Moduli spaces for generic low-dimensional complexes
A space (DGA) \(S\) \(((A,d))\) is marked if there is a specified isomorphism \(H^* (S; \mathbb{Q}) \cong H\) \((H^* (A, d)\) \(\cong H)\) for a given algebra \(H\). In this paper, the authors describe a moduli space classifying marked homotopy types of spaces (DGA's). The moduli space is defined in terms of differentials on the rational homotopy theoretic bigraded model associated to \(H\). Because the structure of the bigraded model is rather complicated in high degrees, the authors focus on situations where only low dimensional information is required. In particular, they introduce notions of aspheric and \({\mathbf L}\)-fibred algebras in order to make low dimensional information (e.g. the Maltsev completion of \(\pi_1\)) deterministic. Indeed, then a `finite part' of the moduli space may be used in place of the full moduli space for certain purposes of classification. A main result proved here is the fact that, given an aspheric and \({\mathbf L}\)-fibred cohomology algebra for marked spaces \(S\) and \(S'\), the `finite' moduli space description of equivalence of \(S\) and \(S'\) corresponds to isomorphism of appropriate lower central series quotients of the Mal'tsev completions of \(\pi_1 (S)\) and \(\pi_1 (S')\). The authors apply these ideas to construct invariants for certain classes of links whose link algebras are \({\mathbf L}\)-fibred and aspheric.
differentials on the rational homotopy theoretic bigraded model
lower central series quotients of the Mal'tsev completions
scientific article; zbMATH DE number 643789
scientific article