# Transversality theorem in highly relative situations and its application

## TL;DR

This paper develops a framework for relative objects in differential topology, introduces arrangements of manifolds, and proves a relative transversality theorem with applications to embedding manifolds with faces into polyhedra.

## Contribution

It introduces arrangements of manifolds and establishes a relative transversality theorem, extending classical differential topology concepts to relative objects.

## Key findings

- Defined the notion of arrangements of manifolds.
- Proved a relative version of the Transversality Theorem.
- Demonstrated an embedding theorem for manifolds with faces.

## Abstract

In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with corners, there seem to be poor general notions to deal with them. Moreover, we want even more direct differential calculus on relative objects and extension of classical notions and theories to relative situations; e.g. functions, vector fields, jet bundles, singularities, and so on. To establish this, the notion of arrangements of manifolds is introduced, which, for example, enables us to control behaviors of smooth maps on manifolds around corners. We construct jet bundles and prove a relative version of Transversality Theorem for some sorts of arrangements. Finally, embedding theorem of manifolds with faces into polyhedra is proved as an application.