On the algebra of cornered Floer homology

TL;DR

This paper explores the algebraic structures underlying bordered Floer homology, introducing a differential graded 2-algebra and an algebra-module, and demonstrates a decomposition theorem for Floer complexes related to surface splittings.

Contribution

It introduces a nilCoxeter sequential 2-algebra and an associated algebra-module, advancing the understanding of algebraic properties under surface splittings in Floer homology.

Findings

Defined a differential graded 2-algebra for the circle

Constructed an algebra-module for surfaces with connected boundary

Proved a decomposition theorem for Floer complexes of planar grid diagrams

Abstract

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is satisfied. Moreover, with a view toward the structure of a potential Floer homology theory of 3-manifolds with codimension-two corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.