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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150102.png" /> be a fibration and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150103.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150104.png" />-dimensional vector bundle over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150105.png" />, classified by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150106.png" />. Then the homotopy class lifting the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150107.png" /> to a mapping in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150108.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b0150109.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501010.png" />, i.e. it is an equivalence class of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501012.png" />, where two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501014.png" /> are said to be equivalent if they are fibrewise homotopic. No method of consistently defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501015.png" />-structures for equivalent fibrations exists, because this consistency depends on the choice of the equivalence.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501016.png" /> of fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501017.png" /> and mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501020.png" /> is the standard mapping). The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501021.png" /> (and sometimes only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501022.png" />) is called a structure series. An equivalence class of sequences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501023.png" />-structures on the [[Normal bundle|normal bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501024.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501025.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501027.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501028.png" />; they coincide beginning from some sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501029.png" />. A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501030.png" /> with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501031.png" />-structure on it is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501033.png" />-manifold.
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501034.png" />, a more general space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501035.png" />, classifying sphere bundles, can be considered and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501036.png" />-structures can be introduced on them.
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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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Lashof,  "Poincaré duality and cobordism"  ''Trans. Amer. Math. Soc.'' , '''109'''  (1963)  pp. 257–277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR></table>
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<table><tr><td valign="top">[1]</td> <td valign="top">  R. Lashof,  "Poincaré duality and cobordism"  ''Trans. Amer. Math. Soc.'' , '''109'''  (1963)  pp. 257–277</td></tr><tr><td valign="top">[2]</td> <td valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</td></tr></table>
  
  
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Here
 
Here
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501037.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501038.png" />-planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015010/b01501039.png" />.
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is the limit of the Grassmann manifolds of $r$-planes in $\mathbf{R} ^ { n + r }$.

Latest revision as of 17:02, 1 July 2020

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.

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

\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 $\mathbf{R} ^ { n + r }$.

How to Cite This Entry:
B-Phi-structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B-Phi-structure&oldid=50466
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article