View source for Difference-element-in-K-theory

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" />.



====Comments====


====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>