A structure on a vector bundle (or sphere bundle, etc.) that is a generalization of the concept of the structure group of a fibration.
Let be a fibration and let be an -dimensional vector bundle over a space , classified by the mapping . Then the homotopy class lifting the mapping to a mapping in is called a -structure on , i.e. it is an equivalence class of mappings such that , where two mappings and are said to be equivalent if they are fibrewise homotopic. No method of consistently defining -structures for equivalent fibrations exists, because this consistency depends on the choice of the equivalence.
Let there be a sequence of fibrations and mappings such that ( is the standard mapping). The family (and sometimes only ) is called a structure series. An equivalence class of sequences of -structures on the normal bundle of a manifold is called a -structure on ; they coincide beginning from some sufficiently large . A manifold with a fixed -structure on it is called a -manifold.
Instead of , a more general space , classifying sphere bundles, can be considered and -structures can be introduced on them.
|||R. Lashof, "Poincaré duality and cobordism" Trans. Amer. Math. Soc. , 109 (1963) pp. 257–277|
|||R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)|
is the limit of the Grassmann manifolds of -planes in .
B-Phi-structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B-Phi-structure&oldid=19275