Editing Spectrum (topology) (section)

=== Suspension spectrum ===
A spectrum may be constructed out of a space. The '''suspension spectrum''' of a space <math>X</math>, denoted <math>\Sigma^\infty X</math> is a spectrum <math>X_n = S^n \wedge X</math> (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 <math>X</math>, so<blockquote><math>\pi_n(\Sigma^\infty X) = \pi_n^\mathbb{S}(X)</math></blockquote>The construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functor<blockquote><math>\Sigma^\infty:h\text{CW} \to h\text{Spectra}</math></blockquote>from the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given by<blockquote><math>[\Sigma^\infty X, \Sigma^\infty Y] = \underset{\to n}{\operatorname{colim}{}}[\Sigma^nX,\Sigma^nY]</math></blockquote>which by the [[Freudenthal suspension theorem]] eventually stabilizes. By this we mean<blockquote><math>\left[\Sigma^N X, \Sigma^N Y\right] \simeq \left[\Sigma^{N+1} X, \Sigma^{N+1} Y\right] \simeq \cdots</math> and <math>\left[\Sigma^\infty X, \Sigma^\infty Y\right] \simeq \left[\Sigma^N X, \Sigma^N Y\right]</math></blockquote>for some finite integer <math>N</math>. For a CW complex <math>X</math> there is an inverse construction <math>\Omega^\infty</math> which takes a spectrum <math>E</math> and forms a space<blockquote><math>\Omega^\infty E = \underset{\to n}{\operatorname{colim}{}}\Omega^n E_n</math></blockquote>called the [[infinite loop space]] of the spectrum. For a CW complex <math>X</math><blockquote><math>\Omega^\infty\Sigma^\infty X = \underset{\to}{\operatorname{colim}{}} \Omega^n\Sigma^nX</math></blockquote>and this construction comes with an inclusion <math>X \to \Omega^n\Sigma^n X</math> for every <math>n</math>, hence gives a map<blockquote><math>X \to \Omega^\infty\Sigma^\infty X</math></blockquote>which is injective. Unfortunately, these two structures, with the addition of the smash product, lead to significant complexity in the theory of spectra because there cannot exist a single category of spectra which satisfies a list of five axioms relating these structures.<ref name=":0" /> The above adjunction is valid only in the homotopy categories of spaces and spectra, but not always with a specific category of spectra (not the homotopy category).