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 The map induces an isomorphism for every. for. There is an additional technical condition which some authors include as an additional axiom: Each map is a fibration, and so the fiber is an Eilenberg–MacLane space,. The first two conditions imply that is also a -space. More generally, if is -connected, then is a -space and all for are contractible.
Existence
Postnikov systems exist on connected CW complexes, 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 obtain a map.
One application of the Postnikov tower is the computation of homotopy groups of spheres. For an -dimensional sphere we can use the Hurewicz theorem to show each is contractible for, 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 -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.
Whitehead tower
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 The lower homotopy groups are zero, so for. The induced map is an isomorphism for. The maps are fibrations with fiber.
The spaces in the Whitehead tower are constructed inductively. If we construct a by killing off the higher homotopy groups in, we get an embedding. If we let for some fixed basepoint, 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, 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 thus, giving the desired result.
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.
