https://topospaces.subwiki.org/w/index.php?action=history&feed=atom&title=K-groupK-group - Revision history2026-05-19T01:25:42ZRevision history for this page on the wikiMediaWiki 1.41.2https://topospaces.subwiki.org/w/index.php?title=K-group&diff=976&oldid=prevVipul: 1 revision2008-05-11T19:47:42Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://topospaces.subwiki.org/w/index.php?title=K-group&diff=975&oldid=prevVipul at 20:30, 24 December 20072007-12-24T20:30:40Z<p></p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
Let <math>X</math> be a [[compact Hausdorff space]]. The (unreduced) K-group of <math>X</math>, denoted <math>K(X)</math> is defined as the [[Grothendieck group]] of the following Abelian monoid:<br />
<br />
* As a set, the set of all [[complex vector bundle]]s over <math>X</math>, upto isomorphism<br />
* The additive operation is direct sum of complex vector bundles<br />
* The zero element is the zero-dimensional trivial vector bundle<br />
<br />
Equivalently <math>K(X)</math> can be viewed as the set of ''formal differences'' of complex vector bundles, upto ''stable'' isomorphism (we need stable isomorphism because the monoid of complex vector bundles ''per se'' is not cancellative).<br />
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<math>K(X)</math> is also a ring, with the multiplication operation that induced by the [[tensor product of complex vector bundles]].<br />
<br />
==Related notions==<br />
<br />
* [[Reduced K-group]]<br />
* [[KO-group]]<br />
* [[KSp-group]]</div>Vipul