When two cycles have a transversal intersection on a smooth manifold , then is a cycle. Moreover, the homology class that represents depends only on the homology class of and . The sign of is determined by the orientations...
A homology class in a singular homology theory is represented by a finite linear combination of geometric subobjects with zero boundary. Such a linear combination is considered to be homologous to zero if it is the boundary of...
Homology is a concept that is used in many branches of algebra and topology. Historically, the term "homology" was first used in a topological sense by Poincaré. To him, it meant pretty much what is now called a bordism,...
The homomorphism which, according to the snake lemma, permits construction of an exact sequence (1) from the above commutative diagram with exact rows. The homomorphism is defined by (2) ...
A circle bundle is a fiber bundle whose fibers are circles. It may also have the structure of a principal bundle if there is an action of that preserves the fibers, and is locally trivial. That is, if every point has a...
A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose and are two bundles, then is a bundle map if there is a map such that for all . In particular, the fiber bundle of...
The term "bundle" is an abbreviated form of the full term fiber bundle. Depending on context, it may mean one of the special cases of fiber bundles, such as a vector bundle or a principal bundle. Bundles are so named because they...
Given a principal bundle , with fiber a Lie group and base manifold , and a group representation of , say , then the associated vector bundle is (1) In particular, it is the quotient space where . This construction...
Given a group action and a principal bundle , the associated fiber bundle on is (1) In particular, it is the quotient space where . For example, the torus has a action given by (2) and the frame bundle...
An anchor is the bundle map from a vector bundle to the tangent bundle TB satisfying 1. and 2. where and are smooth sections of , is a smooth function of , and the bracket is the "Jacobi-Lie bracket"...
