Index of transverse Dirac operator and cohomotopy Seiberg-Witten invariant for codimension $4$ Riemannian foliation

TL;DR

This paper develops a foliated version of the Bauer-Furuta invariant for codimension 4 Riemannian foliations, providing index estimates and vanishing results for the transversal Dirac operator based on basic cohomology conditions.

Contribution

It introduces a finite dimensional approximation of the transversal Seiberg-Witten map and defines a new foliated Bauer-Furuta invariant, linking it to the index theory of the transversal Dirac operator.

Findings

Defined a foliated Bauer-Furuta invariant for codimension 4 foliations.

Provided index estimates for the transversal Dirac operator under certain cohomological conditions.

Proved vanishing of the index when specific basic cohomology groups are of dimension zero or one.

Abstract

For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic cohomological group is of zero dimension, we can give an estimate for the index of transversal Dirac operator of a foliated spin structure. Furthermore, under the condition that $H^\pm_b(M)=1$, we show the vanishing of the index of the transverse Dirac operator. This gives a topological condition for the vanishing of the index of the transverse Dirac operator.