Monopoles, twisted integral homology, and Hirsch algebras

We provide an explicit computation over the integers of the bar version $\overline{HM}_*$ of the monopole Floer homology of a three-manifold in terms of a new invariant associated to its triple cup product called extended cup homology. This refines previous computations over fields of characteristic zero by Kronheimer and Mrowka, who established a relationship to Atiyah and Segal's twisted de Rham cohomology, and characteristic two by Lidman using surgery techniques in Heegaard Floer theory. In order to do so, we first develop a general framework to study the homotopical properties of the cohomology of a dga twisted with respect a particular kind of Maurer-Cartan element called twisting sequence. Then, for dgas equipped with the additional structure of a Hirsch algebra (which consists of certain higher operations that measure the failure of strict commutativity, and related associativity properties), we develop a product on twisting sequences and a theory of rational characteristic classes. These are inspired by Kraines' classical construction of higher Massey products and may be of independent interest. We then compute the most important infinite family of such higher operations explicitly for the minimal cubical realization of the torus. Building on the work of Kronheimer and Mrowka, the determination of $\overline{HM}_*$ follows from these computations and certain functoriality properties of the rational characteristic classes.