Спектры в алгебраической топологии: определение и свойства.

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,there exist spaces such that evaluating the cohomology theory in degree on a space is equivalent to computing the homotopy classes of maps to the space , that is. 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.

There are many variations of the definition: in general, a spectrum is any sequence of pointed topological spaces or pointed simplicial sets together with the structure maps , where is the smash product. The smash product of a pointed space with a circle is homeomorphic to the reduced suspension of , denoted
The following is due to Frank Adams (1974): a spectrum (or CW spectrum) is a sequence of CW complexes together with inclusions of the suspension as a subcomplex of
For other definitions, see symmetric spectrum and simplicial 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 define the homotopy group as the colimitwhere the maps are induced from the composition of the map (that is, given by functoriality of ) and the structure map A spectrum is said to be connective if its are zero for negative k.

Consider singular cohomology with coefficients in an abelian group For a CW complex , the group can be identified with the set of homotopy classes of maps from to , the Eilenberg–MacLane space with homotopy concentrated in degree We write this asThen the corresponding spectrum has th space ; it is called the Eilenberg–MacLane spectrum of Note this construction can be used to embed any ring 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 isomorphismsshowing 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.

As a second important example, consider topological K theory. At least for X compact, is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, 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 while the first space is Here is the infinite unitary group and is its classifying space. By Bott periodicity we get and for all n, so all the spaces in the topological K theory spectrum are given by either or There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8 periodic spectrum.

One of the quintessential examples of a spectrum is the sphere spectrum This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, soWe can write down this spectrum explicitly as where Note the smash product gives a product structure on this spectruminduces a ring structure on Moreover, if considering the category of symmetric spectra, this forms the initial object, analogous to in the category of commutative rings.

Another canonical example of spectra come from the Thom spectra representing various cobordism theories. This includes real cobordism , complex cobordism , framed cobordism, spin cobordism , string cobordism , and so on. In fact, for any topological group there is a Thom spectrum .

A spectrum may be constructed out of a space. The suspension spectrum of a space , denoted is a spectrum (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 , soThe construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functorfrom the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given bywhich by the Freudenthal suspension theorem eventually stabilizes. By this we mean and for some finite integer For a CW complex there is an inverse construction which takes a spectrum and forms a spacecalled the infinite loop space of the spectrum. For a CW complex and this construction comes with an inclusion for every , hence gives a mapwhich 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).

An Ω spectrum is a spectrum such that the adjoint of the structure map (i. e., the map) is a weak equivalence. The K theory spectrum of a ring is an example of an Ω 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" ( 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.

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
.

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.

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.