nLab manifold with boundary

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Contents

Contents

Idea

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

A manifold with boundary is a topological space that is locally isomorphic either to an n\mathbb{R}^n or to a half-space H n={x n|x n0}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 i n={x n|x i+1,,x n0}H^n_i = \{ \vec x \in \mathbb{R}^n | x^{i+1}, \cdots, x^n \geq 0\} for 0in0 \leq i \leq n.

For details see at manifold.

Properties

Collar neighbourhood theorem

Embedding into the category of diffeological spaces

Proposition

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

The evident functor

SmthMfdWBdrCrnAAAADiffeologicalSpaces 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.

(Iglesias-Zemmour 13, 4.16, Gürer & Iglesias-Zemmour 19)

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:

On cobordism theory of manifolds with corners:

Concerning quantum field theory and particularly (quantum)gravity on manifolds with corners (cf. extended field theory):

and references therein.

Last revised on June 12, 2024 at 13:08:52. See the history of this page for a list of all contributions to it.