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. What this looks like is for a space X there is a list of spaces where and there's a series of maps that are fibrations with fibers Eilenberg-MacLane spaces. In short, we are decomposing the homotopy type of X using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X_n with the homotopy type of X for degrees \leq n, these spaces have null homotopy groups for 1 < k < n.

Definition

A Postnikov system of a path-connected space X is an inverse system of spaces with a sequence of maps compatible with the inverse system such that The first two conditions imply that X_1 is also a -space. More generally, if X is (n-1)-connected, then X_n is a -space and all X_{i} for i < n 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 between X and its inverse limit, so showing that X 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 along the boundary map , killing off the homotopy class. For X_{m} this process can be repeated for all n > m, giving a space which has vanishing homotopy groups \pi_n(X_m). Using the fact that X_{n-1}can be constructed from X_n 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 X_n are homotopic to a CW complex which differs from X only by cells of dimension \geq n+2.

Homotopy classification of fibrations

The sequence of fibrations have homotopically defined invariants, meaning the homotopy classes of maps p_n, give a well defined homotopy type. The homotopy class of p_n comes from looking at the homotopy class of the classifying map for the fiber. The associated classifying map is hence the homotopy class [p_n] is classified by a homotopy class called the nth Postnikov invariant of X, 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

One of the special cases of the homotopy classification is the homotopy class of spaces X such that there exists a fibration giving a homotopy type with two non-trivial homotopy groups,, and. Then, from the previous discussion, the fibration map gives a cohomology class in which can also be interpreted as a group cohomology class. This space X can be considered a higher local system.

Examples of Postnikov towers

Postnikov tower of a K(G, n)

One of the conceptually simplest cases of a Postnikov tower is that of the Eilenberg–Maclane space K(G,n). This gives a tower with

Postnikov tower of S2

The Postnikov tower for the sphere S^2 is a special case whose first few terms can be understood explicitly. Since we have the first few homotopy groups from the simply connectedness of S^2, degree theory of spheres, and the Hopf fibration, giving for k \geq 3, hence Then,, and X_3 comes from a pullback sequence which is an element in If this was trivial it would imply. 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. This is the quadratic form on \Z \to \Z coming from the Hopf fibration S^3 \to S^2. Note that each element in 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 S^n we can use the Hurewicz theorem to show each S^n_i is contractible for i < n, 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 We can then form a homological spectral sequence with E^2-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 obtain a computation of. For the case n = 3, this can be computed explicitly using the path fibration for K(\Z, 3), the main property of the Postnikov tower for (giving, and the universal coefficient theorem giving . Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group since is stable for. Note that similar techniques can be applied using the Whitehead tower (below) for computing and, 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 spectra pg 85-86.

Definition

For a spectrum E a postnikov tower of E is a diagram in the homotopy category of spectra,, given by with maps commuting with the p_n maps. Then, this tower is a Postnikov tower if the following two conditions are satisfied: where 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

Given a CW complex X, 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

Implications

Notice X_1 \to X is the universal cover of X since it is a covering space with a simply connected cover. Furthermore, each X_n \to X is the universal n-connected cover of X.

Construction

The spaces X_n in the Whitehead tower are constructed inductively. If we construct a by killing off the higher homotopy groups in X_n, we get an embedding. If we let for some fixed basepoint p, then the induced map is a fiber bundle with fiber homeomorphic to and so we have a Serre fibration Using the long exact sequence in homotopy theory, we have that for, for i < n-1, 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 that the Eilenberg–Maclane space has a cellular decomposition 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 from the Postnikov tower, we get a space X^n 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 because the coverings of the unoriented cobordism spectrum M\text{O} gives other bordism theories such as string bordism.

Whitehead tower and string theory

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 3-connected cover of called the string group, and is the 7-connected cover called the fivebrane group.