Finite dimensional approximations in geometry

TL;DR

This paper explores finite dimensional approximations of invariants in low dimensional topology, using homotopy classes of nonlinear elliptic operators, with applications to the Seiberg-Witten equations on 4-manifolds.

Contribution

It introduces a novel approach to understanding topological invariants via homotopy classes of operators, extending classical invariants into cohomotopy theory.

Findings

Homotopy class approach refines invariants in low dimensional topology.

Application of the method to Seiberg-Witten equations yields new insights.

Potential for new invariants in 4-dimensional topology.

Abstract

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting. Instead of looking at the solutions, if we can regard some kind of homotopy class of the operator itself as an invariant, then the refined version of the invariant is understood as Euler class or degree in cohomotopy theory. This idea can be carried out for the Seiberg-Witten equation on 4-dimensional manifolds and we have some applications to 4-dimensional topology.