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Concept
Postnikov system
Summary
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 agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.
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 .
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. Note the third condition is only included optionally by some authors.
Postnikov systems exist on connected CW complexes, and there is a weak homotopy-equivalence between and its inverse limit, so
showing that 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 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 of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces are homotopic to a CW complex which differs from only by cells of dimension .
The sequence of fibrations have homotopically defined invariants, meaning the homotopy classes of maps , give a well defined homotopy type . The homotopy class of comes from looking at the homotopy class of the classifying map for the fiber . The associated classifying map is
hence the homotopy class is classified by a homotopy class
called the n-th Postnikov invariant of , since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.
Official source
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