Contents

# Contents

## Idea

A manifold is a topological space that is locally isomorphic to a Cartesian space $\mathbb{R}^n$.

A manifold with boundary is a topological space that is locally isomorphic either to an $\mathbb{R}^n$ or to a half-space $H^n = \{ \vec x \in \mathbb{R}^n | x^n \geq 0\}$.

A manifold with corners is a topological space that is locally isomorphic to an $H^n_i = \{ \vec x \in \mathbb{R}^n | x^{i+1}, \cdots, x^n \geq 0\}$ for $0 \leq i \leq n$.

For details see at manifold.

## Properties

### Embedding into the category of diffeological spaces

###### Proposition

(manifolds with boundaries and corners form full subcategory of diffeological spaces)

The evident functor

$SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces$

from the category of smooth manifolds with boundaries and corners to that of diffeological spaces is fully faithful, hence is a full subcategory-embedding.

## References

### On manifolds with boundary

• Manifolds with boundary (pdf, pdf)

On cobordism theory of MUFr-manifolds with boundaries, their e-invariant and their appearance in the first line of the Adams-Novikov spectral sequence:

### On manifolds with corners

The full subcategory-embedding of manifolds with boundaries and corners into that of diffeological spaces is discussed in:

Last revised on January 21, 2021 at 13:39:24. See the history of this page for a list of all contributions to it.