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 $\phi _ { n } : B _ { n } \rightarrow B O _ { n }$ be a fibration and let $\xi $ be an $n$-dimensional vector bundle over a space $X$, classified by the mapping $\xi : X \rightarrow B O _ { n }$. Then the homotopy class lifting the mapping $\xi : X \rightarrow B O _ { n }$ to a mapping in $B _ { n }$ is called a $( B _ { n } , \phi _ { n } )$-structure on $\xi $, i.e. it is an equivalence class of mappings $\xi ^ { * } : X \rightarrow B_n$ such that $\phi _ { n } \circ \xi ^ { * } = \xi$, where two mappings $\xi ^ { * }$ and $\xi ^ { * \prime } : X \rightarrow B _ { n }$ are said to be equivalent if they are fibrewise homotopic. No method of consistently defining $( B _ { n } , \phi _ { n } )$-structures for equivalent fibrations exists, because this consistency depends on the choice of the equivalence.

Let there be a sequence $( B , \phi , g )$ of fibrations $\phi _ { r } : B _ { r } \rightarrow B O _ { r }$ and mappings $g _ { r } : B _ { r } \rightarrow B _ { r + 1}$ such that $j _ { r } \circ \phi _ { r } = \phi _ { r + 1 } \circ g _ { r }$ ($j_\gamma : B O _ { r } \rightarrow B O _ { r + 1}$ is the standard mapping). The family $\{ B _ { r } , \phi _ { r } , g _ { r } \}$ (and sometimes only $( B _ { r } , \phi _ { r } )$) is called a structure series. An equivalence class of sequences of $( B _ { r } , \phi _ { r } )$-structures on the normal bundle $\{ \xi_r\}$ of a manifold $M ^ { n }$ is called a $( B , \phi )$-structure on $M$; they coincide beginning from some sufficiently large $r$. A manifold $M ^ { n }$ with a fixed $( B , \phi )$-structure on it is called a $( B , \phi )$-manifold.

Instead of $B O _ { n }$, a more general space $B G _ { n }$, classifying sphere bundles, can be considered and $( B , \phi )$-structures can be introduced on them.

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Here

\begin{equation*} B O _ { n } = \operatorname { lim } _ { r \rightarrow \infty } \operatorname { inf } \operatorname { Gras } _ { n } ( \mathbf{R} ^ { r + n } ) \end{equation*}

is the limit of the Grassmann manifolds of $r$-planes in $\mathbf{R} ^ { n + r }$.