Editing Spectrum (topology) (section)

== Examples ==

=== Eilenberg–Maclane spectrum ===
{{Main articles|Eilenberg–Maclane spectrum}}

Consider [[singular cohomology]] <math> H^n(X;A) </math> with coefficients in an [[abelian group]] <math>A</math>. For a [[CW complex]] <math>X</math>, the group <math> H^n(X;A) </math> can be identified with the set of homotopy classes of maps from <math>X</math> to <math>K(A,n)</math>, the [[Eilenberg–MacLane space]] with homotopy concentrated in degree <math>n</math>. We write this as<blockquote><math>[X,K(A,n)] = H^n(X;A)</math></blockquote>Then the corresponding spectrum <math>HA</math> has <math>n</math>-th space <math>K(A,n)</math>; it is called the '''Eilenberg–MacLane spectrum''' of <math>A</math>. Note this construction can be used to embed any ring <math>R</math> into the category of spectra. This embedding forms the basis of spectral geometry, a model for [[derived algebraic geometry]]. One of the important properties of this embedding are the isomorphisms<blockquote><math>\begin{align}
  \pi_i( H(R/I) \wedge_R H(R/J) ) &\cong H_i\left(R/I\otimes^{\mathbf{L}}R/J\right)\\
                                  &\cong \operatorname{Tor}_i^R(R/I,R/J)
\end{align}</math></blockquote>showing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the [[derived tensor product]]. Moreover, Eilenberg–Maclane spectra can be used to define theories such as [[topological Hochschild homology]] for commutative rings, a more refined theory than classical Hochschild homology.

=== 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]].

=== Sphere spectrum ===
{{Main|Sphere spectrum}}

One of the quintessential examples of a spectrum is the [[sphere spectrum]] <math>\mathbb{S}</math>. This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, so<blockquote><math>\pi_n(\mathbb{S}) = \pi_n^{\mathbb{S}}</math></blockquote>We can write down this spectrum explicitly as <math>\mathbb{S}_i = S^i</math> where <math>\mathbb{S}_0 = \{0, 1\}</math>. Note the smash product gives a product structure on this spectrum<blockquote><math>S^n \wedge S^m \simeq S^{n+m}</math></blockquote>induces a ring structure on <math>\mathbb{S}</math>. Moreover, if considering the category of [[Symmetric spectrum|symmetric spectra]], this forms the initial object, analogous to <math>\mathbb{Z}</math> in the category of commutative rings.

=== Thom spectra ===
{{Main|Thom spectrum}}
Another canonical example of spectra come from the [[Thom spectrum|Thom spectra]] representing various cobordism theories. This includes real cobordism <math>MO</math>, complex cobordism <math>MU</math>, framed cobordism, spin cobordism <math>MSpin</math>, string cobordism <math>MString</math>, and [[Whitehead tower|so on]]. In fact, for any topological group <math>G</math> there is a Thom spectrum <math>MG</math>.

=== 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).

=== Ω-spectrum ===
An '''Ω-spectrum''' is a spectrum such that the adjoint of the structure map (i.e., the map<math>X_n \to \Omega X_{n+1}</math>) is a weak equivalence. The [[K-theory spectrum]] of a ring is an example of an Ω-spectrum.

=== Ring spectrum ===
A '''[[ring spectrum]]''' is a spectrum ''X'' such that the diagrams that describe [[ring axioms]] in terms of smash products commute "up to homotopy" (<math>S^0 \to X</math> corresponds to the identity.) For example, the spectrum of topological ''K''-theory is a ring spectrum. A '''[[module spectrum]]''' may be defined analogously.

For many more examples, see the [[list of cohomology theories]].