Hopf invariants and differential forms

TL;DR

This paper develops a method using differential forms to create a complete invariant for real homotopy classes of maps between certain smooth manifolds, enhancing understanding of their topological properties.

Contribution

It introduces a novel approach employing differential form integration to classify maps up to real homotopy equivalence between simply-connected manifolds.

Findings

Differential forms can distinguish real homotopy classes of maps.

A complete invariant for these classes is constructed.

The method applies to maps between compact and finite-type manifolds.

Abstract

Let $f,g:M \rightarrow N$ be two maps between simply-connected smooth manifolds $M$ and $N$, such that $M$ is compact and $N$ is of finite $\mathbb{R}$-type. The goal of this paper is to use integration of certain differential forms to obtain a complete invariant of the real homotopy classes of the maps $f$ and $g$.