Functorial Whitehead Tower?

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The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spaces) as the connected component and the universal cover.

Can the remaining spaces be constructed functorially?

For the dual situation the answer is yes. I.e. for the Postnikov tower where we have a tower of spaces where the bottom homotopy groups are intact, but where we have killed off all the higher homotopy groups does have a functorial construction (again for nice spaces). The construction I know passes through simplicial sets. I'm wondering if something similar exists for the Whitehead tower?

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3 Answers

The nth stage of the Whitehead tower of X is the homotopy fiber of the map from X to the nth (or so) stage of its Postnikov tower, so you can use your functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the path space of the target).

The nth stage of the Whitehead tower of X is also the cofibrant replacement for X in the right Bousfield localization of Top with respect to the object Sn (or so). Since Top is right proper and cellular this localization exists by the result of chapter 5 of Hirschhorn's book on localizations of model categories. You might look there to see how the cofibrant replacement functor is constructed. With some care you should be able to define functorially the maps in the tower as well.

(BTW, the Postnikov tower can similarly be obtained functorially by a left Bousfield localization of Top.)

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I have taken the liberty of copying this comment to the nLab entry here ncatlab.org/nlab/show/…  – Urs Schreiber Nov 13 2009 at 9:13

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If we have a functorial Postnikov tower of a pointed space X, the tower inherits a basepoint. Then take the tower over the Postnikov tower which is pointwise the path fibration. This is pointwise a fibration. Pull this tower back along the map from the constant tower on X, then this gives a tower over X, which, by the general treatment in Whitehead's 1952 (I think) article, is a Whitehead tower for X.

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BTW This works for all spaces, not just locally nice ones. In a piece of shameless advertising I approach this problem from a 2-bundle point of view, and get for spaces that satisfy a 2-dimensional version of the universal covering space condition a functorial topological groupoid I claim is a good substitute for the 2-connected stage. – David Roberts Nov 13 2009 at 6:44

$X \leftarrow B\Omega X \leftarrow B^2 \Omega^2 X \leftarrow \ldots$

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