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Difference between revisions of "Difference-element-in-K-theory"

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An element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316702.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316703.png" /> is a pair of spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316704.png" /> is usually supposed to be a finite [[Cellular space|cellular space]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316705.png" /> is a cellular subspace of it), constructed from a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316708.png" /> are vector bundles of the same dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d0316709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167010.png" /> is an isomorphism of vector bundles (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167011.png" /> is the part of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167013.png" /> located above the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167014.png" />). The construction of a difference element can be carried out in the following way. First one supposes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167015.png" /> is the trivial bundle and that some trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167017.png" /> is fixed. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167018.png" /> gives a trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167019.png" /> and hence gives an element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167020.png" />. This element is independent of the choice of the trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167021.png" /> above all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167022.png" />. In the general case one chooses a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167023.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167024.png" /> such that the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167025.png" /> is trivial, and the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167026.png" /> is assigned the same element as the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031670/d03167027.png" />.
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An element of the group  $  K ( X, A) $(
 +
where  $  ( X, A) $
 +
is a pair of spaces and  $  X $
 +
is usually supposed to be a finite [[Cellular space|cellular space]], while  $  A $
 +
is a cellular subspace of it), constructed from a triple  $  ( \xi , \eta , \zeta ) $,
 +
where  $  \xi $
 +
and  $  \eta $
 +
are vector bundles of the same dimension over  $  X $
 +
and  $  \zeta :  \xi | _ {A} \rightarrow \eta | _ {A} $
 +
is an isomorphism of vector bundles (here  $  \sigma \mid  _ {A} $
 +
is the part of the vector bundle  $  \sigma $
 +
over  $  X $
 +
located above the subspace  $  A $).
 +
The construction of a difference element can be carried out in the following way. First one supposes that  $  \eta $
 +
is the trivial bundle and that some trivialization of  $  \eta $
 +
over  $  X $
 +
is fixed. Then  $  \zeta $
 +
gives a trivialization of  $  \xi \mid  _ {A} $
 +
and hence gives an element of the group  $  \widetilde{K}  ( X/A) = K ( X, A) $.
 +
This element is independent of the choice of the trivialization of  $  \eta $
 +
above all of  $  X $.
 +
In the general case one chooses a bundle  $  \sigma $
 +
over  $  X $
 +
such that the bundle  $  \eta \oplus \sigma $
 +
is trivial, and the triple  $  ( \xi , \eta , \zeta ) $
 +
is assigned the same element as the triple  $  ( \xi \oplus \sigma , \eta \oplus \sigma , \zeta \oplus  \mathop{\rm id}  \sigma ) $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,  F. Hirzebruch,  "Analytic cycles on complex manifolds"  ''Topology'' , '''1'''  (1961)  pp. 28–45</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.F. Atiyah,  R. Bott,  A. Shapiro,  "Clifford modules"  ''Topology'' , '''3. Suppl. 1'''  (1964)  pp. 3–38</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.F. Atiyah,  F. Hirzebruch,  "Analytic cycles on complex manifolds"  ''Topology'' , '''1'''  (1961)  pp. 28–45</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.F. Atiyah,  R. Bott,  A. Shapiro,  "Clifford modules"  ''Topology'' , '''3. Suppl. 1'''  (1964)  pp. 3–38</TD></TR></table>

Revision as of 17:33, 5 June 2020


An element of the group $ K ( X, A) $( where $ ( X, A) $ is a pair of spaces and $ X $ is usually supposed to be a finite cellular space, while $ A $ is a cellular subspace of it), constructed from a triple $ ( \xi , \eta , \zeta ) $, where $ \xi $ and $ \eta $ are vector bundles of the same dimension over $ X $ and $ \zeta : \xi | _ {A} \rightarrow \eta | _ {A} $ is an isomorphism of vector bundles (here $ \sigma \mid _ {A} $ is the part of the vector bundle $ \sigma $ over $ X $ located above the subspace $ A $). The construction of a difference element can be carried out in the following way. First one supposes that $ \eta $ is the trivial bundle and that some trivialization of $ \eta $ over $ X $ is fixed. Then $ \zeta $ gives a trivialization of $ \xi \mid _ {A} $ and hence gives an element of the group $ \widetilde{K} ( X/A) = K ( X, A) $. This element is independent of the choice of the trivialization of $ \eta $ above all of $ X $. In the general case one chooses a bundle $ \sigma $ over $ X $ such that the bundle $ \eta \oplus \sigma $ is trivial, and the triple $ ( \xi , \eta , \zeta ) $ is assigned the same element as the triple $ ( \xi \oplus \sigma , \eta \oplus \sigma , \zeta \oplus \mathop{\rm id} \sigma ) $.

Comments

References

[a1] M.F. Atiyah, F. Hirzebruch, "Analytic cycles on complex manifolds" Topology , 1 (1961) pp. 28–45
[a2] M.F. Atiyah, R. Bott, A. Shapiro, "Clifford modules" Topology , 3. Suppl. 1 (1964) pp. 3–38
How to Cite This Entry:
Difference-element-in-K-theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Difference-element-in-K-theory&oldid=46651
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article