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Coefficient ring: The coefficient groups π n (S) are the stable homotopy groups of spheres, which are notoriously hard to compute or understand for n > 0. (For n < 0 they vanish, and for n = 0 the group is Z.) Stable homotopy is closely related to cobordism of framed manifolds (manifolds with a trivialization of the normal bundle).
A Poincaré space X does not have a tangent bundle, but it does have a well-defined stable spherical fibration, which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to X having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a ...
If M is a paracompact manifold and P → M is a principal G-bundle, then there exists a map f : M → BG, unique up to homotopy, such that P is isomorphic to f ∗ (EG), the pull-back of the G-bundle EG → BG by f. Proof. On one hand, the pull-back of the bundle π : EG → BG by the natural projection P × G EG → BG is the bundle P × EG.
The vector bundles associated to these principal bundles via the natural action of G on are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold V k ( F n ) {\displaystyle V_{k}(\mathbb {F} ^{n})} is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian.
In all dimensions, the fundamental group of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take ...
Smale's classification of immersions of spheres as the homotopy groups of Stiefel manifolds, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of frame bundles are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the ...
In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differential structure. In dimension at most 2 (Rado), and 3 (Moise), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same.
Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps , and the class of fiber bundles forms ...