topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
For a vector space and a cardinal number (generally taken to be a natural number), the Grassmannian is the space of all -dimensional linear subspaces of .
For , write for the orthogonal group acting on . More generally, say that a Euclidean vector space is an inner product space such that there exists a linear isometry to equipped with its canonical inner product. Then write for the group of linear isometric automorphisms of . For the following we regard these groups as topological groups in the canonical way.
For and , then the th Grassmannian of is the coset topological space.
where the action of the product group is via its canonical embedding .
Generally, for an inclusion of Euclidean vector spaces, then
where denotes the orthogonal complement of in .
The group acts transitively on the set of -dimensional linear subspaces, and given any such, then its stabilizer subgroup in is isomorphic to . In this way the underlying set of is in natural bijection to the set of -dimensional linear subspaces in . The realization as a coset as above serves to equip this set naturally with a topological space.
Similarly, the real Stiefel manifold is the coset
The quotient projection
is an -principal bundle, with associated bundle a vector bundle of rank . In the limit (colimit) that is this gives a presentation of the -universal principal bundle and of the universal vector bundle of rank , respectively.. The base space is the classifying space for -principal bundles and rank vector bundles.
The real Grassmannians and the complex Grassmannians admit the structure of CW-complexes. Moreover the canonical inclusions
are subcomplex incusion (hence relative cell complex inclusions).
Accordingly there is an induced CW-complex structure on the classifying space
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
(complex projective space is Oka manifold)
Every complex projective space , , is an Oka manifold. More generally every Grassmannian over the complex numbers is an Oka manifold.
The Grassmannian (def. ) is
for : the point;
for : the real projective space ;
for : the point;
for : the empty space .
If is an inner product space, then the orthogonal complement defines an isomorphism between and .
Textbook accounts:
Stanley Kochmann, section 1.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, American Mathematical Society (1996) [ams:fim-7, cds:2264210]
Allen Hatcher, section 1.2 of: Vector bundles & K-theory [web]
SL Kleiman, Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publications Mathématiques de l’IHÉS (1969) [numdam:PMIHES_1969__36__281_0, pdf]
Karin Baur, Grassmannians and cluster structures, Bull. Iranian Math. Soc. 47 (2021) (Suppl 1) S5-S33 [doi:10.1007/s41980-021-00542-6]
Lecture notes:
See also:
Arguments that some effects in topological phases of matter are “unstable” or “fragile” in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an unstable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:
Influential precursor discussion:
More explicit highlighting of the role of the unstable case (and coinage of the term “fragile” for “unstable”):
Hoi Chun Po, Haruki Watanabe, Ashvin Vishwanath: Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121 (2018) 126402 [doi:10.1103/PhysRevLett.121.126402]
Adrien Bouhon, Tomáš Bzdušek, Robert-Jan Slager: Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102 (2020) 115135 [doi:10.1103/PhysRevB.102.115135, arXiv:2005.02044]
Applications:
Adrien Bouhon, Robert-Jan Slager, around equation (3) in: Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]
Zory Davoyan, Wojciech J. Jankowski, Adrien Bouhon, Robert-Jan Slager, section II.A in: Three-dimensional -symmetric topological phases with Pontryagin index [doi:10.1103/PhysRevB.109.165125 arXiv:2308.15555]
Expositions with an eye towards non-abelian braiding of band nodes in momentum space:
Gunnar F. Lange: Multi-gap topology & non-abelian braiding in -space, talk at University of Oslo (Feb 2023) pdf]
Adrien Bouhon: Non-Abelian and Euler multi-gap topologies in crystalline materials, talk at: Quantum Information and Quantum Matter, CQTS @ NYU Abu Dhabi (May 2023) pdf]
Last revised on June 3, 2025 at 17:50:44. See the history of this page for a list of all contributions to it.