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 more context on the following cf. at classifying space.
For , write for the orthogonal group regarded as a topological group and acting canonically on .
For , the th real Stiefel manifold of is the coset topological space.
where the action of is via the standard subgroup inclusion as an upper right block.
The canonical group action induces a transitive action on the set of -dimensional linear subspaces equipped with an orthonormal basis, and given any such subspace , then its stabilizer subgroup in is isomorphic to , the symmetries of the orthogonal subspace . In this way the underlying set of is in natural bijection to the set of -dimensional linear subspaces in equipped with orthonormal bases. The realization of this set as a coset (1) serves to equip it naturally with the structure of a topological space.
More generally, given any finite-dimensional inner product space , there is a corresponding orthogonal group , and one can repeat the above definition more or less verbatim.
By def. there are canonical inclusions that are compatible with the actions of the respective orthogonal groups. The colimit (in Top, see there) over these inclusions is denoted
This is a model for the total space of the -universal principal bundle.
The Stiefel manifold (Def. ) is -connected.
Observe that
the coset coprojection is a Serre fibration
(by this prop. and this corollary)
its fiber (hence: homotopy fiber) is
so that we have a homotopy fiber sequence of this form:
The corresponding long exact sequence of homotopy groups has, has the following structure in degrees strictly bounded above by , by this prop (taking in the notation there):
This implies the claim. (Exactness of the sequence says that every element in is in the kernel of zero, hence in the image of 0, hence is 0 itself.)
The colimiting space from Def. is weakly contractible.
This follows from Prop. , together with the fact that the sequence of maps are closed -inclusions. See this Prop. together with the accompanying commentary there.
The Stiefel manifold admits the structure] of a CW-complex.
(e.g. James 1959 p. 3, James 1976 p. 5 with p. 21, Hatcher 2002 p. 302, Blaszczyk 2007)
And it should be true that with that cell structure the inclusions are subcomplex inclusions:
According to Yokota 1956, the inclusions are cellular such that this is compatible with the group action (reviewed here in 3.3 and 3.3.1). This implies that also the projection is cellular (e.g. Hatcher 2002 p. 302).
Similarly, the Grassmannian manifold is the coset
The quotient coprojection
is an -principal bundle, with associated bundle a vector bundle of rank . In the limit (colimit) that 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 for rank vector bundles.
Named after:
Further discussion:
Yoshihiro Saito: On the homotopy groups of Stiefel manifolds, J. Inst. Polytech. Osaka City Univ. Ser. A 6 1 (1955) 39-45 [euclid:ojm/1353054734]
I. Yokota, On the cells of symplectic groups, Proc. Japan Acad. 32 (1956) 399-400
Ioan Mackenzie James: Spaces associated with Stiefel manifolds, Proc. Lond. Math. Soc. (3) 9 (1959) [doi:10.1112/plms/s3-9.1.115, pdf]
Ioan Mackenzie James: On the homotopy type of Stiefel manifolds, Proceedings of the AMS, 29 1 (1971)
Ioan Mackenzie James: The topology of Stiefel manifolds, Cambridge University Press (1976)
Stanley Kochmann, section 1.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS (1996)
Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Zbigniew Błaszczyk, On cell decompositions of (2007) [pdf]
See also:
Last revised on July 8, 2025 at 02:34:30. See the history of this page for a list of all contributions to it.