Quantum methods in algebraic topology (Q2765027)

In one of the standard approaches to rational homotopy theory, we consider the deRham complex on the affine algebraic \(n\)-simplex \( \text{Spec}({\mathbb Q}[x_0,\ldots,x_n]/(x_0 + \cdots + x_n = 1)).\) By letting \(n\) vary over the ordinal number category, we obtain a simplicial differential graded algebra. Then if \(X\) is a simplicial set, the simplicial maps of \(X\) to this dga yield a dga, and this is a good model for the cochains on the space \(X\). The reason this works, of course, is that the Poincaré Lemma holds: the deRham cohomology of the affine \(n\)-simplex over the rationals is trivial. NEWLINENEWLINENEWLINEIf the ground ring is not a \({\mathbb Q}\)-algebra, however, the Poincaré Lemma fails; therefore, attempts to use these methods to study non-rational homotopy have failed, and one has to resort to more exotic methods [\textit{M. A. Mandell}, Topology 40, No. 1, 43-94 (2001; Zbl 0974.55004)]. The point of the paper under review is to find some context where the Poincaré Lemma does hold non-rationally. The main idea is to introduce a variant of the deRham complex; in particular, one uses a ``braided commutative'' deRham complex that arises from the study of quantum groups. Then the program indicated in the previous paragraph can be begun. For example, the author has a result (Theorem 3.6) which says that the resulting differential graded objects detect weak equivalence types of spaces.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00054].