Editing Spectrum (topology) (section)

=== Topological complex K-theory ===
As a second important example, consider [[topological K-theory]]. At least for ''X'' compact, <math> K^0(X) </math> is defined to be the [[Grothendieck group]] of the [[monoid]] of complex [[vector bundles]] on ''X''. Also, <math> K^1(X) </math> is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zeroth space is <math> \mathbb{Z} \times BU </math> while the first space is <math>U</math>. Here <math>U</math> is the infinite [[unitary group]] and <math>BU</math> is its [[classifying space]]. By [[Bott periodicity]] we get <math> K^{2n}(X) \cong K^0(X) </math> and <math> K^{2n+1}(X) \cong K^1(X) </math> for all ''n'', so all the spaces in the topological K-theory spectrum are given by either <math> \mathbb{Z} \times BU </math> or <math>U</math>. There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-[[periodic spectrum]].