Fibre Bundles

a₁ algebra B₁ bijection bundle morphism calculate characteristic classes Chern class clutching map cofunctor cohomology commutative diagram compact complex vector bundle Corollary cross section CW-complex defined Definition denote elements equals exact sequence exists fibre bundle fibre homotopy finite CW-complexes following commutative diagram following diagram functor G-module G-morphism G-space homeomorphism homotopy equivalence Hopf invariant inclusion induced KO(X Let f linear locally trivial manifold map f maximal torus module monomorphism Moreover multiplication n-dimensional notations orientation orthogonal phism principal G-bundle Proof properties proves the proposition proves the theorem quadratic form quotient real vector bundle relation restriction result semigroup SO(n Spin splitting map statement Stiefel-Whitney class structure subgroup subspace tangent bundle topology transition functions unique V₁ Vect vector bundle morphism vector fields vector space Weyl group x₁ Z2-graded Ꮓ Ꮓ