Postnikov system
natural system, homotopic resolution, -decomposition of general type
A sequence of fibrations
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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.
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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
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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:
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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.
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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:
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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
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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) |
Comments
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 |
Postnikov system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Postnikov_system&oldid=43407