A relative h-principle via cobordism-like categories

TL;DR

This paper establishes an h-principle for certain sheaves using cobordism category techniques, providing a new proof of the Madsen-Weiss theorem relating cobordism categories to Thom spaces.

Contribution

It introduces a novel approach to the h-principle via cobordism-like categories and offers an alternative proof of a fundamental theorem in the topology of cobordism categories.

Findings

Proves an h-principle with boundary conditions for topological sheaves.

Provides a new proof of the Madsen-Weiss theorem.

Connects cobordism categories with the homotopy type of Thom spaces.

Abstract

We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism categories, and they are of simplicial and categorical nature. Applying the main result of this paper to a certain sheaf we find another proof of the Madsen-Weiss theorem, that describes the homotopy type of the classifying space of the cobordism category as the loop space of the Thom space of the complement of the tautological bundle over the Grassmannians.