nLab
Whitehead tower

Description

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 $i\le n$).

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

Construction by co-killing of homotopy groups

The whitehead tower may be constructed by iteratively co-killing? homotopy groups:

start with the map

$$X\to {\pi }_0(X)$$X \to \pi_0(X)

that maps each point to its path-connected component. Then the homotopy fiber

$$\begin{array}{ccc}X⟨0⟩& \to & *\\ \downarrow & & {\downarrow }^{[x]}\\ X& \to & {\pi }_0(X)\end{array}$$\array{ X\langle 0\rangle &\to& {*} \\ \downarrow && \downarrow^{[x]} \\ X &\to& \pi_0(X) }

is just an ordinary pullback and in fact $X⟨0⟩$ is just the path-connected component of the base point $x\in X$, as desired.

Remark: It is not entirely unreasonable to notice at this step that instead of the set ${\pi }_0(X)$ we could use here just $\{\mathrm{0,1}\}$ and send each point in the path connected component of $x$ to 1 and all others to 0. Then $*\stackrel{1}{\hookrightarrow }\{\mathrm{0,1}\}$ is the subobject classifier in Set and co-killing the 0-th homotopy “group” is the same as taking the subobject classified by the characteristic connected-component map $X\to {\mathbb{Z}}_2$.

Next let ${\pi }_1(X⟨0⟩,x)$ be the first homotopy group of $X⟨0⟩$ at $x$ and $ℬ{\pi }_1(X⟨0⟩,x)$ its delooping. There is then naturally a map $X⟨0⟩\to ℬ{\pi }_1(X⟨0⟩)$ and the homotopy fiber

$$\begin{array}{ccc}X⟨1⟩& \to & *\\ \downarrow & & \downarrow \\ X⟨0⟩& \to & ℬ{\pi }_1(X⟨0⟩,x)\end{array}$$\array{ X\langle 1\rangle &\to& {*} \\ \downarrow && \downarrow \\ X\langle 0\rangle &\to& \mathcal{B} \pi_1(X\langle 0\rangle,x) }

is the universal cover (universal 1-connected cover) of $X⟨0⟩$.

And so on.

Notice that after the second step for $X⟨n⟩$ with $n\ge 1$ the Hurewicz theorem? ensures that the first non-vanishing homotopy group ${\pi }_{n+1}(X⟨n⟩)$ is isomorphic to the homology group? $H_{n+1}(X⟨n⟩,\mathbb{Z})$. This in turn, if it has no nontrivial torsion subgroup, is isomorphic to the cohomology group $H^{n-1}(X⟨n⟩,\mathbb{Z})$.

Whitehead’s construction

In

• Fiber Spaces and the Eilenberg Homology Groups, PNAS 38, No. 5 (1952)

G. W. Whitehead answers the question, posed by 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? $\mathrm{PX}=P_{\mathrm{xX}}$. 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 ${\pi }_n(X,x)$ 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.

In other $(\mathrm{\infty }\mathrm{,1})$-toposes

Given an $(\mathrm{\infty }\mathrm{,1})$-topos $C$ we can talk about n-truncated objects in $C$. The above construction then has an analogue in $C$ (may need something like properness so as to get the pullback of a fibration to be a fibration – I’m making this section up as I go along -DMR). The path fibration is replaced by the trivial cofibration-fibration factorisation of the inclusion of the basepoint $*\to X$ (probably want this to be functorial for my liking -DMR).

For example, in $\mathrm{sSet}$, we have a functoral $n$-truncation operation, via the coskeleton endofunctor ${\mathrm{Cosk}}_{n+1}:\mathrm{sSet}\to \mathrm{sSet}$ (I think I have the indexing right - it may be ${\mathrm{Cosk}}_n$. -DMR). We also have the decalage functor, which gives us the factorisation $*\to {\mathrm{Dec}}^{1}X\to X$. The pullback of ${\mathrm{Dec}}^1{\mathrm{Cosk}}_{n+1}X\to {Cosk}_{n+1}X$ (again, indexing) along the unit $X\to {\mathrm{Cosk}}_{n+1}X$ gives the $n$-connected cover of $X$, and now we are justified in saying the. This idea of the canonical Postnikov tower is explored in low dimensions by Duskin in

• Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories, TAC 9 no. 2, (2002).

In Duskin’s treatment the inclusion $X\to {\mathrm{Cosk}}_{3}X$ for $X$ a Kan complex is just the map sending an $\mathrm{\infty }$-groupoid into the nerve of the bigroupoid representing its 2-type. This hopefully motivates somewhat the details of the construction in the next section.

Non-traditional approach in $\mathrm{Top}$

The following is some semblance of current research, except the stuff about the string 2-group. All errors and stupid ideas are mine - David Roberts

For locally nice spaces (say locally contractible), it is desirable to have a functorial construction of the Whitehead tower. A shadow of a low-dimensional case of this can be seen in the construction of the string 2-group given by Baez-Crans-Schreiber-Stevenson. Since finite-dimensional Lie groups have non-trivial third homotopy group, it is not possible to form the 3-connected cover in the category $\mathrm{fin}.\dim .\mathrm{LieGrp}$, like it is possible to take the 0-, and 1-connected covers. While most people give up the smoothness and make do with the topological group ${\mathrm{String}}_G$, The BCSS construction leaps out of that category into that of strict (Frechet) Lie 2-groups. The construction is also functorial.

David Roberts: As an aside, I know of two classes of spaces with explicitly constructed (i.e. not via co-killing homotopy groups) 2-connected covers: the 2-sphere and its quotients by $Z/n$ (lens spaces), and the loop group? $\Omega G$ of a compact, simple, simply-connected Lie group $G$ (the latter is the level-1 central extension by $U(1)$, the first needs no introduction). Does anybody know of any other $n$-connected covers (other than, say, the Hopf fibrations) that are canonically given?

However, without one could demand a conceptually similar approach to the $n$-connected cover of a general (locally nice) space or smooth manifold. For $n=2$, and taking only topological spaces for consideration, this is contained in my thesis work. The rough result is that one gets a 2-connected topological groupoid $X^{(2)}$ equipped with a map to $X$ that factors through the universal covering space $X^{(1)}$. This is functorial, and generalises to higher connected covers (at least heuristically - I don’t have $n$-categorical superpowers).

But the general idea is that one would get an $(n-1)$-groupoid $X^{(n)}$ over $X$ which is $n$-connected and such that the map to $X$ factors through the $(n-2)$-groupoid $X^{(n-1)}$. The map to $X$ should induce isomorphisms on homotopy groups ${\pi }_i$ for $i>n$, as in the usual Whitehead tower.

If we want to consider arbitrary $n$ and retain some sort of local triviality on our connected covers we cannot get away from the assumption that the space in question is locally contractible. An assumption of this sort appears in

• B. Toen, Vers une interpretation Galoisienne de la theorie de l’homotopie, Cahiers de Top. et Geom. Diff. Cat., Vol. XLIII-4 (2002), 257-312.

in the context of locally constant $\mathrm{\infty }$-stacks and their monodromy (I haven’t got this article at the moment, and I suspect they may be $(\mathrm{\infty }\mathrm{,1})$-stacks, but I’m not 100 percent certain -DMR). Technically speaking, we don’t need local contractibility but the existence of a basis of open sets $U\hookrightarrow X$ such that this inclusion map is null-homotopic. But I will continue to call this local contractibilty, for lack of a beter term (if there is such a term, I’d like to know -DMR).

Construction using topological $n$-groupoids

Consider the fundamental $n$-groupoid ${\Pi }_n(X)$ of the locally contractible space $X$ (as a Trimble n-groupoid, say), or at least for now its underlying globular set. We can take the compact open-topology on the set of $k$-morphisms for $k<n$. As the space is locally contracible, in particular semi-locally $n$-connected, the space $\mathrm{Hom}(S^{n-1},X)$ is semi-locally simply-connected (I have a fragment of a paper saying this is true for the 'absolute case' - that is, locally $n$-connected implies the mapping space locally simply connected, but I expect it to be true for the relative case -DMR). In particular, we can take the fundamental groupoid ${\Pi }_1(\mathrm{Hom}(S^{n-1},X))$, which has a topology given in the usual way?. The arrow space of this fundamental groupoid is then non other than the space of $n$-arrows of ${\Pi }_n(X)$. It needs to be checked that the $n$ compositions ${\#}_k$, $k=0,\dots ,n-1$ are continuous, as well as a bunch of other stuff, but I think this should follow from (unique) lifting theorems for covering spaces.

The object space of ${\Pi }_n(X)$ is just $X$, and so there is an inclusion $X\hookrightarrow {\Pi }_n(X)$, and it is this that replaces the Postnikov section in the Whitehead construction outlined above. The topological fundamental $n$-groupoid, even though it contains apparently more homotopical information than the untopologised fundamental $n$-groupoid ${\Pi }_n(X{)}^{\delta }$ ($\delta =$discrete topology), I posit that under the assumptions on $X$, the inclusion ${\Pi }_n(X{)}^{\delta }\to {\Pi }_n(X)$ has an ana-n-functor pseudoinverse (taking the Grothendieck pretopology of open covers should be enough). On passing to the homotopy colimit this span should become a span weak homotopy equivalences, and so we can consider the topologised and the untopologised to be different representatives of the $n$-type of $X$.

Given a basepoint $x\in X$, we can form the tangent $n$-groupoid $T_x{\Pi }_n(X)$, which is equivalent to the trivial $n$-groupoid $*$ (even as a topological $n$-groupoid), and gives us what should be in any sensible definition a fibration $T_x{\Pi }_n(X)\to {\Pi }_n(X)$. Pull back this fibration to $X$, and call the resulting thing $X^{(n)}$. It is fairly easy to see that $X^{(n)}$ is a topological $(n-1)$-groupoid over $X$. This then should be the $n$-connected cover of $X$. For $n=1$ this is precisely the classical construction of the universal covering space of a pointed space. For $n=2$ this is treated in

• D.M. Roberts, Fundamental bigroupoids and 2-covering spaces, PhD thesis, available here

and the two-dimensional homotopical tools developed there can be used to show that $X^{(2)}$ is 2-connected.

A word is probably in order about the notion of $k$-connectedness for topological $n$-groupoids. This has its usual meaning, once homotopy groups ${\pi }_i$ have been defined. The reader should be warned that these have nothing to do with the groups ${\pi }_0\mathrm{Eq}(1_{{.}_{{.}_{1_x}}})/\sim$ obtained from considering the autoequivalence $n-i+1$-group of the identity $(i-1)$-arrow on the identity $(i-2)$-arrow on … on the object $x$. These should be defined in such a way as to agree with the homotopy colimit $\mathrm{hocolim}X$ of $X$ considered as a truncated simplicial space. In particular, ${\pi }_1$ of a topological groupoid is not the group of automorphisms of the basepoint, but a quotient of the set of generalised paths, an idea going back to Haefliger (for an intro see the early parts of chapter 2 of the above thesis, available at HomePage, or the preprint H. Colman, On the 1-homotopy type of Lie groupoids, arXiv:math/0612257).

There are other interpretations of $X^{(n)}$:

• The (0-)source fibre of ${\Pi }_n(X)$, which is:
• The pullback of $(s_0,t_0):{\mathrm{Hom}}_{{\Pi }_n(X)}\to X\times X$ along the inclusion $\{x\}\times X\to X\times X$, where ${\mathrm{Hom}}_{{\Pi }_n(X)}$ is the (internalised) hom-$(n-1)$-groupoid familiar from the definition of a Trimble $n$-groupoid.
• The 'vertical fundamental $(n-1)$-groupoid' of $\mathrm{PX}\to X$, the path fibration.

The last can be thought of as a families version of the usual fundamental $(n-1)$-groupoid: take vertical paths, vertical homotopies between paths etc.

Examples

Whitehead tower of the orthogonal group

The Whitehead tower of the orthogonal group $O(n)$ starts out as

$$\cdots \to \mathrm{Fivebrane}(n)\to \mathrm{String}(n)\to \mathrm{Spin}(n)\to \mathrm{SO}(n)\to O(n)$$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to O(n)

where the terms are

… $\to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group

References

For instance:

example 4.20 in

• Alan Hatcher, Algebraic Topology (pdf)