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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.


[1] R. Lashof, "Poincaré duality and cobordism" Trans. Amer. Math. Soc. , 109 (1963) pp. 257–277
[2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)



is the limit of the Grassmann manifolds of -planes in .

How to Cite This Entry:
B-Phi-structure. Yu.B. Rudyak (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098