Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations

TL;DR

This paper characterizes stable cohomotopy groups in codimensions two and three using algebraic and geometric methods, providing bordism interpretations and applications to vector bundle sections.

Contribution

It offers a complete algebraic characterization in codimension two and a partial one in codimension three, along with bordism-theoretic interpretations for specific manifolds.

Findings

Complete algebraic characterization of stable cohomotopy in codimension two.

Partial characterization in codimension three up to a 3-primary parameter.

Derived conditions for the existence of nowhere-vanishing vector bundle sections.

Abstract

This paper investigates stable cohomotopy groups in codimensions two and three from complementary algebraic and geometric viewpoints. For general CW complexes, we give a complete characterization of stable cohomotopy in codimension two and a characterization in codimension three up to a $3$-primary parameter. Geometrically, we provide bordism-theoretic interpretations of these stable cohomotopy groups for oriented manifolds in codimension two and string manifolds in codimension three. As an application, we derive necessary and sufficient conditions for the existence of nowhere-vanishing sections of vector bundles, extending the foundational codimension-one results of Konstantis.