manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
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.
(manifolds with boundaries and corners form full subcategory of diffeological spaces)
The evident functor
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)
On cobordism theory of MUFr-manifolds with boundaries, their e-invariant and their appearance in the first line of the Adams-Novikov spectral sequence:
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.
Last revised on June 12, 2024 at 13:08:52. See the history of this page for a list of all contributions to it.