Postnikov system
Encyclopedia
In homotopy theory, a branch of algebraic topology
, a Postnikov system (or Postnikov tower) is a way of constructing a topological space
from its homotopy groups. Postnikov systems were introduced by, and named after, Mikhail Postnikov
.
The Postnikov system of a path-connected space X is a tower of spaces …→ Xn →…→ X1→ X0 with the following properties:
Every path-connected space has such a Postnikov system, and it is unique up to homotopy. The space X can be reconstructed from the Postnikov system as its inverse limit
: X = limn Xn. By the long exact sequence for the fibration Xn→Xn−1, the fiber (call it Kn) has at most one non-trivial homotopy group, which will be in degree n; it is thus an Eilenberg–Mac Lane space of type K(πn(X), n). The Postnikov system can be thought of as a way of constructing X out of Eilenberg–Mac Lane spaces.
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, a Postnikov system (or Postnikov tower) is a way of constructing a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
from its homotopy groups. Postnikov systems were introduced by, and named after, Mikhail Postnikov
Mikhail Postnikov
Mikhail Mikhailovich Postnikov was a Soviet mathematician, known for his work in algebraic and differential topology....
.
The Postnikov system of a path-connected space X is a tower of spaces …→ Xn →…→ X1→ X0 with the following properties:
- each map Xn→Xn−1 is a fibrationFibrationIn topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
; - πk(Xn) = πk(X) for k ≤ n;
- πk(Xn) = 0 for k > n.
Every path-connected space has such a Postnikov system, and it is unique up to homotopy. The space X can be reconstructed from the Postnikov system as its inverse limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
: X = limn Xn. By the long exact sequence for the fibration Xn→Xn−1, the fiber (call it Kn) has at most one non-trivial homotopy group, which will be in degree n; it is thus an Eilenberg–Mac Lane space of type K(πn(X), n). The Postnikov system can be thought of as a way of constructing X out of Eilenberg–Mac Lane spaces.