nLab
Whitehead tower

Contents

Idea

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

*X (2)X (1)X (0)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)X^{(n)} is (n1)(n-1)-connected

  • and each morphism X (n+1)X (n)X^{(n+1)} \to X^{(n)} induces an isomorphism on all homotopy groups in degree k(n+1)k \geq (n+1) (and the inclusion 1π n(X (n))1 \to \pi_n(X^{(n)}) in degree nn as well as the identity 1=11 = 1 in degree k<nk \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*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)X (n)X^{(n+1)} \to X^{(n)} to be the homotopy fiber of the corresponding map into the (n+1)(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 XX is a sequence of homotopy types

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

where the space X (n)X^{(n)} is the homotopy fiber of the map XX (n+1)X \to X_{(n+1)} into the item X (n+1)X_{(n+1)} in the Postnikov tower of XX.

Here each homotopy pullback

X (n) * X X (n+1) \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 PX (n+1)*P X_{(n+1)} \simeq *, which is a Hurewicz fibration PX (n+1)X (n+1)P X_{(n+1)} \to X_{(n+1)}. In this case also the ordinary pullback X (n)XX^{(n)}\to X

X (n) PX (n+1) X X (n+1) \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)(X,x) is a sequence of fibrations

XnX1X0X \ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X

where each XnXn1X\langle n\rangle \to X\langle n-1 \rangle induces isomorphisms on homotopy groups π i\pi_i for i>ni\gt n and such that XnX\langle n\rangle is nn-connected (has trivial homotopy groups π i\pi_i for ini \leq n). The homotopy long exact sequence then shows that the fiber of XnXn1X\langle n\rangle \to X\langle n-1 \rangle is a K(π n1(X,x),n1)K(\pi_{n-1}(X,x),n-1) Eilenberg-Mac Lane space. One has a model for K(π n1(X,x),n1)K(\pi_{n-1}(X,x),n-1) which is an abelian topological group; this has a remarkable consequence when (X,x)=(G,e)(X,x)=(G,e) is a topological group. Indeed, in this case one sees inductively that GnG\langle n\rangle has a model which is a topological group, which is an abelian group extension:

1K(π n1(X,x),n1)GnGn11 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(,2)K(\mathbb{Z},2)-extension of the spin group.

For n=0n=0 we require that X0XX\langle 0 \rangle \hookrightarrow X is the inclusion of the path-component of xx. Really this is defined up to homotopy, but we have a canonical model. If XX is locally connected and semilocally path-connected, then X1X\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’s construction

Whitehead 1952 answered the question, posed by Witold Hurewicz, of the existence of what we would now call nn-connected 'covers' of a given space XX, taking this to mean a fibration XnXX\langle n\rangle \to X with XnX\langle n\rangle nn-connected and otherwise inducing isomorphisms on homotopy groups.

The construction proceeds as follows (using modern terminology). Given a pointed space (X,x)(X,x),

  • Choose a representative for the Postnikov section X nX_n such that XX nX \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 nX_n, i.e. the path fibration PX nP X_n. This is a Hurewicz fibration.

  • Pull this back to XX, to get p:XnXp\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 pp has the desired properties.

This gives us a single nn-connected cover, but by considering the Postnikov tower

X(X nX n1X 1X 0) X \to (\ldots \to X_n \to X_{n-1} \to \ldots \to X_1 \to X_0)

of XX, where each map XX nX \to X_n is the inclusion of a closed subspace, it is simple to see there are induced maps XnXn1X\langle n\rangle \to X\langle n-1\rangle over XX for all nn.

One way of obtaining a Postnikov section as above is to choose representatives ϕ g:S n+1X\phi_g\colon S^{n+1} \to X of generators gg of π n+1(X,x)\pi_{n+1}(X,x) and attaching cells: X(1)B n+2 {ϕ g}XX(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X. We then choose representatives for the generators of π n+2(X(1),x)\pi_{n+2}(X(1),x) and attach cells and so on. The colimit lim nX(n)\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 nn-connected cover (except in special cases, like when n=1n=1 and XX is a well-connected space).

Functorial constructions

The nnth stage of the Whitehead tower of XX is the homotopy fiber of the map from XX to the nnth (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 nnth stage of the Whitehead tower of XX is also the cofibrant replacement for XX in the right Bousfield localization of Top with respect to the object S nS^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)O(n) starts out as

Whiteheadtower BFivebrane * secondfracPontr.class BString 16p 2 B 8 * firstfracPontr.class BSpin 12p 1 B 4 * secondSWclass BSO w 2 B 2 2 * firstSWclass BO τ 8BO τ 4BO τ 2BO w 1 τ 1BOB 2 Postnikovtower \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

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 2w_2 can be identified as such by representing BOτ 2BOBO/ nB 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 sSetsSet is a simplicial model category, sSet(S 2,)sSet(S^2,-) can be applied and preserves the pullback as well as the homotopy pullback, hence sends BOτ 2BO B O \to \tau_{\leq 2} B O to an isomorphism on connected components. This identifies BSOB 2B 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(, 2)H^2(-,\mathbb{Z}_2). Analogously for the other characteristic maps.

In summary, more concisely, the tower is

BFivebrane BString 16p 2 B 7U(1) B 8 BSpin 12p 1 B 3U(1) B 4 BSO w 2 B 2 2 BO w 1 B 2 BGL, \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:

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

Whitehead tower in general (,1)(\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

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Whitehead towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Cech nerve of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

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

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)

Revised on November 18, 2013 12:42:32 by Urs Schreiber (82.169.114.243)