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.
using an inverse system of topological spaces whose homotopy type at degree
agrees with the truncated homotopy type of the original space
There is a similar construction called the Whitehead tower (defined below) where instead of having spaces
Note the third condition is only included optionally by some authors.
Postnikov systems exist on connected CW complexes,[1]: 354 and there is a weak homotopy-equivalence between
They can be constructed on a CW complex by iteratively killing off homotopy groups.
One of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces
[2] have homotopically defined invariants, meaning the homotopy classes of maps
comes from looking at the homotopy class of the classifying map for the fiber
is classified by a homotopy class called the nth Postnikov invariant of
, since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.
One of the conceptually simplest cases of a Postnikov tower is that of the Eilenberg–Maclane space
, degree theory of spheres, and the Hopf fibration, giving
comes from a pullback sequence which is an element in If this was trivial it would imply
In fact, this is responsible for why strict infinity groupoids don't model homotopy types.
[3] Computing this invariant requires more work, but can be explicitly found.
One application of the Postnikov tower is the computation of homotopy groups of spheres.
, since the theorem implies that the lower homotopy groups are trivial.
Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group
Note that similar techniques can be applied using the Whitehead tower (below) for computing
, giving the first two non-trivial stable homotopy groups of spheres.
In addition to the classical Postnikov tower, there is a notion of Postnikov towers in stable homotopy theory constructed on spectra[6]pg 85-86.
, and finally, there is an exact sequence where if the middle morphism is an isomorphism, the other two groups are zero.
and noting that the Eilenberg–Maclane space has a cellular decomposition giving the desired result.
Another way to view the components in the Whitehead tower is as a homotopy fiber.
which has 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[8][9][10] in bordism theory because the coverings of the unoriented cobordism spectrum
There are physically relevant interpretations for the higher parts in this tower, which can be read as