In
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
, a branch of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a Postnikov system (or Postnikov tower) is a way of decomposing a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
's
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s using an
inverse system
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
of topological spaces whose
homotopy type
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
at degree
agrees with the truncated homotopy type of the original space
. Postnikov systems were introduced by, and are named after,
Mikhail Postnikov.
Definition
A Postnikov system of a
path-connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
is an inverse system of spaces
:
with a sequence of maps
compatible with the inverse system such that
# The map
induces an isomorphism
for every
.
#
for
.
# Each map
is a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
, and so the fiber
is an
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
,
.
The first two conditions imply that
is also a
-space. More generally, if
is
-connected, then
is a
-space and all
for
are
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
. Note the third condition is only included optionally by some authors.
Existence
Postnikov systems exist on connected
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
es,
and there is a
weak homotopy-equivalence between
and its inverse limit, so
:
,
showing that
is a CW approximation of its inverse limit. They can be constructed on a CW complex by iteratively killing off homotopy groups. If we have a map
representing a homotopy class
, we can take the
pushout
A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...
along the boundary map
, killing off the homotopy class. For
this process can be repeated for all
, giving a space which has vanishing homotopy groups
. Using the fact that
can be constructed from
by killing off all homotopy maps
, we obtain a map
.
Main property
One of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces
are homotopic to a CW complex
which differs from
only by cells of dimension
.
Homotopy classification of fibrations
The sequence of fibrations
have homotopically defined invariants, meaning the homotopy classes of maps
, give a well defined homotopy type
. The homotopy class of
comes from looking at the homotopy class of the
classifying map for the fiber
. The associated classifying map is
:
,
hence the homotopy class