the derived information of commutative rings, where the smash product acts as the derived tensor product. Moreover, the Eilenberg–Maclane spectrum can be used to define theories such as Topological Hochschild homology for commutative rings, which gives a more refined theory of the classical Hochschild homology. Topological complex K-theory 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 zeroth 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. Sphere spectrum One of the quintessential examples of a spectrum is the sphere spectrum . This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, soWe can write down this spectrum explicitly as where . Note the smash product gives a product structure on this spectruminduces a ring structure on . Moreover, if considering the category of symmetric spectra, this forms the initial object, analogous to in the category of commutative rings. Thom spectra Another canonical example of spectra come from the Thom spectra representing various cobordism theories. This includes real cobordism , complex cobordism , framed cobordism, spin cobordism , string cobordism , and so on. In fact, for any topological group there is a Thom spectrum . Suspension spectrum A spectrum may be constructed out of a space. The suspension spectrum of a space , denoted is a spectrum (the structure maps are the identity.) For example, the suspension spectrum of the 0-sphere is the sphere spectrum discussed above. The homotopy groups of this spectrum are then the stable homotopy groups of , soThe construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functorfrom the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given bywhich by the Freudenthal suspension theorem eventually stabilizes. By this we mean and for some