nLab
Whitehead tower

Idea

The Whitehead tower of a pointed homotopy type $X$ is an interpolation of the point inclusion $*\to X$ be a sequence of homotopy types

$$*\to \cdots \to X^{(2)}\to X^{(1)}\to X^{(0)}\simeq X$$* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X

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 $k\ge (n+1)$ (and the inclusion $1\to {\pi }_n(X^{(n)})$ in degree $n$ as well as the identity $1=1$ in degree $k<n$).

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.

Definition

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

$$*\to \cdots \to X^{(2)}\to X^{(1)}\to X^{(0)}\simeq X$$* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X

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

$$\begin{array}{ccc}{X}^{(n)}& \to & *\\ \downarrow & & \downarrow \\ X& \to & X_{(n+1)}\end{array}$$\array{ X^{(n)} &\to& * \\ \downarrow && \downarrow \\ X &\to& X_{(n+1)} }

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 $X^{(n)}\to X$

$$\begin{array}{ccc}{X}^{(n)}& \to & P{X}_{(n+1)}\\ \downarrow & & \downarrow \\ X& \to & X_{(n+1)}\end{array}$$\array{ X^{(n)} &\to& P X_{(n+1)} \\ \downarrow && \downarrow \\ X &\to& X_{(n+1)} }

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

$$\dots \to X⟨n⟩\to \dots \to X⟨1⟩\to X⟨0⟩\to X$$\ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X

where each $X⟨n⟩\to X⟨n-1⟩$ induces isomorphisms on homotopy groups ${\pi }_i$ for $i>n$ and such that $X⟨n⟩$ is $n$-connected (has trivial homotopy groups ${\pi }_i$ for $i\le n$). The homotopy long exact sequence then shows that the fiber of $X⟨n⟩\to X⟨n-1⟩$ 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 $G⟨n⟩$ has a model which is a topological group, which is an abelian group extension:

$$1\to K({\pi }_{n-1}(X,x),n-1)\to G⟨n⟩\to G⟨n-1⟩\to 1$$1\to K(\pi_{n-1}(X,x),n-1) \to G\langle n\rangle \to G\langle n-1 \rangle \to 1

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 $X⟨0⟩\hookrightarrow X$ 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 $X⟨1⟩$ 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.

Constructions

Whitehead’s construction

In WHitehead 1952 is 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 $X⟨n⟩\to X$ with $X⟨n⟩$ $n$-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 $\mathrm{\infty }$-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 $p:X⟨n⟩\to X$, 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

$$X\to (\dots \to X_n\to X_{n-1}\to \dots \to X_1\to X_0)$$X \to (\ldots \to X_n \to X_{n-1} \to \ldots \to X_1 \to X_0)

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 $X⟨n⟩\to X⟨n-1⟩$ over $X$ for all $n$.

One way of obtaining a Postnikov section as above is to choose representatives ${\varphi }_g:S^{n+1}\to X$ of generators $g$ of ${\pi }_{n+1}(X,x)$ and attaching cells: $X(1)≔B^{n+2}{\cup }_{\{{\varphi }_g\}}X$. 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).

Functorial constructions

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.

Examples

Whitehead tower of the orthogonal group

The Whitehead tower of the classifying space/delooping of the orthogonal group $O(n)$ starts out as

$$\begin{array}{cc}& \mathrm{Whitehead}\mathrm{tower}\\ & ⋮\\ & B\mathrm{Fivebrane}& \to & \cdots & \to & *\\ & \downarrow & & & & \downarrow \\ \mathrm{second}\mathrm{frac}\mathrm{Pontr}.\mathrm{class}& B\mathrm{String}& \to & \cdots & \stackrel{\frac{1}{6}{p}_2}{\to }& B^8\mathbb{Z}& \to & *\\ & \downarrow & & & & \downarrow & & \downarrow \\ \mathrm{first}\mathrm{frac}\mathrm{Pontr}.\mathrm{class}& B\mathrm{Spin}& & & & & \stackrel{\frac{1}{2}{p}_1}{\to }& B^4\mathbb{Z}& \to & *\\ & \downarrow & & & & \downarrow & & \downarrow & & \downarrow \\ \mathrm{second}\mathrm{SW}\mathrm{class}& BSO& \to & \cdots & \to & & \to & & \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2& \to & *\\ & \downarrow & & & & \downarrow & & \downarrow & & \downarrow & & \downarrow \\ \mathrm{first}\mathrm{SW}\mathrm{class}& BO& \to & \cdots & \to & {\tau }_{\le 8}BO& \to & {\tau }_{\le 4}BO& \to & {\tau }_{\le 2}BO& \stackrel{w_1}{\to }& {\tau }_{\le 1}BO\simeq B{\mathbb{Z}}_2& \mathrm{Postnikov}\mathrm{tower}\end{array}$$\array{ & Whitehead tower \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ second frac Pontr. class & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ first frac Pontr. class & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ second SW class & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ first SW class & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & Postnikov tower }

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

• orthogonal structure

• orientation

• spin structure

• string structure

• fivebrane structure

and where the obstruction classes are the universal characteristic classes

• first Stiefel-Whitney class $w_1$

• second Stiefel-Whitney class $w_2$

• first fractional Pontryagin class $\frac{1}{2}{p}_1$

• second fractional Pontryagin class $\frac{1}{6}{p}_2$

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 $BO\to {\tau }_{\le 2}BO\simeq \mathrm{BO}{/}_{{\sim }_n}$ 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 $\mathrm{sSet}$ is a simplicial model category, $\mathrm{sSet}(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $BO\to {\tau }_{\le 2}BO$ to an isomorphism on connected components. This identifies $B\mathrm{SO}\to B^2\mathbb{Z}$ 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

$$\begin{array}{c}⋮\\ \downarrow \\ B\mathrm{Fivebrane}\\ \downarrow \\ B\mathrm{String}& \stackrel{\frac{1}{6}{p}_2}{\to }& B^{7}U(1)& \simeq B^8\mathbb{Z}\\ \downarrow \\ B\mathrm{Spin}& \stackrel{\frac{1}{2}{p}_1}{\to }& B^{3}U(1)& \simeq B^4\mathbb{Z}\\ \downarrow \\ B\mathrm{SO}& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2\\ \downarrow \\ BO& \stackrel{w_1}{\to }& B{\mathbb{Z}}_2\\ {\downarrow }^{\simeq }\\ B\mathrm{GL}\end{array},$$\array{ \vdots \\ \downarrow \\ B Fivebrane \\ \downarrow \\ B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z} \\ \downarrow \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z} \\ \downarrow \\ B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2 \\ \downarrow \\ B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ B GL } \,,

where each “hook” is a fiber sequence.

Whitehead tower in general $(\mathrm{\infty },1)$-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

• Whitehead tower in an (∞,1)-topos.

References

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

• Allen Hatcher, Algebraic Topology (pdf)

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)