that are obtained from right to left by removing homotopy groups from below, hence such that
each is -connected
The construction of Whitehead towers is traditionally done for topological spaces regarded up to weak homotopy equivalence, hence as objects of the (∞,1)-category Top. The discussion directly generalizes to any (∞,1)-topos.
Here each homotopy pullback
in the (∞,1)-category Top may be computed (as described at homotopy pullback) as an ordinary pullback in the 1-category Top of a fibrantly replaced diagram, for instance with the point replaced by the path fibration , which is a Hurewicz fibration . In this case also the ordinary pullback
is a fibration, and this is often taken as part of the definition of the Whitehead tower.
From this perspective the Whitehead tower of a pointed space is a sequence of fibrations
where each induces isomorphisms on homotopy groups for and such that is -connected (has trivial homotopy groups for ). The homotopy long exact sequence then shows that the fiber of is a Eilenberg-Mac Lane space. One has a model for which is an abelian topological group; this has a remarkable consequence when is a topological group. Indeed, in this case one sees inductively that has a model which is a topological group, which is an abelian group extension:
For we require that is the inclusion of the path-component of . Really this is defined up to homotopy, but we have a canonical model. If is locally connected and semilocally path-connected, then can be chosen as the universal covering space.
In traditional models this construction is highly non-functorial, except for nice spaces in low dimensions as remarked above.
Whitehead 1952 answered the question, posed by Witold Hurewicz, of the existence of what we would now call -connected 'covers' of a given space , taking this to mean a fibration with -connected and otherwise inducing isomorphisms on homotopy groups.
The construction proceeds as follows (using modern terminology). Given a pointed space ,
Choose a representative for the Postnikov section such that is a closed subspace (I would be tempted to make it a closed cofibration, but I don’t know any reason for this to be necessary -DMR).
Pull this back to , to get , which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that has the desired properties.
This gives us a single -connected cover, but by considering the Postnikov tower
of , where each map is the inclusion of a closed subspace, it is simple to see there are induced maps over for all .
One way of obtaining a Postnikov section as above is to choose representatives of generators of and attaching cells: . We then choose representatives for the generators of and attach cells and so on. The colimit is then a Postnikov section with the properties we require.
Understandably, this process is unbelievably non-canonical, and so we are generally reduced to existence theorems using this method – unless there is a functorial way to construct Postnikov sections. Strictly speaking we can only say an -connected cover (except in special cases, like when and is a well-connected space).
The th stage of the Whitehead tower of is the homotopy fiber of the map from to the th (or so) stage of its Postnikov tower, so one can use a 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 th stage of the Whitehead tower of is also the cofibrant replacement for in the right Bousfield localization of Top with respect to the object (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.
where the stages are the deloopings of
where lifts through the stages correspond to
For instance can be identified as such by representing by a Kan fibration (see at Postnikov tower) between Kan complexes so that then the homotopy pullback (as discussed there) is given by an ordinary pullback. Since is a simplicial model category, can be applied and preserves the pullback as well as the homotopy pullback, hence sends to an isomorphism on connected components. This identifies as being an isomorphism on the second homotopy group. Therefore, by the Hurewicz theorem, it is also an isomorphism on the cohomology group . Analogously for the other characteristic maps.
In summary, more concisely, the tower is
where each “hook” is a fiber sequence.
|Whitehead tower of orthogonal group||orientation||spin||string||fivebrane||ninebrane|
|homotopy groups of stable orthogonal group||0||0||0||0||0||0||0||0|
|stable homotopy groups of spheres||0||0||0|
|image of J-homomorphism||0||0||0||0||0||0||0||0||0|
While a notion of Postnikov tower in an (∞,1)-category depends on the categorical homotopy groups in an (∞,1)-category, the notion of Whitehead tower makes good sense with respect to the geometric homotopy groups.
Applying the Hurewicz theorem stagewise to a Whitehead tower yields an method for computing the homotopy groups of the original space. This process, or rather the refinement thereof for Whitehead towers generalized to Adams resolutions, is formalized by the Adams spectral sequence, see there for more.
|tower diagram/filtering||spectral sequence of a filtered stable homotopy type|
|filtered chain complex||spectral sequence of a filtered complex|
|Postnikov tower||Atiyah-Hirzebruch spectral sequence|
|chromatic tower||chromatic spectral sequence|
|skeleta of simplicial object||spectral sequence of a simplicial stable homotopy type|
|skeleta of Sweedler coring of E-∞ algebra||Adams spectral sequence|
|filtration by support||…|
|slice filtration||slice spectral sequence|
The original reference is
A textbook account is around example 4.20 in
A more detailed useful discussion happens to be in section 2.A, starting on p. 11 of