# HM-bar, twisted integral homology, and Hirsch algebras

# HM-bar, twisted integral homology, and Hirsch algebras

**Online Talk**

**Zoom link: ****https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09**

We give a completely algebraic formula for HM-bar(Y, s; A) for any 3-manifold Y and spin^c structure s, as well as any local coefficient system A (it is isomorphic to an algebraically determined group we call *extended cup homology*). This formula resembles, but is somewhat more complicated than, the expected formula in terms of "cup homology"; however, computer calculations suggest these may still be noncanonically isomorphic.

The proof requires we pass through three ideas: *twisting sequences*, a simple kind of Maurer-Cartan element which lets us define a Z/2-graded twisted homology group, and their homotopical properties; *Hirsch algebras*, dg-algebras with operations asserting that the product is homotopy-commutative, and this homotopy commutator operation is associative up to coherent homotopy; and *combinatorics of subsets of {1, ..., n} *which corresponds to a Hirsch algebra structure on a minimal model for the n-torus.