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Wikipedia / Postnikov system

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type where that are fibrations with Eilenberg-MacLane spaces as fibers. In short, we are decomposing the homotopy type using an inverse system of topological spaces whose homotopy type. Postnikov systems were introduced by, and are named after, Mikhail Postnikov.

There is a similar construction called the Whitehead tower.

Definition

A Postnikov system of a path-connected space is an inverse system of spaces

compatible with the inverse system such that

  1. .

  2. .

  3. is a fibration is an Eilenberg–MacLane space.

are contractible. Note the third condition is only included optionally by some authors.

Existence

Postnikov systems exist on connected CW complexes, and there is a weak homotopy-equivalence and its inverse limit, so

  1. ,

is a CW approximation, we can take the pushout.

Main property

.

Homotopy classification of fibrations

comes from looking at the homotopy class of the classifying map. The associated classifying map is

  1. ,

is classified by a homotopy class

called the nth Postnikov invariant, since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.

Fiber sequence for spaces with two nontrivial homotopy groups

such that there exists a fibration

giving a homotopy type gives a cohomology class in

  1. ,

which can also be interpreted as a group cohomology class can be considered a higher local system.

Examples of Postnikov towers

Postnikov tower of a K(G, n)

. This gives a tower with

Postnikov tower of S2

is a special case whose first few terms can be understood explicitly. Since we have the first few homotopy groups from the simply connectedness, degree theory of spheres, and the Hopf fibration, hence

comes from a pullback sequence

which is an element in

  1. .

. But, this is not the case! In fact, this is responsible for why strict infinity groupoids don't model homotopy types. Computing this invariant requires more work, but can be explicitly found. gives a different homotopy 3-type.

Homotopy groups of spheres

One application of the Postnikov tower is the computation of homotopy groups of spheres. For an n-dimensional sphere we can use the Hurewicz theorem, since the theorem implies that the lower homotopy groups are trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration

  1. .

-terms

  1. .

,

  1. ,

equivalently written as

  1. .

, and the universal coefficient theorem. Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group.

, giving the first two non-trivial stable homotopy groups of spheres.

Postnikov towers of spectra

In addition to the classical Postnikov tower, there is a notion of Postnikov towers in stable homotopy theory constructed on spectrapg 85-86.

Definition

, given by

  1. ,

with maps

maps. Then, this tower is a Postnikov tower if the following two conditions are satisfied:

  1. ,

  2. ,

are stable homotopy groups of a spectrum. It turns out every spectrum has a Postnikov tower and this tower can be constructed using a similar kind of inductive procedure as the one given above.

Whitehead tower

, there is a dual construction to the Postnikov tower called the Whitehead tower. Instead of killing off all higher homotopy groups, the Whitehead tower iteratively kills off lower homotopy groups. This is given by a tower of CW complexes,

  1. ,

where

  1. .

  2. .

  3. .

Implications

since it is a covering space.

Construction

,. If we let

for some fixed basepoint is a fiber bundle with fiber homeomorphic to

  1. ,

and so we have a Serre fibration

  1. .

Using the long exact sequence, and finally, there is an exact sequence

  1. ,

and noting that the Eilenberg–Maclane space has a cellular decomposition

  1. ; thus,

  2. ,

giving the desired result.

As a homotopy fiber

Another way to view the components in the Whitehead tower is as a homotopy fiber. If we take

which has

Whitehead tower of spectra

The dual notion of the Whitehead tower can be defined in a similar manner using homotopy fibers in the category of spectra. If we let

then this can be organized in a tower giving connected covers of a spectrum. This is a widely used construction in bordism theory gives other bordism theories

such as string bordism.

Whitehead tower and string theory

In Spin geometry group is constructed as the universal cover of the Special orthogonal group is a fibration, giving the first term in the Whitehead tower. There are physically relevant interpretations for the higher parts in this tower, which can be read as called the string group-connected cover called the fivebrane group.

See also

References