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Vector Bundles & Classifying Constructions

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Abstractions about vector bundles and constructions built from them — bundle types and structure (holomorphic, line, flat and inverse bundles, bundle metrics), bundle gerbes, and classifying spaces such as BO(n), alongside looser invariants and theorems (cohomological dimension, AD+, Arakelian's theorem).

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • AD+ — Woodin’s strengthening of the Axiom of Determinacy that combines dependent choice for reals, ordinal determinacy below Theta, and infinity-Borel definability for every set of reals.
  • Arakelian's Theorem — Arakelian's theorem makes connectedness and local connectedness of a one-point complement equivalent to universal uniform holomorphic approximation on a closed plane-domain set.
  • Bundle Gerbe — A geometric gerbe presentation built from a surjective submersion, a line bundle on paired fibers, and associative multiplication over triples.
  • Bundle metric — A smoothly varying nondegenerate bilinear form on each fibre of a vector bundle.
  • Classifying space for O(n) — The classifying space BO(n) represents rank-n real vector bundles: homotopy classes of maps into BO(n) classify such bundles over suitable base spaces.
  • Cohomological dimension — Cohomological dimension denotes invariant of a group within group cohomology.
  • Flat Vector Bundle — A vector bundle with a zero-curvature linear connection has homotopy-invariant parallel transport, locally constant transition data, and a monodromy representation.
  • Holomorphic vector bundle — A complex vector bundle over a complex manifold whose total space is complex and whose projection map is holomorphic.
  • Inverse Bundle — A finite-rank vector bundle whose Whitney sum with a specified same-base bundle is isomorphic to a finite-rank trivial bundle.
  • Line Bundle — A locally trivial rank-one vector bundle whose one-dimensional fibers are glued over a base by invertible scalar transition functions, allowing local products to carry nontrivial global twisting.