B-Phi-structure
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.
References
[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) |
Comments
Here
![]() |
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=50466