# Modulo 2 counting of Klein-bottle leaves in smooth taut foliations

## TL;DR

This paper proves that the parity of Klein-bottle leaves in smooth taut foliations remains invariant under smooth deformations, given certain holonomy conditions, contributing to the understanding of foliation topology.

## Contribution

It establishes a new invariance property of Klein-bottle leaves in taut foliations under smooth deformations with holonomy constraints.

## Key findings

- Parity of Klein-bottle leaves is invariant under deformation
- Invariance holds when Klein-bottle leaves have non-trivial linear holonomy
- Advances understanding of foliation topology and leaf counting

## Abstract

This article proves that the parity of the number of Klein-bottle leaves in a smooth cooriented taut foliation is invariant under smooth deformations within taut foliations, provided that every Klein-bottle leaf involved in the counting has non-trivial linear holonomy.