Spectrum (topology)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory (which follows from Brown's representability theorem). This means given a cohomology theoryformula_1There exist spaces formula_2 such that evaluating the cohomology theory in degree formula_3 on a space formula_4 is equivalent to computing the homotopy classes of maps to the space formula_2, that isformula_6Note 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 formula_7 of pointed topological spaces or pointed simplicial sets together with the structure maps formula_8 giving homotopy equivalences.
The treatment here is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence formula_9 of CW complexes together with inclusions formula_10 of the suspension formula_11 as a subcomplex of formula_12.
For other definitions, see symmetric spectrum and simplicial spectrum.
Homotopy groups of a spectrum.
One of the most important invariants of spectra are the homotopy groups of the spectrum. 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 formula_13 define the homotopy group formula_14 as the colimitformula_15where the maps are induced from the composition of the suspension mapformula_16and the structure mapformula_17A spectrum is said to be connective if its formula_18 are zero for negative "k".
Examples.
Eilenberg–Maclane spectrum.
Consider singular cohomology formula_19 with coefficients in an abelian group formula_20. For a CW complex formula_4, the group formula_19 can be identified with the set of homotopy classes of maps from formula_4 to formula_24, the Eilenberg–MacLane space with homotopy concentrated in degree formula_25. We write this asformula_26Then the corresponding spectrum formula_27 has formula_25-th space formula_24; it is called the Eilenberg–MacLane spectrum. Note this construction can be used to embed any ring formula_30 into the category of spectra. This embedding forms the basis of used Spectral geometry as a model for Derived algebraic geometry. One of the important properties found in this embedding are the isomorphismsformula_31showing the category of spectra keeps track of 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, formula_32 is defined to be the Grothendieck group of the monoid of complex vector bundles on "X". Also, formula_33 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 formula_34 while the first space is formula_35. Here formula_35 is the infinite unitary group and formula_37 is its classifying space. By Bott periodicity we get formula_38 and formula_39 for all "n", so all the spaces in the topological K-theory spectrum are given by either formula_34 or formula_35. 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 formula_42. This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, soformula_43We can write down this spectrum explicitly as formula_44 where formula_45. Note the smash product gives a product structure on this spectrumformula_46induces a ring structure on formula_42. Moreover, if considering the category of symmetric spectra, this forms the initial object, analogous to formula_48 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 formula_49, complex cobordism formula_50, framed cobordism, spin cobordism formula_51, string cobordism formula_52, and so on. In fact, for any topological group formula_53 there is a Thom spectrum formula_54.
Suspension spectrum.
A spectrum may be constructed out of a space. The suspension spectrum of a space formula_4, denoted formula_56 is a spectrum formula_57 (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 formula_4, soformula_59The construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functorformula_60from the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given byformula_61which by the Freudenthal suspension theorem eventually stabliizes. By this we meanformula_62 and
formula_63for some finite integer formula_64. For a CW complex formula_4 there is an inverse construction formula_66 which takes a spectrum formula_13 and forms a spaceformula_68called the infinite loop space of the spectrum. For a CW complex formula_4formula_70and this construction comes with an inclusion formula_71 for every formula_25, hence gives a mapformula_73which 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. 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 (formula_74) 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" (formula_75 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.
Functions, maps, and homotopies of spectra.
There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.
A function between two spectra "E" and "F" is a sequence of maps from "E""n" to "F""n" that commute with the 
maps Σ"E""n" → "E""n"+1 and Σ"F""n" → "F""n"+1.
Given a spectrum formula_76, a subspectrum formula_77 is a sequence of subcomplexes that is also a spectrum. As each "i"-cell in formula_78 suspends to an ("i" + 1)-cell in formula_79, a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a map of spectra formula_80 to be a function from a cofinal subspectrum formula_53 of formula_13 to formula_83, where two such functions represent the same map if they coincide on some cofinal subspectrum. Intuitively such a map of spectra does not need to be everywhere defined, just "eventually" become defined, and two maps that coincide on a cofinal subspectrum are said to be equivalent. 
This gives the category of spectra (and maps), which is a major tool. There is a natural embedding of the category of pointed CW complexes into this category: it takes formula_84 to the "suspension spectrum" in which the "n"th complex is formula_85.
The smash product of a spectrum formula_13 and a pointed complex formula_4 is a spectrum given by formula_88 (associativity of the smash product yields immediately that this is indeed a spectrum). A homotopy of maps between spectra corresponds to a map formula_89, where formula_90 is the disjoint union formula_91 with formula_92 taken to be the basepoint. 
The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
Finally, we can define the suspension of a spectrum by formula_93. This translation suspension is invertible, as we can desuspend too, by setting formula_94.
The triangulated homotopy category of spectra.
The stable homotopy category is additive: maps can be added by using a variant of the track addition used to define homotopy groups. Thus homotopy classes from one spectrum to another form an abelian group. Furthermore the stable homotopy category is triangulated (Vogt (1970)), the shift being given by suspension and the distinguished triangles by the mapping cone sequences of spectra 
Smash products of spectra.
The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy. Some more recent definitions of spectra, such as symmetric spectra, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes.
The smash product is compatible with the triangulated category structure. In particular the smash product of a distinguished triangle with a spectrum is a distinguished triangle.
Generalized homology and cohomology of spectra.
We can define the (stable) homotopy groups of a spectrum to be those given by 
where formula_42 is the sphere spectrum and formula_98 is the set of homotopy classes of maps from formula_4 to formula_100. 
We define the generalized homology theory of a spectrum "E" by
and define its generalized cohomology theory by
Here formula_4 can be a spectrum or (by using its suspension spectrum) a space.
Technical complexities with spectra.
One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum formula_104formula_105sendingformula_106a pair of adjoint functors formula_107, the and the smash product formula_108 in both the category of spaces and the category of spectra. If we let formula_109 denote the category of based, compactly generated, weak Hausdorff spaces, and formula_110 denote a category of spectra, the following five axioms can never be satisfied by the specific model of spectra:
formula_121where formula_122 is the unit map in the adjunction.
Because of this, the study of spectra is fractured based upon the model being used. For an overview, check out the article cited above.
History.
A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. Spectra were adopted by Michael Atiyah and George W. Whitehead in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of J. Michael Boardman gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case. (This is essentially the category described above, and it is still used for many purposes: for other accounts, see Adams (1974) or Rainer Vogt (1970).) Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified definitions of spectrum: see Michael Mandell "et al." (2001) for a unified treatment of these new approaches.