Examples
Consider singular cohomology with coefficients in an abelian group A. By Brown representability is the set of homotopy classes of maps from X to K(A,n), the Eilenberg-MacLane space with homotopy concentrated in degree n. Then the corresponding spectrum HA has n'th space K(A,n); it is called the Eilenberg–MacLane spectrum.
As a second important example, consider topological K-theory. At least for X compact, is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, 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 zero'th space is while the first space is . Here is the infinite unitary group and is its classifying space. By Bott periodicity we get and for all n, so all the spaces in the topological K-theory spectrum are given by either or . There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum.
For many more examples, see the list of cohomology theories.
Read more about this topic: Spectrum (homotopy Theory)
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