Whitehead Tower

In mathematics , the Whitehead tower of a topological space is an aid in the calculation of homotopy groups .

definition

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

${\displaystyle \ldots \to X_{n}\to X_{n-1}\to \ldots \to X_{2}\to X_{1}\to X_{0}=X}$

of mappings of topological spaces with the following properties:

• for all is a fibrillation , a fiber whose Eilenberg-MacLane space is ${\displaystyle n}$ ${\displaystyle X_{n}\to X_{n-1}}$ ${\displaystyle K(\pi _{n}X,n-1)}$
• ${\displaystyle X_{n}}$is -connected, ie for all is ${\displaystyle n}$ ${\displaystyle k\leq n}$ ${\displaystyle \pi _{k}X_{n}=0}$
• for all is . ${\displaystyle k>n}$ ${\displaystyle \pi _{k}X_{n}=\pi _{k}X}$

construction

${\displaystyle X_{1}}$is the universal overlay of . ${\displaystyle X}$

${\displaystyle X_{n}}$is constructed from as follows. First, one can in a room by the weak homotopy type of embedding by successively all homotopy groups of dimensions by adhesion of cells of dimensions "kills". Then one defines the space of all paths in which start in a base point and end in. ${\displaystyle X_{n-1}}$ ${\displaystyle X_{n-1}}$ ${\displaystyle Y_{n}}$ ${\displaystyle K(\pi _{n}X,n)=K(\pi _{n}X_{n-1},n)}$ ${\displaystyle n+1,n+2,\ldots }$ ${\displaystyle n+2,n+3,\ldots }$ ${\displaystyle X_{n}=\Omega _{*}^{X_{n-1}}\subset \Omega K(\pi _{n}X,n)}$ ${\displaystyle K(\pi _{n}X,n)}$ ${\displaystyle *}$ ${\displaystyle X_{n-1}}$

The "endpoint" projection is a fiber whose fiber is the loop space . This one has the weak homotopy type . ${\displaystyle X_{n}\to X_{n-1}}$ ${\displaystyle \Omega Y_{n}}$ ${\displaystyle K(\pi _{n}X,n-1)}$

If there is a CW complex , then the fiber is a CW complex and in particular according to Whitehead's theorem one . If, in addition, the higher homotopy groups are finitely generated , then the homotopyu-equivalent to a topological Abelian group and the construction can be carried out in such a way that principal bundles with an Abelian structural group are for the fibers . ${\displaystyle X}$ ${\displaystyle K(\pi _{n}X,n-1)}$ ${\displaystyle \pi _{k}X,k\geq 2}$ ${\displaystyle K(\pi _{n}X,n-1)}$ ${\displaystyle n>1}$ ${\displaystyle X_{n}\to X_{n-1}}$

See also

• Postnikov Tower

literature

• H. Cartan, J.-P. Serre: Espaces fibers et groupes d'homotopie. I. Constructions generales. CR Acad. Sci. Paris 234, (1952).
• GW Whitehead: Fiber spaces and the Eilenberg homology groups. Proc. Nat. Acad. Sci. USA 38, (1952). 426-430. PMC 1063578 (free full text)
• R. Bott, L. Tu: Differential forms in algebraic topology. Graduate Texts in Mathematics, 82nd Springer-Verlag, New York-Berlin, 1982. ISBN 0-387-90613-4 .