Postnikov system

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In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agree's with the truncated homotopy type of the original space . Postnikov systems were introduced by, and named after, Mikhail Postnikov.

Definition[edit]

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

with a sequence of maps compatible with the inverse system such that

(1) induces an isomorphism for every

(2) for [1]

There is an additional technical condition which some authors include as an additional axiom:

(3) Each map is a fibration, and so the fiber is an Eilenberg–Maclane space, .

Note that the first two conditions imply is also a -space; more generally, if is -connected, then is a -space and all for are contractible.

Existence[edit]

Postnikov systems exist on connected CW complexes [2] and there is a weak homotopy-equivalence between and showing is a CW approximation of . 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 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 get a map .

Homotopy groups of spheres[edit]

One application of the Postnikov tower is the computation of homotopy groups of spheres. For we can use the Hurewicz theorem to show each is contractible for since the theorem implies the lower homotopy groups are trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration

We can then form a homological spectral sequence with -terms

.

And the first non-trivial map to

equivalently written as

If it's easy to compute and , then we can get information about what this map looks like. In particular, if it's an isomorphism, we get a computation of .

Whitehead tower[edit]

Given a CW complex 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

where

(1) The lower homotopy groups are zero, so for

(2) The induced map is an isomorphism for

(3) The maps are fibrations with fiber

Implications[edit]

Notice is the universal cover of since it is a covering space with a simply connected cover. Furthermore each is the universal -connected cover of .

Construction[edit]

The spaces in the Whitehead tower are constructed inductively. If we construct a by killing off the higher homotopy groups in [3], we get an embedding . If we let

for some fixed base point , then the induced map is a fiber bundle with fiber homeomorphic to

so we have a Serre fibration

Using the long exact sequence in homotopy theory, we have for , for , and finally, there is an exact sequence

where if the middle morphism is an isomorphism, the other two groups are zero. This can be checked by looking at the inclusion and noting the Eilenberg-maclane space has a cellular decomposition

so

giving the desired result.

Whitehead tower and string theory[edit]

In Spin geometry the group is constructed as the universal cover of the Special orthogonal group , so 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

where is the -connected cover of , and is the -connected cover.[4][5]

See also[edit]

References[edit]

  1. ^ Hatcher, Allen. Algebraic Topology (PDF). p. 410.
  2. ^ Hatcher, Allen. Algebraic Topology (PDF). p. 354.
  3. ^ Maxim, Laurentiu. "Lecture Notes on Homotopy Theory and Applications" (PDF). p. 66. Archived (PDF) from the original on 16 Feb 2020.
  4. ^ "Mathematical physics – Physical application of Postnikov tower, String(n) and Fivebrane(n)". Physics Stack Exchange. Retrieved 2020-02-16.
  5. ^ "at.algebraic topology – What do Whitehead towers have to do with physics?". MathOverflow. Retrieved 2020-02-16.