nLab manifold with boundary

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

Collar neighbourhood theorem

• 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

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

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)

Related concepts

• asymptotic boundary

• manifold with corners

• closed manifold

• cobordism category

• MOFr, MUFr, MSUFr

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:

• Pierre Conner, Edwin Floyd, Section 16 of: The relation of cobordism to $K$-theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

On manifolds with corners

• Klaus Jänich, Section 1.2 of: On the classification of $O(n)$-manifolds, Math. Ann. 176, 53–76 (1968) (doi:10.1007/BF02052956)

• Dominic Joyce, On manifolds with corners (arXiv:0910.3518)

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

• Patrick Iglesias-Zemmour, section 4.16 of Diffeology, Mathematical Surveys and Monographs, AMS (2013) (web, publisher)

• Serap Gürer, Patrick Iglesias-Zemmour, Differential forms on corners, 2017 (pdf)

• Serap Gürer, Patrick Iglesias-Zemmour, Differential forms on manifolds with boundary and corners, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 (doi:10.1016/j.indag.2019.07.004)

On cobordism theory of manifolds with corners:

• Gerd Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000) (doi:10.1090/S0002-9947-00-02676-3)

  (their f-invariant and their appearance in the second line of the Adams-Novikov spectral sequence)

• Josh Genauer, Cobordism categories of manifolds with corners, Transactions of the American Mathematical Society Vol. 364, No. 1 (2012), pp. 519-550 (arXiv:0810.0581, jstor:41407770, doi:10.1090/S0002-9947-2011-05474-7)

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

• Giovanni Canepa, Alberto S. Cattaneo, Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism, Annales Henri Poincaré 25 (2024) 2585-2639 [arXiv:2202.08684, doi:10.1007/s00023-023-01360-8]

• Alberto Cattaneo, Poisson Structures from Corners of Field Theories, talk at CQTS (22 Nov 2023) [slides:pdf, video:YT, Zm]

• Luca Ciambelli, Jerzy Kowalski-Glikman, Ludovic Varrin. Quantum Corner Symmetry: Representations and Gluing [arXiv:2406.07101]

and references therein.