Paths and cylinders
The Whitehead tower of a pointed homotopy type is an interpolation of the point inclusion by a sequence of homotopy types
that are obtained from right to left by removing homotopy groups from below, hence such that
each is -connected
and each morphism induces an isomorphism on all homotopy groups in degree (and the inclusion in degree as well as the identity in degree ).
The notion of Whitehead tower is dual to the notion of Postnikov tower, which instead is a factorization of the terminal morphism into a tower, where homotopy groups are added from right to left.
In fact, the Whitehead tower may be constructed by taking each stage to be the homotopy fiber of the corresponding map into the st stage of the Postnikov tower.
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.
The Whitehead tower of a homotopy type is a sequence of homotopy types
where the space is the homotopy fiber of the map into the item in the Postnikov tower of .
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 instance, the string group can be realized as a topological group as a -extension of the spin group.
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).
Form the -connected cover of , i.e. the path fibration . This is a Hurewicz fibration.
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.
Whitehead tower of the orthogonal group
The Whitehead tower of the classifying space/delooping of the orthogonal group starts out as
where the stages are the deloopings of
… fivebrane group string group spin group special orthogonal group orthogonal group,
where lifts through the stages correspond to
and where the obstruction classes are the universal characteristic classes
and where every possible square in the above is a homotopy pullback square (using the pasting law).
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.
Via the J-homomorphism this corresponds to the stable homotopy groups of spheres:
|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|
Whitehead tower in general -toposes
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.
A good notion of geometric homotopy groups in an (∞,1)-topos exist in a locally contractible (∞,1)-topos. The notion of Whitehead tower in this context is discussed at
The original reference is
- George Whitehead Fiber Spaces and the Eilenberg Homology Groups, PNAS 38, No. 5 (1952)
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
- Linus Kramer, Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface Memoirs of the American Mathematical Society number 752 (web) also (arXiv)