Postnikov system

natural system, homotopic resolution, -decomposition of general type

A sequence of fibrations

whose fibres are the Eilenberg–MacLane spaces (cf. Eilenberg–MacLane space), where is some group (Abelian for ). This system was introduced by M.M. Postnikov [1]. The space is called the -th term (or the -th layer) of the Postnikov system . The Postnikov system is said to converge to a space if its inverse limit is weakly homotopy equivalent to . In this case is called the limit of the Postnikov system .

A morphism of a Postnikov system into a Postnikov system is a sequence of continuous mappings such that the diagram below is homotopy commutative. A morphism induces a mapping : , which is called its limit.

The definition of a Postnikov system implies that for any the mapping is an -equivalence (see Homotopy type). In particular, for , and for . The spaces and are of the same -type. In particular, if the Postnikov system is finite, i.e. if for some number for all the group is trivial, then and are homotopy equivalent. In the general case, for there are isomorphisms and , i.e. the homology groups and the homotopy groups stabilize when tends to infinity. For any CW-complex of dimension the sets and coincide. The characteristic class of the fibration , i.e. the image under the transgression

of the fundamental class , is called the -th -invariant (or the -th Postnikov factor) of the Postnikov system or of its limit . For any the -th term of the Postnikov system, and hence the -type of , are completely determined by the groups and the -invariants . Often the double sequence below is called a Postnikov system:

A space is the limit of a Postnikov system if and only if there exist -equivalences such that for any . Limits of morphisms of a Postnikov system are characterized analogously.

There exists a version of the notion of a Postnikov system which sometimes turns out to be more useful. In this version the spaces are assumed to be CW-complexes such that and , and the mappings are taken to be cellular mappings (which are not fibrations any more) such that, first, and, secondly, the homotopy fibre of the mapping (i.e. the fibre of this mapping turned into a fibration) is the space . Such Postnikov systems are called cellular. The limit of a cellular Postnikov system is a CW-complex for which for any . An arbitrary Postnikov system is homotopy equivalent to a cellular Postnikov system.

The fundamental theorem in the theory of Postnikov systems states (see [1], [6]) that each space is the limit of some unique (up to isomorphism) Postnikov system . This Postnikov system is called the Postnikov system of the space . A version of the fundamental theorem for mappings holds: Any mapping is the limit of some morphism of the Postnikov system of into the Postnikov system of . This morphism is called the Postnikov system of the mapping (it is also called the homotopic resolution, the -system of general type or the Moore–Postnikov system of the mapping). For a constant mapping of a path-connected space its Postnikov system coincides with the Postnikov system of the space .

In applications the so-called standard Postnikov systems (often called just Postnikov systems) are widely spread. These systems are Postnikov systems which consist of the principal fibrations induced from the standard Serre fibrations by the Postnikov factors interpreted as mappings by virtue of the representation of the cohomology group as . All spaces which are homotopy simple in all dimensions (Abelian spaces in the terminology of [2]) and only these spaces have standard Postnikov systems (see [3], [4]).

Standard Postnikov systems are applied to solve extension and lifting problems to which numerous problems in algebraic topology are reduced. The combined formulation of these problems is the following. Let a (homotopy) commutative square of spaces and mappings be given in which the mapping is a closed cofibration with cofibre and is a fibration with fibre . The question is whether there exists a mapping such that both triangles obtained are (homotopy) commutative.

Further, if such a mapping does exist, then one is expected to determine the set of homotopy classes of mappings "below A" (i.e. ) and "above B" . Suppose that for the fibration there exists a standard Postnikov system (for this purpose, for example, it is sufficient for the spaces and to be simply connected). The problem of relative lifting is solved step by step.

Consider the "elementary" problem of the relative lifting of a mapping from the -st term of the Postnikov system to its -th term:

The mappings and define a mapping , i.e. a cohomology class , called an obstruction. The mapping can be lifted to if and only if . Two liftings and determine an element , called a difference, which is equal to zero if and only if the liftings and are homotopic.

Thus, the problem of relative lifting is solved if the sequentially occurring obstructions vanish (for example, if ). A lifting is unique if the sequentially occurring differences vanish (for example, if ). In the case when the cofibration is an imbedding of CW-complexes, the obstruction and the difference coincide with usual "cell-wise" obstruction and difference (cf. Difference cochain and chain).

For simply-connected spaces with finitely-generated homology groups the Postnikov system is effectively computable [5] and, hence, the homotopy type of is effectively computable as well. However, in practice, for the majority of spaces one succeeds to compute only initial segments of Postnikov systems, which is due to the sharply increasing complexity of the computations. For computations one uses the method of cohomology operations (cf. Cohomology operation).

The dual of the Postnikov system is the Cartan–Serre system

of a space , consisting of fibrations whose fibres are the Eilenberg–MacLane spaces . The space is called the ()-st killing space for . The terms of the Cartan–Serre system are homotopy fibres of ()-equivalences for the Postnikov system of , and the terms of a Postnikov system are loop spaces over the fibres of .

A split Postnikov system is a sequence of principal fibrations

whose fibres are the Eilenberg–MacLane spaces , . Split Postnikov systems are the principal tool in the studies of so-called nilpotent spaces and, in particular, of their localizations (see Localization in categories, [2], [6], [7]). There also exist other versions of Postnikov systems (see [6]).

References

[1]	M.M. Postnikov, "Studies on the homotopy theory of continuous mappings" , 1–2 , Moscow (1955) (In Russian)
[2]	M.M. Postnikov, "Localization of topological spaces" Russian Math. Surveys , 32 : 6 (1977) pp. 121–184 Uspekhi Mat. Nauk , 32 : 6 (1977) pp. 117–181
[3]	R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) pp. Chapt. 13
[4]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[5]	E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20
[6]	H.J. Baues, "Obstruction theory of homotopy classification of maps" , Springer (1977)
[7]	P. Hilton, G. Mislin, J. Roitberg, "Localization of nilpotent groups and spaces" , North-Holland (1975)

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. Chapt. IX
[a2]	B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. Chapt. 17