bundles

• (stable) parameterized homotopy theory

• fiber bundles in physics

• Chern-Weil theory

Context

• slice topos, slice (∞,1)-topos

• (linear) dependent type theory

Classes of bundles

• covering space

• retractive space

• fiber bundle, fiber ∞-bundle

  numerable bundle

  • sphere bundle

  • projective bundle

• principal bundle, principal ∞-bundle

  • principal 2-bundle, principal 3-bundle

  • circle bundle, circle n-bundle

  • orientation bundle

  • spinor bundle, stringor bundle

• associated bundle, associated ∞-bundle

  • gerbe, 2-gerbe, ∞-gerbe,

  • local coefficient bundle

• vector bundle, 2-vector bundle, (∞,1)-vector bundle

  real, complex/holomorphic, quaternionic

  topological, differentiable, algebraic

  with connection

  • line bundle

    complex, holomorphic, algebraic

    cubical structure

  • Chern-Weil theory

  • topological K-theory

  • tensor category of vector bundles

    (VectBund)

  • direct sum, tensor product, external tensor product, inner product

  • dual vector bundle

  • stable vector bundle

  • virtual vector bundle

• bundle of spectra

• natural bundle

• equivariant bundle

Universal bundles

• universal principal bundle, universal principal ∞-bundle

• universal vector bundle, universal complex line bundle

• subobject classifier, object classifier

Presentations

• bundle gerbe

• groupal model for universal principal ∞-bundles

• microbundle

Examples

• empty bundle, zero bundle

• trivial vector bundle

• tangent bundle, normal bundle

• tautological line bundle

  • basic line bundle on the 2-sphere

• Hopf fibration

• canonical line bundle

• prequantum circle bundle, prequantum circle n-bundle

Constructions

• clutching construction

cohomology

• cocycle, coboundary, coefficient

• homology

  • chain, cycle, boundary

• characteristic class

  • universal characteristic class

  • secondary characteristic class

  • differential characteristic class

• fiber sequence/long exact sequence in cohomology

• fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle, twisted ∞-bundle

• ∞-group extension

• obstruction

Special and general types

• cochain cohomology

• ordinary cohomology, singular cohomology

• group cohomology, nonabelian group cohomology, Lie group cohomology

• Galois cohomology

• groupoid cohomology, nonabelian groupoid cohomology

• generalized (Eilenberg-Steenrod) cohomology

  • cobordism cohomology theory

  • integral cohomology

  • K-theory

  • elliptic cohomology, tmf

  • taf

• abelian sheaf cohomology

  • Deligne cohomology

    • de Rham cohomology

    • Dolbeault cohomology

  • etale cohomology

    • group of units, Picard group, Brauer group

  • crystalline cohomology

  • syntomic cohomology

  • motivic cohomology

• cohomology of operads

• Hochschild cohomology, cyclic cohomology

  • string topology

• nonabelian cohomology

  • principal ∞-bundle

    • universal principal ∞-bundle, groupal model for universal principal ∞-bundles

    • principal bundle, Atiyah Lie groupoid

    • principal 2-bundle/gerbe

  • covering ∞-bundle/local system

  • (∞,1)-vector bundle / (∞,n)-vector bundle

  • quantum anomaly

    • orientation, Spin structure, Spin^c structure, String structure, Fivebrane structure

• cohomology with constant coefficients / with a local system of coefficients

• ∞-Lie algebra cohomology

  • Lie algebra cohomology, nonabelian Lie algebra cohomology, Lie algebra extensions, Gelfand-Fuks cohomology,

• bialgebra cohomology

Special notions

• Čech cohomology

• hypercohomology

Variants

• equivariant cohomology

  • equivariant homotopy theory

  • Bredon cohomology

• twisted cohomology

  • twisted bundle

    • twisted K-theory, twisted spin structure, twisted spin^c structure

  • twisted differential c-structures

    • twisted differential string structure, twisted differential fivebrane structure

• differential cohomology

  • differential generalized (Eilenberg-Steenrod) cohomology

    • differential cobordism cohomology

    • Deligne cohomology

    • differential K-theory

    • differential elliptic cohomology

  • differential cohomology in a cohesive topos

    • Chern-Weil theory

    • ∞-Chern-Weil theory

• relative cohomology

Extra structure

• Hodge structure

• orientation, in generalized cohomology

Operations

• cohomology operations

• cup product

• connecting homomorphism, Bockstein homomorphism

• fiber integration, transgression

• cohomology localization

Theorems

• universal coefficient theorem

• Künneth theorem

• de Rham theorem, Poincare lemma, Stokes theorem

• Hodge theory, Hodge theorem

  nonabelian Hodge theory, noncommutative Hodge theory

• Brown representability theorem

• hypercovering theorem

• Eckmann-Hilton-Fuks duality