In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.
Contents
The definition of a spectrum
There are many variations of the definition: in general, a "spectrum" is any sequence
The treatment here is due to Adams (1974): a spectrum (or CW-spectrum) is a sequence
For other definitions, see symmetric spectrum and simplicial spectrum.
Examples
Consider singular cohomology
As a second important example, consider topological K-theory. At least for X compact,
For many more examples, see the list of cohomology theories.
Invariants
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 En to Fn that commute with the maps ΣEn → En+1 and ΣFn → Fn+1.
Given a spectrum
The smash product of a spectrum
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
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
and define its generalized cohomology theory by
Here
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 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 Mandell et al. (2001) for a unified treatment of these new approaches.