Holomorphic triangles and invariants for smooth four-manifolds

TL;DR

This paper introduces new invariants for smooth four-manifolds using Floer homology and holomorphic disks, providing a richer structure for three- and four-dimensional theories in topology.

Contribution

It develops a novel four-dimensional invariant based on holomorphic triangles and Floer homology, extending the structure of three-dimensional Floer theories.

Findings

Defined new invariants for smooth four-manifolds

Established an absolute grading for Floer homology groups

Linked holomorphic disks in symmetric products to topological invariants

Abstract

The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in two earlier papers (math.SG/0101206 and math.SG/0105202). This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces.