Postnikov Tower

In the mathematical sub-area of algebraic topology, a Postnikow tower or Postnikow system is a method of dividing a given topological space into Eilenberg-MacLane spaces , which enables, for example, the calculation of its homology groups using spectral sequences .

definition

Let it be a given topological space . A postnikov tower from is a consequence ${\ displaystyle X}$ ${\ displaystyle X}$

{\ displaystyle \ ldots \ to Y_ {n} \ to Y_ {n-1} \ to \ ldots \ to Y_ {2} \ to Y_ {1}}

of mappings of topological spaces with the following properties:

there is a grain for everyone ${\ displaystyle n}$ ${\ displaystyle Y_ {n} \ to Y_ {n-1}}$

for all is

{\ displaystyle k> n}

{\ displaystyle \ pi _ {k} Y_ {n} = 0}

for all is .

{\ displaystyle k \ leq n}

{\ displaystyle \ pi _ {k} Y_ {n} = \ pi _ {k} X}

A postnikov tower exists for every contiguous room . For CW complexes , the rooms in the Postnikow tower are clearly defined except for homotopy equivalence , otherwise only except for weak homotopy equivalence . ${\ displaystyle X}$

One obtains (with the exception of homotopy equivalence) by "killing" in degrees by sticking cells of the dimensions to the homotopy groups . (By replacing the inclusion with its imaging cone , one obtains a grain without changing the homotopy type.) ${\ displaystyle Y_ {n}}$ ${\ displaystyle X}$ ${\ displaystyle \ geq n + 2}$ ${\ displaystyle X}$ ${\ displaystyle \ geq n + 1}$ ${\ displaystyle Y_ {n + 1} \ to Y_ {n}}$

If there is a CW complex, then it is an Eilenberg-MacLane space and the fiber of the fiber is an Eilenberg-MacLane space . ${\ displaystyle X}$ ${\ displaystyle Y_ {1}}$ ${\ displaystyle K (\ pi _ {1} X, 1)}$ ${\ displaystyle Y_ {n} \ to Y_ {n-1}}$ ${\ displaystyle K (\ pi _ {n} X, n)}$

The mapping from in the projective limes is a weak homotopy equivalence . ${\ displaystyle X}$ ${\ displaystyle \ varprojlim _ {n \ in \ mathbb {N}} Y_ {n}}$

If the effect of on for is trivial, the fibers can be implemented as main fiber bundles. ${\ displaystyle \ pi _ {1} X}$ ${\ displaystyle \ pi _ {n} X}$ ${\ displaystyle n> 1}$ ${\ displaystyle Y_ {n} \ to Y_ {n-1}}$

MM Postnikow : Determination of the homology groups of a space by means of the homotopy invariants. In: Doklady Akad. Nauk SSSR. (NS) 76, 1951, 359-362. (Russian)
R. Bott , LW Tu: Differential forms in algebraic topology. (= Graduate Texts in Mathematics. 82). Springer-Verlag, New York / Berlin 1982, ISBN 0-387-90613-4 .
Allen Hatcher : Algebraic topology. Cambridge University Press, Cambridge 2002, ISBN 0-521-79160-X .
P. Griffiths , J. Morgan : Rational homotopy theory and differential forms. (= Progress in Mathematics. 16). 2nd Edition. Springer, New York 2013, ISBN 978-1-4614-8467-7 .