Relative K-area homology and applications

TL;DR

This paper investigates the properties of K-area homology, establishing estimates, exact sequences, and obstructions related to scalar curvature on manifolds, with implications for topology and geometry.

Contribution

It introduces an Uhlenbeck type estimate for K-area homology and explores its behavior under surgery, also providing an index theoretic obstruction to positive scalar curvature.

Findings

Established an Uhlenbeck type estimate for K-area homology.

Derived exact sequences in K-area homology for certain manifolds.

Provided an index theoretic obstruction to positive scalar curvature on spin manifolds with boundary.

Abstract

We show an Uhlenbeck type estimate for closed simply connected manifolds which provides the existence of certain exact sequences in K-area homology. This leads to the behavior of the K-area homology under surgery. Moreover, we give an index theoretic obstruction to positive scalar curvature on compact spin manifolds with boundary.