# On the whistle cobordism operation in string topology of classifying spaces

## TL;DR

This paper investigates the non-triviality of the whistle cobordism operation within string topology of classifying spaces, revealing its significance in the context of 2D topological quantum field theories for Lie groups.

## Contribution

It proves the non-triviality of the whistle cobordism operation for classifying spaces of Lie groups with labels in maximal closed subgroups, advancing understanding of cobordism operations.

## Key findings

- Whistle cobordism operation is non-trivial for certain labels.
- Gluing operations preserve non-triviality under specific conditions.
- Results apply to the string topology of classifying spaces of Lie groups.

## Abstract

In this manuscript, we consider cobordism operations in the $2$-dimensional labeled open-closed topological quantum field theory for the classifying space of a connected compact Lie group in the sense of Guldberg. In particular, it is proved that the whistle cobordism operation is non-trivial in general provided the labels are in the set of maximal closed subgroups of the given Lie group. The non-triviality of cobordism operations induced by gluing the whistle with the opposite and other labeled cobordisms is also discussed.