The Whitehead tower of a pointed homotopy type $X$ is an interpolation of the point inclusion $* \to X$ by a sequence of homotopy types
that are obtained from right to left by removing homotopy groups from below, hence such that
each $X^{(n)}$ is $(n-1)$-connected
and each morphism $X^{(n+1)} \to X^{(n)}$ induces an isomorphism on all homotopy groups in degree $k \geq (n+1)$ (and the inclusion $1 \to \pi_n(X^{(n)})$ in degree $n$ as well as the identity $1 = 1$ in degree $k \lt n$).
The notion of Whitehead tower is dual to the notion of Postnikov tower, which instead is a factorization of the terminal morphism $X \to *$ into a tower, where homotopy groups are added from right to left.
In fact, the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X^{(n)}$ to be the homotopy fiber of the corresponding map into the $(n+1)$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 $X$ is a sequence of homotopy types
where the space $X^{(n)}$ is the homotopy fiber of the map $X \to X_{(n+1)}$ into the item $X_{(n+1)}$ in the Postnikov tower of $X$.
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 $P X_{(n+1)} \simeq *$, which is a Hurewicz fibration $P X_{(n+1)} \to X_{(n+1)}$. In this case also the ordinary pullback $X^{(n)}\to X$
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 $(X,x)$ is a sequence of fibrations
where each $X\langle n\rangle \to X\langle n-1 \rangle$ induces isomorphisms on homotopy groups $\pi_i$ for $i\gt n$ and such that $X\langle n\rangle$ is $n$-connected (has trivial homotopy groups $\pi_i$ for $i \leq n$). The homotopy long exact sequence then shows that the fiber of $X\langle n\rangle \to X\langle n-1 \rangle$ is a $K(\pi_{n-1}(X,x),n-1)$ Eilenberg-Mac Lane space. One has a model for $K(\pi_{n-1}(X,x),n-1)$ which is an abelian topological group; this has a remarkable consequence when $(X,x)=(G,e)$ is a topological group. Indeed, in this case one sees inductively that $G\langle n\rangle$ 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 $K(\mathbb{Z},2)$-extension of the spin group.
For $n=0$ we require that $X\langle 0 \rangle \hookrightarrow X$ is the inclusion of the path-component of $x$. Really this is defined up to homotopy, but we have a canonical model. If $X$ is locally connected and semilocally path-connected, then $X\langle 1\rangle$ 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 $n$-connected 'covers' of a given space $X$, taking this to mean a fibration $X\langle n\rangle \to X$ with $X\langle n\rangle$ $n$-connected and otherwise inducing isomorphisms on homotopy groups.
The construction proceeds as follows (using modern terminology). Given a pointed space $(X,x)$,
Choose a representative for the Postnikov section $X_n$ such that $X \hookrightarrow X_n$ 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 $\infty$-connected cover of $X_n$, i.e. the path fibration $P X_n$. This is a Hurewicz fibration.
Pull this back to $X$, to get $p\colon X\langle n\rangle \to X$, which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that $p$ has the desired properties.
This gives us a single $n$-connected cover, but by considering the Postnikov tower
of $X$, where each map $X \to X_n$ is the inclusion of a closed subspace, it is simple to see there are induced maps $X\langle n\rangle \to X\langle n-1\rangle$ over $X$ for all $n$.
One way of obtaining a Postnikov section as above is to choose representatives $\phi_g\colon S^{n+1} \to X$ of generators $g$ of $\pi_{n+1}(X,x)$ and attaching cells: $X(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X$. We then choose representatives for the generators of $\pi_{n+2}(X(1),x)$ and attach cells and so on. The colimit $\lim_{\to n} X(n)$ 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 $n$-connected cover (except in special cases, like when $n=1$ and $X$ is a well-connected space).
The $n$th stage of the Whitehead tower of $X$ is the homotopy fiber of the map from $X$ to the $n$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 $n$th 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 $S^n$ (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.
The Whitehead tower of the classifying space/delooping of the orthogonal group $O(n)$ starts out as
where the stages are the deloopings of
… $\to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group,
where lifts through the stages correspond to
and where the obstruction classes are the universal characteristic classes
first fractional Pontryagin class $\tfrac{1}{2}p_1$
second fractional Pontryagin class $\tfrac{1}{6}p_2$
and where every possible square in the above is a homotopy pullback square (using the pasting law).
For instance $w_2$ can be identified as such by representing $B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}$ 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 $sSet$ is a simplicial model category, $sSet(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $B O \to \tau_{\leq 2} B O$ to an isomorphism on connected components. This identifies $B SO \to B^2 \mathbb{Z}$ as being an isomorphism on the second homotopy group. Therefore, by the Hurewicz theorem, it is also an isomorphism on the cohomology group $H^2(-,\mathbb{Z}_2)$. 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:
$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Whitehead tower of orthogonal group | orientation | spin | string | fivebrane | ninebrane | |||||||||||||
homotopy groups of stable orthogonal group | $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ |
stable homotopy groups of spheres | $\pi_n(\mathbb{S})$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | 0 | $\mathbb{Z}_3$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ |
image of J-homomorphism | $im(\pi_n(J))$ | 0 | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{24}$ | 0 | 0 | 0 | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{504}$ | 0 | 0 | 0 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |
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
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