(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
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
The clutching construction is the construction of a $G$-principal bundle on an n-sphere from a cocycle in $G$-Cech cohomology given by the covering of the $n$-sphere by two hemi-n-spheres that overlap a bit at the equator, and one single transition function on that equator $S^{n-1} \to G$.
The Möbius strip is the result of the single non-trivial clutching construction for real line bundle over the circle.
In physics, in gauge theory, the clutching construction plays a central role in the discussion of Yang-Mills instantons, and monopoles (Dirac monopole). Here the discussion is usually given in terms of gauge fields on $n$-dimensional Minkowski spacetime such that they vanish at infinity. Equivalently this means that one has gauge fields on the one-point compactification of Minkowski spacetime, which is the n-sphere. The transition function of the clutching construction then appears as the asymptotic gauge transformation.
Reviews include
Allen Hatcher, page 22 of Vector bundles and K-theory (web)
Klaus Wirthmüller, p. 17-28 of Vector bundles and K-theory, 2012 (pdf)
Wikipedia, Clutching construction
Last revised on June 21, 2017 at 06:05:57. See the history of this page for a list of all contributions to it.