# Three applications of delooping to H-principles

## TL;DR

This paper introduces a method for establishing h-principles on closed manifolds, demonstrating its versatility through three applications including a homotopical version of Vassiliev's principle, contractibility of framed functions, and Mather-Thurston theory.

## Contribution

It presents a unified approach to prove various h-principles under different conditions, expanding their applicability.

## Key findings

- Proves a homotopical version of Vassiliev's h-principle.
- Shows the contractibility of the space of framed functions.
- Provides a version of Mather-Thurston theory.

## Abstract

In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contractibility of the space of framed functions, and a version of Mather-Thurston theory.