In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory<blockquote><math>\mathcal{E}^*:\text{CW}^{op} \to \text{Ab}</math>,</blockquote>there exist spaces <math>E^k</math> such that evaluating the cohomology theory in degree <math>k</math> on a space <math>X</math> is equivalent to computing the homotopy classes of maps to the space <math>E^k</math>, that is<blockquote><math>\mathcal{E}^k(X) \cong \left[X, E^k\right]</math>.</blockquote>Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
The definition of a spectrum
There are many variations of the definition: in general, a spectrum is any sequence <math>X_n</math> of pointed topological spaces or pointed simplicial sets together with the structure maps <math>S^1 \wedge X_n \to X_{n+1}</math>, where <math>\wedge</math> is the smash product. The smash product of a pointed space <math>X</math> with a circle is homeomorphic to the reduced suspension of <math>X</math>, denoted <math>\Sigma X</math>.
The following is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence <math>E:= \{E_n\}_{n\in \mathbb{N </math> of CW complexes together with inclusions <math> \Sigma E_n \to E_{n+1} </math> of the suspension <math> \Sigma E_n </math> as a subcomplex of <math> E_{n+1} </math>.
For other definitions, see symmetric spectrum and simplicial spectrum.
Homotopy groups of a spectrum
Some of the most important invariants of a spectrum are its homotopy groups. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum <math>E</math> define the homotopy group <math>\pi_n(E)</math> as the colimit<blockquote><math>\begin{align}
\pi_n(E) &= \lim_{\to k} \pi_{n+k}(E_k) \\
&= \lim_\to \left(\cdots \to \pi_{n+k}(E_k) \to \pi_{n+k+1}(E_{k+1}) \to \cdots\right)
\end{align}</math></blockquote>where the maps are induced from the composition of the map <math>\Sigma: \pi_{n+k}(E_n) \to \pi_{n+k+1}(\Sigma E_n)</math> (that is, <math> [S^{n+k}, E_n] \to [S^{n+k+1}, \Sigma E_n]</math> given by functoriality of <math>\Sigma</math>) and the structure map <math>\Sigma E_n \to E_{n+1}</math>. A spectrum is said to be connective if its <math>\pi_k</math> are zero for negative k.
Examples
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. 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{LR/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
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 spectra, this forms the initial object, analogous to <math>\mathbb{Z}</math> 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 <math>MO</math>, complex cobordism <math>MU</math>, framed cobordism, spin cobordism <math>MSpin</math>, string cobordism <math>MString</math>, and 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.