors recall the notion of a positive weight decomposition on a model (i.e. essentially a decomposition of each of the constituent vector spaces in each degree of the commutative differential graded algebra model that is compatible with the differential and the product in an appropriate sense) and discover a homogeneity condition on the Euler class that implies the Thom space has a positive weight decomposition when the base space does. Furthermore, they show that a smooth complex variety has a positive weight decomposition so that the Thom space \(\mathrm{Th}(\xi)\) of a vector bundle \(\xi\colon E \to B\) with \(B\) a smooth complex variety has positive weights. The point of all this is that spaces (or more precisely, their models) with positive weights may be identified as so-called universal spaces with the property that, under some finiteness assumptions, maps on rationalizations may be lifted to integral space maps up to multiplication by an integer. Thom’s original criterion for representing a cohomology class \(c \in H^k(M;\mathbb{Z})\) asked for a lift of \(c\colon M \to K(\mathbb{Z},k)\) to \(M \to\mathrm{Th}(\xi_u)\) where \(\xi_u\) is the appropriate universal bundle. Here the authors start with a vector bundle \(\xi\colon E \to B\) with \(B\) a smooth complex variety and \(\theta\colon B \to BSO(n)\) a classifying map. They show that a class \(c \in H^k(M;\mathbb{Z})\) is represented by a submanifold via the structure map \(\theta\) exactly when the lifting (as above) is only through \(M \to\mathrm{Th}(\xi)_\mathbb{Q}\), the rationalization of the Thom space of \(\xi\). The point is that, since \(\mathrm{Th}(\xi)_\mathbb{Q}\) has positive weights, a rational map may be lifted to the integral space \(\mathrm{Th}(\xi)\), thus obtaining the Thom criterion.