Postnikov system

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 formula_1 agrees with the truncated homotopy type of the original space formula_2. Postnikov systems were introduced by, and are named after, Mikhail Postnikov.
Definition.
A Postnikov system of a path-connected space formula_2 is an inverse system of spaces
with a sequence of maps formula_5 compatible with the inverse system such that
The first two conditions imply that formula_14 is also a formula_15-space. More generally, if formula_2 is formula_17-connected, then formula_18 is a formula_13-space and all formula_20 for formula_21 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 formula_2 and its inverse limit, so
showing that formula_2 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 formula_25 representing a homotopy class formula_26, we can take the pushout along the boundary map formula_27, killing off the homotopy class. For formula_28 this process can be repeated for all formula_29, giving a space which has vanishing homotopy groups formula_30. Using the fact that formula_31can be constructed from formula_18 by killing off all homotopy maps formula_33, we obtain a map formula_34.
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 formula_18 are homotopic to a CW complex formula_36 which differs from formula_2 only by cells of dimension formula_38.
Homotopy classification of fibrations.
The sequence of fibrations formula_39 have homotopically defined invariants, meaning the homotopy classes of maps formula_40, give a well defined homotopy type formula_41. The homotopy class of formula_40 comes from looking at the homotopy class of the classifying map for the fiber formula_43. The associated classifying map is
hence the homotopy class formula_45 is classified by a homotopy class
called the n-th Postnikov invariant of formula_2, 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 formula_2 such that there exists a fibration
giving a homotopy type with two non-trivial homotopy groups, formula_50, and formula_51. Then, from the previous discussion, the fibration map formula_52 gives a cohomology class in
which can also be interpreted as a group cohomology class. This space formula_2 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 formula_55. This gives a tower with
Postnikov tower of S2.
The Postnikov tower for the sphere formula_57 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 formula_57, degree theory of spheres, and the Hopf fibration, giving formula_59 for formula_60, hence
Then, formula_62, and formula_63 comes from a pullback sequence
which is an element in
If this was trivial it would imply formula_66. 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 formula_67 on formula_68 coming from the Hopf fibration formula_69. Note that each element in formula_70 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 formula_71-dimensional sphere formula_72 we can use the Hurewicz theorem to show each formula_73 is contractible for formula_21, 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 formula_76-terms
And the first non-trivial map to formula_78,
equivalently written as
If it's easy to compute formula_81 and formula_82, then we can get information about what this map looks like. In particular, if it's an isomorphism, we obtain a computation of formula_78. For the case formula_84, this can be computed explicitly using the path fibration for formula_85, the main property of the Postnikov tower for formula_86 (giving formula_87, and the Universal coefficient theorem giving formula_88. Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group formula_89 since formula_90 is stable for formula_91.
Note that similar techniques can be applied using the Whitehead tower (below) for computing formula_92 and formula_93, 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.
For a spectrum formula_94 a postnikov tower of formula_94 is a diagram in the homotopy category of spectra, formula_96, given by
with maps
commuting with the formula_40 maps. Then, this tower is a Postnikov tower if the following two conditions are satisfied:
where formula_104 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 formula_2, 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 formula_113 is the universal cover of formula_2 since it is a covering space with a simply connected cover. Furthermore, each formula_115 is the universal formula_71-connected cover of formula_2.
Construction.
The spaces formula_18 in the Whitehead tower are constructed inductively. If we construct a formula_119 by killing off the higher homotopy groups in formula_18, we get an embedding formula_121. If we letformula_122for some fixed basepoint formula_123, then the induced map formula_124 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 formula_127 for formula_128, formula_129 for formula_130, 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 formula_121 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 takeformula_135from the Postnikov tower, we get a space formula_136 which hasformula_137
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 letformula_138then 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 formula_139 gives other bordism theoriesformula_140such as String bordism.
Whitehead tower and string theory.
In Spin geometry the formula_141 group is constructed as the universal cover of the Special orthogonal group formula_142, so formula_143 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 asformula_144where formula_145 is the formula_146-connected cover of formula_142 called the string group, and formula_148 is the formula_149-connected cover called the fivebrane group.