Editing K-theory (section)

=== Grothendieck group for compact Hausdorff spaces ===
Given a compact [[Hausdorff space]] <math>X</math> consider the set of isomorphism classes of finite-dimensional vector bundles over <math>X</math>, denoted <math>\text{Vect}(X)</math> and let the isomorphism class of a vector bundle <math>\pi:E \to X</math> be denoted <math>[E]</math>. Since isomorphism classes of vector bundles behave well with respect to [[direct sum]]s, we can write these operations on isomorphism classes by

:<math>[E]\oplus[E'] =[E\oplus E'] </math>

It should be clear that <math>(\text{Vect}(X),\oplus)</math> is an abelian monoid where the unit is given by the trivial vector bundle <math>\R^0\times X \to X</math>. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of <math>X</math> and is denoted <math>K^0(X)</math>.

We can use the [[Serre–Swan theorem]] and some algebra to get an alternative description of vector bundles over <math>X</math> as [[projective module]]s over the ring <math>C^0(X;\Complex)</math> of continuous complex-valued functions. Then, these can be identified with [[Idempotence|idempotent]] matrices in some ring of matrices <math>M_{n\times n}(C^0(X;\Complex))</math>. We can define equivalence classes of idempotent matrices and form an abelian monoid <math>\textbf{Idem}(X)</math>. Its Grothendieck completion is also called <math>K^0(X)</math>. One of the main techniques for computing the Grothendieck group for topological spaces comes from the [[Atiyah–Hirzebruch spectral sequence]], which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group <math>K^0</math> for the spheres <math>S^n</math>.<ref>{{Cite book|last=Park, Efton.|url=https://www.worldcat.org/oclc/227161674|title=Complex topological K-theory|date=2008|publisher=Cambridge University Press|isbn=978-0-511-38869-9|location=Cambridge|oclc=227161674}}</ref><sup>pg 51-110</sup>