Contents

# Contents

## Idea

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

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

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 $k \geq (n+1)$ (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 $X^{(n+1)} \to X$ 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

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

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

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

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

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

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

$\ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X$

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:

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

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 $X\langle 0 \rangle \hookrightarrow X$ 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.

## Constructions

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

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

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 $X\langle n\rangle \to X\langle n-1\rangle$ 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).

### 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

$\array{ & \mathbf{\text{Whitehead tower}} \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ \mathbf{\text{second frac Pontr. class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ \mathbf{\text{first frac Pontr. class}} & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ \mathbf{\text{second SW class}} & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ \mathbf{\text{first SW class}} & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & \mathbf{\text{Postnikov tower}} }$

where each square and each composite rectangle is a homotopy pullback square (all controled by the pasting law),

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

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

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

where each “hook” is a fiber sequence.

Via the J-homomorphism this corresponds to the stable homotopy groups of spheres:

$n$012345678910111213141516
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\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$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\mathbb{Z}_2$

## Whitehead tower in general $(\infty,1)$-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

The original reference is

• George WhiteheadFiber 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)