# The higher structure of unstable homotopy groups

Samik Basu, David Blanc, and Debasis Sen  Stat-Math Unit, Indian Statistical
Institute, Kolkata 700108, India<EMAIL_ADDRESS>Department of
Mathematics
University of Haifa
3498838 Haifa
Israel<EMAIL_ADDRESS>Department of Mathematics & Statistics
Indian Institute of Technology, Kanpur
Uttar Pradesh 208016
India<EMAIL_ADDRESS>
###### Abstract.

We construct certain unstable higher order homotopy operations indexed by the
simplex categories of $\mathbf{\Delta}^{n}$ for $n\geq 2$, and prove that all
elements in the homotopy groups of a wedge of spheres are generated under such
operations by Whitehead products and the group structure. This provides a
stronger unstable analogue of Cohen’s theorem on the decomposition of stable
homotopy.

###### Key words and phrases:

Higher homotopy operations, Toda brackets, unstable homotopy groups of
spheres, spiral spectral sequence

###### 2020 Mathematics Subject Classification:

Primary: 55Q35; secondary: 55P99, 55Q40

## Introduction

In [Co], Joel Cohen showed how all elements in the stable homotopy groups of
the sphere spectrum can be generated under composition and higher order
homotopy operations from certain atomic elements: the three Hopf classes and
their odd-primary analogues. Although Cohen also describes an algorithm for
determining the decomposition of a given class, the main importance of this
result is conceptual, since it is one of the few global facts known about the
stable homotopy groups as a whole (see [Lin, N, Sc1, Sc2]).

Our goal here is to show that a similar but stronger result holds for the
unstable homotopy groups of wedges of spheres. This consists of two distinct
ingredients:

I. There are a number of competing definitions of higher homotopy operations,
particularly in the unstable setting, not all of which agree: these generally
arise as the obstruction to rectifying certain homotopy-commutative diagrams
(see [BM] and [BJT2]), but in order to qualify as higher homotopy operations
sensu stricto, by analogy with the secondary cohomology operations of [Ada],
all but the last object in the diagram should be (wedges of) spheres. To
achieve this, we are forced to use diagrams indexed by simplex categories, as
in [BJT1]. These will be called _higher order simplicial operations_ ; they
include as a special case the long Toda brackets of [BJT3, BBS2].

II. Next, we identify the classes $\varphi\in\pi_{n}{\mathbf{X}}$ which cannot
be decomposed in either the primary sense (that is, writing
$\varphi=\zeta\circ(\psi_{1}\bot\dotsc\bot\psi_{k})$ for
$\psi_{i}\in\pi_{m_{i}}{\mathbf{X}}$ and
$\zeta\in\pi_{n}(\bigvee_{i=1}^{k}{\mathbf{S}}^{m_{i}})$ a non-trivial primary
operation), nor in the higher order sense (i.e., presenting $\varphi$ as the
value of a higher order simplicial operation). This is done in two stages:

We first define the _atomic_ classes $[f]\in\pi_{n}{\mathbf{X}}$ to be those
which induce a non-zero map
$f_{\ast}:H_{n}({{\mathbf{S}}^{n}};{\mathbb{F}_{p}})\to
H_{n}({{\mathbf{X}}};{\mathbb{F}_{p}})$ for some prime $p$, and show:

######  Theorem A.

If ${\mathbf{X}}$ is simply-connected, the elements of
$\pi_{\ast}{\mathbf{X}}$ are generated by the atomic maps under primary
composition and higher order simplicial operations.

See Theorem 5.2 below.

Our main technical result identifies explicitly the atomic classes for
spheres; more precisely, we show:

######  Theorem B.

If ${\mathbf{X}}$ is the $k$-connected cover of the $k$-sphere for $k\geq 2$
the atomic maps in $\pi_{\ast}{\mathbf{X}}$ consist of the Hopf maps
$\eta_{k}\in\pi_{k+1}{\mathbf{S}}^{k}$, $\nu_{k}\in\pi_{k+3}{\mathbf{S}}^{k}$,
and $\sigma_{k}\in\pi_{k+7}{\mathbf{S}}^{k}$, and the first $p$-torsion
classes $\alpha_{1}(p)\in\pi_{k+2p-3}{\mathbf{S}}^{k}$, for odd primes $p$.

See Theorem 6.1 and Corollary 6.13 below. This is an unstable version of [Co,
Theorem 4.2].

Finally, a further analysis of the atomic maps above yields:

######  Theorem C.

All elements in the homotopy groups of a finite wedge of simply-connected
spheres are generated (under primary composition and higher order homotopy
operations) by the group operation and Whitehead products.

See Theorem 7.16 below.

######  0.1 Remark.

As noted above, Cohen provides an algorithm for finding decompositions of
stable classes, in terms of the Adams spectral sequence, although it is not
clear how workable this is (see, e.g., [Ada, WX] for the use of such Toda
decompositions, and the resulting calculations of differentials in the Adams
spectral sequence).

Such an algorithm seems very unlikely to be achievable unstably with the
present state of our knowledge of the unstable homotopy groups of spheres.
However, Theorem C implies that in some sense we have a “combinatorial
description” of homotopy groups in terms of Whitehead products and simplicial
identities. While this is in no sense algorithmic, it differs from the
original approach of [Ka] in that the calculations themselves are homotopy
invariants. We hope to show in the future that the differentials in the
spectral sequences we use are themselves given by higher homotopy operations
(compare [BBS1] and [BJT3]).

In addition to the sample calculations of Section 7, used in the proof of
Theorem C, we illustrate this point, in Section 8 by providing a higher
operation description of the Hopf fibrations
${\mathbf{S}}^{2n+1}\to\mathbb{C}\mathbf{P}^{n}$ for all complex projective
spaces.

######  0.2 Notation.

Let $\Delta$ denote the category of non-empty finite ordered sets and order-
preserving maps (cf. [May, §2]), and $\Delta_{\operatorname{res}}$ the
subcategory with the same objects but only monic maps. Similarly, $\Delta_{+}$
denotes the category of all finite ordered sets,
$\Delta_{\operatorname{res},+}$ the corresponding subcategory of monic maps,
and $\Delta_{\langle{n}\rangle}$ the subcategory of
$\Delta_{\operatorname{res},+}$ with consisting of ordered sets with at most
$n+1$ elements.

A _simplicial object_ $G_{\bullet}$ in a category $\mathcal{C}$ is a functor
$\Delta^{\operatorname{op}}\to\mathcal{C}$, a _restricted_ simplicial object
is a functor $\Delta_{\operatorname{res}}^{\operatorname{op}}\to\mathcal{C}$,
an _augmented simplicial object_ is a functor
$\Delta_{+}^{\operatorname{op}}\to\mathcal{C}$, a _restricted augmented_
simplicial object is a functor
$\Delta_{\operatorname{res},+}^{\operatorname{op}}\to\mathcal{C}$, and an _$n$
-truncated restricted augmented_ simplicial object is a functor
$\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\mathcal{C}$.

In each case write $G_{n}$ for the value of $G_{\bullet}$ at
$[\mathbf{n}]=(0<1<\dotsc<n)$. There is a natural embedding
$c({-})_{\bullet}:\mathcal{C}\to\mathcal{C}^{\Delta^{\operatorname{op}}}$,
with $c({A})_{\bullet}$ the constant simplicial object and similarly
$c_{+}({A})_{\bullet}$ for the constant augmented simplicial object. The
inclusion of categories $\Delta\to\Delta_{+}$ induces a functor
$\sigma^{\ast}(-):\mathcal{C}^{\Delta_{+}^{\operatorname{op}}}\to\mathcal{C}^{\Delta^{\operatorname{op}}}$
forgetting the augmentation.

The category of topological spaces will be denoted by Top, that of pointed
spaces by $\mbox{\sf Top}_{\ast}$, and that of pointed connected spaces by
$\mbox{\sf Top}_{0}$. The category of simplicial sets will be denoted by
$\mathcal{S}=\mbox{\sf Set}^{\Delta^{\operatorname{op}}}$, that of pointed
simplicial sets by $\mathcal{S}_{\ast}=\mbox{\sf
Set}_{\ast}^{\Delta^{\operatorname{op}}}$, that of reduced simplicial sets by
$\mathcal{S}^{\operatorname{red}}$ (see [GJ, III, §3]), and that of simplicial
groups by $\mathcal{G}=\mbox{\sf Gp}^{\Delta^{\operatorname{op}}}$. Write
$\operatorname{map}_{\ast}({\mathbf{X}},{\mathbf{Y}})$ for the standard
function complex in $\mathcal{S}_{\ast}$, $\mbox{\sf Top}_{0}$, or
$\mathcal{G}$ (see [GJ, I, §1.5]).

######  0.3 Acknowledgements.

The research of the first author was supported by SERB MATRICS grant
2018/000845, and that of the second author by Israel Science Foundation grant
770/16.

##  1\. Simplicial higher order operations

Many versions of higher order operations have been studied in homotopy theory,
starting with Adem’s secondary cohomology operations (see [Ade]), Massey’s
triple products (see [Mas]), and the Toda brackets of [T1]. What these
constructions have in common appears to be that they serve as obstructions to
rectifying certain homotopy-commutative diagrams (see [BM, BJT2]).

Here we shall be concerned only with a particular manifestation of this
phenomenon, associated to a restricted augmented simplicial space up to
homotopy. For this purpose we recall the following concepts from [Bl]:

######  1.1 Definition.

For every $0<k\leq n$, let

$D(k,n)~{}:=~{}\\{0,1,\ldots,k\\}\times\\{0,1,\ldots,k,k+1\\}\times\ldots\times\\{0,1,\ldots,n\\}$

where $I=(i_{k},\ldots,i_{n})\in D(k,n)$ represents
$\delta:[\mathbf{n}]\to[\mathbf{k}-\mathbf{1}]$ in
$\Delta^{\operatorname{op}}$ given by the composite of face maps
$d_{I}:=d_{i_{k}}\circ\ldots\circ d_{i_{n}}$. There is thus an equivalence
relation on $D(k,n)$, with $I\sim I^{\prime}$ if and only if
$d_{I}=d_{I^{\prime}}$ (generated by
$(i_{k},\ldots,i_{j},i_{j+1},\ldots,i_{n})\sim(i_{k},\ldots,i_{j+1}-1,i_{j},\ldots,i_{n})$
if $i_{j}<i_{j+1}$).

To every $\delta:[\mathbf{n}]\to[\mathbf{k}-\mathbf{1}]$ (or equivalence class
$[I]\in D(k,n)/\sim$) we associate an $(n-k)$-dimensional abstract polyhedral
complex $\mathcal{P}_{n-k}[{\delta}]$, called the _face-map polyhedron_ for
$\delta$.

This is defined by induction on $d=|I|=n-k\geq 0$: when $d=0$, so
$\delta=d_{i_{n}}$, $\mathcal{P}_{0}[{\delta}])$ is a point. For each
representative $I=(i_{k},\ldots,i_{n})$ of $[I]$ and each partition of
$(i_{k},\ldots,i_{n})$ into $r$ consecutive blocks $\langle~{}i_{k},\ldots
i_{\ell_{1}}~{}|~{}i_{\ell_{1}+1},\ldots
i_{\ell_{2}}~{}|~{}\ldots|i_{\ell_{r-1}+1},\ldots i_{n}~{}\rangle$ of sizes
$s_{1}=(\ell_{1}-k),~{}\ldots$, $s_{r}=(n-\ell_{r-1}+1)$, respectively,
$\mathcal{P}_{n-k}[{\delta}]$ will have a sub-polyhedron of dimension
$(s_{1}-1)\cdot(s_{2}-1)\cdot\ldots\cdot(s_{r}-1)$ equal to the product

$\mathcal{P}_{s_{1}-1}[{i_{k},\ldots
i_{\ell_{1}}}]\times\mathcal{P}_{s_{2}-1}[{i_{\ell_{1}+1},\ldots
i_{\ell_{2}}}]\times\ldots\times\mathcal{P}_{s_{r}-1}[{i_{\ell_{r-1}+1},\ldots
i_{n}}].$

The union of all sub-polyhedra of $\mathcal{P}_{n}[{\delta}]$ of dimension
$\leq k$ is denoted by $\mathcal{P}_{n}[{\delta}]^{(k)}$.

In particular, if $r=n$ we see that $\mathcal{P}_{n-k}[{\delta}]$ has
$(n-k+1)!$ vertices, corresponding to the different ways of decomposing the
composite face map $d_{I}:[\mathbf{n}]\to[\mathbf{k}]$. In fact:

######  1.2 Lemma ([Bl, Lemma 1.7]).

For any $I\in D(k,k+n)$, the face-map polyhedra $\mathcal{P}_{n}[{\delta}]$ is
combinatorially isomorphic to the $d$-dimensional permutohedron for $d=n-k$
(i.e., the convex hull in $\mathbb{R}^{d+1}$ of the $(d+1)!$ points
$(\sigma(a_{0}),\sigma(a_{2}),\ldots,\sigma(a_{n}))\in\mathbb{R}^{n+1}$,
indexed by permutations $\sigma\in\Sigma_{n+1}$, determined by any $n+1$
distinct real numbers $0<a_{0}<a_{1}<\ldots<a_{n}$).

######  1.3 Corollary.

The subcomplex $\mathcal{P}_{n}[{\delta}]^{(n-1)}$ is homeomorphic to
${\mathbf{S}}^{n-1}$.

######  1.4 Example.

The $1$-face-map polyhedron is a $1$-simplex $\mathbf{\Delta}^{1}$, and the
$2$-face-map polyhedron is a hexagon:

(1.5)
$[0,1]\times\\{2\\}$$(0,0,2)$$\\{0\\}\times[0,2]$$(0,1,0)$$[0,1]\times\\{0\\}$$(0,0,0)$$\\{0\\}\times[0,1]$$(0,0,1)$$(0,1,1)$$[0,1]\times\\{1\\}$$(0,1,2)$$\\{0\\}\times[1,2]$

The $3$-face-map polyhedron is a truncated cuboctohedron, whose facets consist
of eight hexagons $\mathcal{P}_{2}[{\gamma}]$ and six squares of the form
$\mathcal{P}_{1}[{\beta}]\times\mathcal{P}_{1}[{\beta^{\prime}}]$:

(1.6)
$(0,1,2,2)$$(0,1,1,2)$$(0,1,1,1)$$(0,1,2,1)$$(0,1,2,3)$$(0,1,1,3)$$(0,0,1,2)$$(0,0,1,1)$$(0,0,0,1)$$(0,1,0,1)$$(0,0,2,1)$$(0,0,0,0)$$(0,1,0,0)$$(0,0,1,3)$$(0,0,0,3)$$(0,0,2,0)$$(0,0,2,3)$$(0,1,0,3)$$(0,0,2,2)$$(0,1,0,2)$$(0,1,1,0)$$(0,1,2,0)$$(0,0,0,2)$$(0,0,1,0)$

######  1.7 Remark.

As noted in [Bl, Lemma 4.11], the face-map polyhedra are a more economical way
to describe the cofibrant replacement
$\mathcal{F}_{\bullet}\Delta_{\operatorname{res}}$ for the indexing category
$\Delta_{\operatorname{res}}$ (as a constant simplicial category), obtained by
iterating the free category functor (see [DKSm]). In fact, the simplicial
mapping spaces in $\mathcal{F}_{\bullet}\Delta_{\operatorname{res}}$ are
triangulations of the corresponding face-map polyhedra.

######  1.8 Definition.

Let $\mathcal{C}$ be a pointed simplicial model category, and
$\widehat{{\mathbf{X}}}_{\bullet}:\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\operatorname{ho}\mathcal{C}$
an $n$-truncated restricted augmented simplicial object in its homotopy
category. We would like to lift it to $\mathcal{C}$ itself by first extending
$\widehat{{\mathbf{X}}}_{\bullet}$ to an $\infty$-homotopy commuting diagram
$\widetilde{{\mathbf{X}}}:\mathcal{F}_{\bullet}\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\mathcal{C}$,
and then strictifying this to a lift
${\mathbf{X}}_{\bullet}:\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\mathcal{C}$
of $\widehat{{\mathbf{X}}}_{\bullet}$, using [BV, Corollary 4.21 & Theorem
4.49] or [DKSm, Theorem 2.4]. At this stage we are not interested in the
strictification itself, but rather in the following construction, which serves
as the last obstruction to obtaining $\widetilde{{\mathbf{X}}}$:

Assuming that (as for $\mbox{\sf Top}_{\ast}$) our pointed simplicial category
$\mathcal{C}$ has an underlying unpointed simplicial category
$\hat{\mathcal{C}}$, the object ${\mathbf{X}}\otimes A\in\mathcal{C}$ (for
${\mathbf{X}}\in\mathcal{C}$ and any simplicial set $A$) is the half-smash
${\mathbf{X}}\ltimes A:={\mathbf{X}}\times A/(\ast\times A)$, where
${\mathbf{X}}\times A$ is the corresponding construction in
$\hat{\mathcal{C}}$. If $\Sigma{\mathbf{X}}$ is a cofibrant suspension in
$\mathcal{C}$ and $A\in\mathcal{S}_{\ast}$ is connected, with basepoint $a$,
there is a canonical weak equivalence

(1.9) ${}\varphi:\Sigma{\mathbf{X}}\otimes
A~{}\xrightarrow{\simeq}~{}\Sigma{\mathbf{X}}\otimes
A~{}/(\Sigma{\mathbf{X}}\otimes\\{a\\})\amalg~{}\Sigma{\mathbf{X}}\otimes\\{a\\}~{}\simeq~{}\Sigma{\mathbf{X}}\wedge
A\vee\Sigma{\mathbf{X}}$

where $\varphi$ onto the first summand is the natural projection, and the
reverse weak equivalence on the second summand is induced by
$\\{a\\}\hookrightarrow A$, where we write ${\mathbf{X}}\wedge A$ for
${\mathbf{X}}\otimes A/({\mathbf{X}}\otimes\\{a\\})$. See [BJ] and compare
[BJT3, Lemma 5.21].

Now fix a morphism $\delta:[\mathbf{m}]\to[\mathbf{k}-\mathbf{1}]$ in
$\Delta_{\langle{n}\rangle}^{\operatorname{op}}$ for some $0\leq k<m\leq n$,
and let $\mathcal{C}(\delta)$ denote the collection of all proper factors
$\gamma$ of $\delta$ – that is, morphisms of the form
$\gamma=[(i_{j},\ldots,i_{\ell})]$ for some $k\leq j<\ell\leq n$ and
$I=(i_{k},\ldots,i_{m})\in D(k,m)$ representing $\delta$. A _compatible
collection_ for $\mathcal{C}(\delta)$ and $\widehat{{\mathbf{X}}}$ is a set
$\\{g^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$ of maps
$g^{\gamma}:\mathcal{P}_{n-k}[{\gamma}]\ltimes\widehat{{\mathbf{X}}}_{n}\to\widehat{{\mathbf{X}}}_{k-1}$
for each such factor $\gamma$, satisfying the following condition:

If $\langle{i_{k},\ldots i_{\ell_{1}}~{}|~{}i_{\ell_{1}+1},\ldots
i_{\ell_{2}}~{}|~{}\ldots|i_{\ell_{r-1}+1},\ldots i_{m}}\rangle$ is a
partition of some sequence $I=(i_{k},\ldots,i_{m})\in D(k,m)$ representing a
factor $\gamma$ into $r$ blocks, with $\gamma_{1}:=[(i_{k},\ldots
i_{\ell_{1}})],~{}\ldots~{},$ $\gamma_{r}=[(i_{\ell_{r-1}+1},\ldots i_{m})]$,
and we set

$P=\mathcal{P}_{\ell_{1}-k-1}[{\gamma_{1}}]\times\mathcal{P}_{\ell_{2}-\ell_{1}}[{\gamma_{2}}]\ldots\mathcal{P}_{n-\ell_{r-1}}[{\gamma_{r}}]~{},$

then we require that
$g^{\gamma}\lvert_{P_{i}\ltimes\widehat{{\mathbf{X}}}{n}}$ be the composite of
the corresponding maps $g^{\gamma_{i}}$ in the sense that

(1.10)
${}g^{\gamma}(x_{1},\ldots,x_{r},y)=g^{\gamma_{1}}(x_{1},g^{\gamma_{2}}(x_{2},(\ldots,g^{\gamma_{r}}(x_{r},y)\ldots)))$

for $x_{i}\in\mathcal{P}_{\ell_{i}-\ell_{i-1}}[{\gamma_{i}}]$ and
$y\in\widehat{{\mathbf{X}}}_{n}$ (assuming here for simplicity of notation
that $\mathcal{C}$ has a faithful forgetful functor to $\mbox{\sf
Set}_{\ast}$).

We further require that if $\gamma=[i_{j}]$ is of length $0$, then
$g^{\gamma}$ must be in the prescribed homotopy class of
$[d_{i_{j}}]\in[\widehat{{\mathbf{X}}}_{j+1},\,\widehat{{\mathbf{X}}}_{j}]$.
Thus in particular, for each vertex $v$ of $\mathcal{P}_{k}[{\gamma}]$ indexed
by $(i_{n-k},\ldots,i_{n})$, the map
$g^{\gamma}\lvert_{\\{v\\}\times\widehat{{\mathbf{X}}}_{n}}$ represents the
class $[d_{i_{n-k}}\circ\ldots\circ d_{i_{n}}]$.

######  1.11 Definition.

If $\\{g^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$ and
$\\{\bar{g}^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$ are two compatible
collections for $\mathcal{C}(\delta)$, a _compatible homotopy_ between them is
a collection of homotopies
$\\{H^{\gamma}:g^{\gamma}\sim\bar{g}^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$
such that for each $0\leq t\leq 1$, the maps
$\\{H^{\gamma}_{t}\\}_{\gamma\in\mathcal{C}(\delta)}$ constitute a compatible
collection for $\mathcal{C}(\delta)$ (using standard topological notation for
the homotopies).

We then have:

######  1.12 Lemma.

Given
$\widehat{{\mathbf{X}}}_{\bullet}:\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\operatorname{ho}\mathcal{C}$
and $\delta:[\mathbf{m}]\to[\mathbf{k}-\mathbf{1}]$ in
$\Delta_{\langle{n}\rangle}^{\operatorname{op}}$ with $\ell=m-k+1$, any
compatible collection $\\{g^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$ for
$\mathcal{C}(\delta)$ induces a map
$f^{\prime}_{(\delta)}:\mathcal{P}_{\ell}[{\delta}]^{(\ell-1)}\ltimes\widehat{{\mathbf{X}}}_{n}\to\widehat{{\mathbf{X}}}_{n-k-1}$.
Moreover, compatibly homotopic collections induce homotopic maps
$f^{\prime}_{(\delta)}\sim\bar{f}^{\prime}_{(\delta)}$.

###### Proof.

All faces of $\mathcal{P}_{\ell}[{\delta}]^{(\ell-1)}$ are products of face-
map polyhedra $\mathcal{P}_{j}[{\gamma}]$ for $\gamma\in\mathcal{C}(\delta)$,
and condition (1.10) guarantees that the maps $g^{\gamma}$ agree on
intersections. ∎

######  1.13 Definition.

Under the canonical splitting of (1.9), the map $f^{\prime}_{(\delta)}$
determines a unique homotopy class
$[f_{(\delta)}]\in[\mathcal{P}_{\ell}[{\delta}]^{(\ell-1)}\wedge\widehat{{\mathbf{X}}}_{n},\widehat{{\mathbf{X}}}_{n-k-1}]$,
Note that the projection of a class on the other summand
$[\widehat{{\mathbf{X}}}_{n},\,\widehat{{\mathbf{X}}}_{n-k-1}]$ coming from
the splitting (1.9) is just the class of $\delta$ (considered as a composite
face map for $\widehat{{\mathbf{X}}}_{\bullet}$).

This class is called a _value_ of the _$k$ -th order homotopy operation_
associated to
$\widehat{{\mathbf{X}}}_{\bullet}:\Delta_{\langle{n}\rangle}^{\operatorname{op}}\to\operatorname{ho}\mathcal{C}$
and $\delta\in D(m-k,m)$ as above. The operation itself is usually thought of
as the collection of all possible values, as we vary the homotopy classes of
the compatible collections $\\{g^{\gamma}\\}_{\gamma\in\mathcal{C}(\delta)}$
for $\mathcal{C}(\delta)$.

######  1.14 Remark.

In order for the subset $\langle\langle{\delta}\rangle\rangle$ to be non-
empty, it is clearly a necessary condition that all the lower order operations
(for proper subclasses $\gamma\subset\delta$) _vanish_ – that is, contain the
null class – because otherwise the various maps
$g^{\gamma}:\mathcal{P}_{j}[{\gamma}]^{(j-1)}\ltimes\widehat{{\mathbf{X}}}_{m}\to\widehat{{\mathbf{X}}}_{m-j-1}$
cannot even extend over the interior of $\mathcal{P}_{j}[{\gamma}]$. A
sufficient condition is that they do so _coherently_ , in the sense of [Bl,
§5.7]. We shall not pursue this point any further, since it is not needed for
our purpose in this generality.

##  2\. Toda brackets

The most familiar type of higher homotopy operation is the Toda bracket of
[T1, T2] and their higher order analogues (see, e.g., [Wal, BBG, BJT2]). We
shall show that these are special cases of the simplicial higher order
operations of Section 1. We start first with the classical version:

######  2.1.

The ordinary Toda bracket.  Assume that we are given a sequence of maps

(2.2)
${}{\mathbf{X}}~{}\xrightarrow{h}~{}{\mathbf{Y}}~{}\xrightarrow{g}~{}{\mathbf{Z}}~{}\xrightarrow{f}~{}{\mathbf{W}}$

in a pointed model category $\mathcal{C}$, with $f\circ g\sim\ast$ and $g\circ
h\sim\ast$. This can be rewritten as a $2$-truncated restricted augmented
simplicial object in $\operatorname{ho}\mathcal{C}$:

(2.3)
$\textstyle{{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=h}$$\scriptstyle{d_{1}=\ast}$$\scriptstyle{d_{0}=\ast}$$\textstyle{{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=g}$$\scriptstyle{d_{1}=\ast}$$\textstyle{{\mathbf{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon=f}$$\textstyle{\mathbf{W}}$

We can try to make (2.3) into a strict restricted simplicial object as
follows:

(2.4) |
---|---
$\textstyle{{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=h}$$\scriptstyle{d_{1}=\operatorname{inc}}$$\scriptstyle{d_{2}=\operatorname{inc}}$$\textstyle{{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=g}$$\scriptstyle{d_{1}=\operatorname{inc}}$$\textstyle{{\mathbf{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon=f}$$\textstyle{C{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=G}$$\scriptstyle{d_{1}=\operatorname{inc}_{1}}$$\textstyle{C{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon=F}$$\textstyle{\mathbf{W}}$$\textstyle{C{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=Ch}$$\scriptstyle{d_{1}=\operatorname{inc}_{2}}$$\textstyle{\Sigma{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon=\langle{f,g,h}\rangle}$

where the space in each simplicial dimension, from left to right, is the wedge
of the relevant column, and
$\operatorname{inc}_{i}:C{\mathbf{X}}\hookrightarrow\Sigma{\mathbf{X}}$
($i=1,2$) denote the two inclusions of the upper and lower cones into
$\Sigma{\mathbf{X}}$ (the pushout of $C{\mathbf{X}}\leftarrow{\mathbf{X}}\to
C{\mathbf{X}}$). Here $\langle{f,g,h}\rangle$ is by definition the value of
the Toda bracket associated to the two choices of nullhomotopies $F:f\circ
g\sim\ast$ and $G:g\circ h\sim\ast$ (uniquely defined by the requirement that
(2.4) satisfies the simplicial identities on the nose).

We see that (2.4) realizes (2.3) up to homotopy if and only if the Toda
bracket vanishes, so that we can choose a nullhomotopy
$K:\langle{f,g,h}\rangle\sim\ast$, allowing us to replace $\Sigma{\mathbf{X}}$
by $C\Sigma{\mathbf{X}}$, with $\varepsilon=K$ on the cone.

######  2.5.

Higher Toda brackets.  One can generalize the ideas of §2.1 in several ways
(see, e.g., [BJT2, §7.A]). However, in our setting the most germane is that of
[BBS2, §1], which we briefly recapitulate:

Let $\mathcal{C}$ be a pointed model category, and let

(2.6)
${}{\mathbf{A}}_{0}~{}\xrightarrow{[f_{0}]}~{}{\mathbf{A}}_{1}~{}\xrightarrow{[f_{1}]}~{}{\mathbf{A}}_{2}~{}\to~{}\dotsc~{}\to~{}{\mathbf{A}}_{n-1}~{}\xrightarrow{[f_{n-1}]}~{}{\mathbf{A}}_{n}$

be a diagram in $\operatorname{ho}\mathcal{C}$ with $[f_{k}]\circ[f_{k-1}]=0$
for each $1\leq k<n$. This will be called a _Toda diagram_ of length $n$.

Our goal is to _strictify_ (2.6) by lifting it to $\mathcal{C}$ (up to weak
equivalence) with all composites strictly null. The obstructions to doing so
may be described as follows:

Let $\mathcal{I}^{n}$ denote $n$-dimensional cube with vertices labelled by
$J=(\varepsilon_{1},\varepsilon_{2},\dotsc,\varepsilon_{n})$ with each
$\varepsilon_{i}\in\\{0,1\\}$, partially ordered in the usual way, with a
unique map $J\to J^{\prime}$ whenever
$\varepsilon_{i}\leq\varepsilon^{\prime}_{i}$ for all $1\leq i\leq n$. Write
$J_{k}$ for the vertex labelled by $k$ ones followed by $n-k$ zeros
($k=0,1,\dotsc,n$).

Any strictification of (2.6) yields a functor
$D:\mathcal{I}^{n}\to\mathcal{C}$ with $D(J_{k})\simeq{\mathbf{A}}_{k}$, all
other vertices of $\mathcal{I}^{n}$ sent to $0$, and the unique morphism
$J_{k}\to J_{k+1}$ sent to a map weakly equivalent to $f_{k}$. A functor
$\widetilde{D}:\mathcal{I}^{n}\to\mathcal{C}$ weakly equivalent to such a $D$
will be called an _enhanced strictification_ of (2.6), and [BBS2, Lemma 1.13]
shows that (2.6) can be strictified if and only if some such enhanced
strictification exists.

There is a model category structure on the category
$\mathcal{C}^{\mathcal{I}^{n}}$ of all such diagrams, with weak equivalences
and fibrations defined fibrewise (see [Hir, 11.6.1]). If we attempt to
construct a cofibrant enhanced strictification
$\widetilde{D}:\mathcal{I}^{n}\to\mathcal{C}$, the obstructions to doing so
are the (higher) Toda brackets associated to (2.6) (see [BBS2, Definition
1.12]).

In order to construct a cofibrant enhanced strictification
$\widetilde{D}:\mathcal{I}^{n}\to\mathcal{C}$ of (2.6), we shall require the
following

######  2.7 Notation.

For $J=(\varepsilon_{1},\varepsilon_{2},\dotsc,\varepsilon_{n})$ with each
$\varepsilon_{i}\in\\{0,1\\}$ as above, we write:

  1. (a)

$\ell(J)$ for the number of initial ones in $J$, with $0\leq\ell(J)\leq n$;

  2. (b)

$R(J)$ for (possibly empty) _remainder_ of $J$ after the initial bloc of ones
and the zero ending it.

  3. (c)

$\lambda(J)$ for the total number of ones in $J$.

If $\mathbb{I}^{r}:=[0,1]^{r}$ is the $r$-dimensional cube, thought of as a
topological space (or simplicial set), then for any
$\vec{\eta}=(\varepsilon_{1},\dotsc,\varepsilon_{r})$
($\varepsilon_{i}\in\\{0,1\\}$) as above, we let

$L(\vec{\eta}):=\\{(t_{1},\dotsc,t_{r})\in\mathbb{I}^{r}~{}:\ \forall i,1\leq
i\leq r\ \Rightarrow 0\leq t_{i}\leq\varepsilon_{i}\\}~{}.$

This is a $k$-dimensional subcube of $\mathbb{I}^{r}$, where
$k:=\lambda(\vec{\eta})$. Its boundary $\partial L(\vec{\eta})$ is the union
of two sub-cubical complexes:

$\begin{split}\partial_{-}L(\vec{\eta})~{}:=&~{}\\{(t_{1},\dotsc,t_{r})\in\mathbb{I}^{r}~{}:\
\exists i,1\leq i\leq r\ \ t_{i}=\varepsilon_{i}=0\\}\hskip
5.69054pt\text{and}\\\
\partial_{+}L(\vec{\eta})~{}:=&~{}\\{(t_{1},\dotsc,t_{r})\in\mathbb{I}^{r}~{}:\
\exists i.1\leq i\leq r\ \ t_{i}=\varepsilon_{i}=1\\}~{}.\end{split}$

For any cofibrant ${\mathbf{A}}\in\mathcal{C}$,
$\vec{\eta}=(\varepsilon_{1},\dotsc,\varepsilon_{r})$, and
$k=\lambda(\vec{\eta})$ as above, we denote the cofiber of
${\mathbf{A}}\otimes\partial_{+}L(\vec{\eta})\hookrightarrow{\mathbf{A}}\otimes
L(\vec{\eta})$ by ${\mathbf{A}}\Join L(\vec{\eta})$. Note that $L(\vec{\eta})$
is a $k$-fold cylinder, so ${\mathbf{A}}\otimes
L(\vec{\eta})\simeq{\mathbf{A}}$, while ${\mathbf{A}}\Join L(\vec{\eta})$ is a
$k$-fold cone, and thus contractible for $k\geq 1$. When $k=0$,
$L(\vec{\eta})$ is a point, and by convention ${\mathbf{A}}\Join
L(\vec{\eta}):={\mathbf{A}}$. Moreover,

(2.8)
${}{\mathbf{A}}\otimes\partial_{+}L(\vec{\eta})~{}/~{}\left({\mathbf{A}}\otimes\partial_{-}L(\vec{\eta})\cap{\mathbf{A}}\otimes\partial_{+}L(\vec{\eta})\right)~{}\cong~{}\Sigma^{k-1}{\mathbf{A}}$

is a $(k-1)$-fold suspension of ${\mathbf{A}}$.

Finally, for each $n\geq 2$ we write $J^{n}_{\operatorname{fin}}$ for the
final object $(1,1,\dotsc,1)$ in $\mathcal{I}^{n}$,
$\widehat{J}^{n}_{\operatorname{fin}}$ for $(0,1,\dotsc,1)$, and
$\widehat{I}^{n}$ for the subcategory of $\mathcal{I}^{n}$ containing all
objects, but omitting the morphism $\widehat{J}^{n}_{\operatorname{fin}}\to
J^{n}_{\operatorname{fin}}$.

######  2.9 Example.

For $J=(11101010)$ we have $\ell(J)=3$ and $\lambda(J)=5$, with
$\vec{\eta}=R(J)=(1010)$ and

$\begin{split}L(\vec{\eta})~{}=&~{}[0,1]\otimes\\{0\\}\otimes[0,1]\otimes\\{0\\}\subseteq[0,1]^{4}~{},\\\
\partial_{-}L(\vec{\eta})~{}=&~{}(\\{0\\}\otimes\\{0\\}\otimes[0,1]\otimes\\{0\\})\cup([0,1]\otimes\\{0\\}\otimes\\{0\\}\otimes\\{0\\})~{},\hskip
5.69054pt\text{and}\\\
\partial_{+}L(\vec{\eta})~{}=&~{}(\\{1\\}\otimes\\{0\\}\otimes[0,1]\otimes\\{0\\})\cup([0,1]\otimes\\{0\\}\otimes\\{1\\}\otimes\\{0\\})~{}.\end{split}$

######  2.10 Definition.

Let $\mathcal{C}$ be a pointed simplicial model category $n\geq 3$, and (2.6)
a diagram in $\operatorname{ho}\mathcal{C}$. _Initial data_ for the associated
$(n-2)$-nd order _Toda bracket_ is a cofibrant diagram
$\widehat{D}:\widehat{I}^{n}\to\mathcal{C}$ such that:

  1. (a)

$\widehat{D}(J_{k})$ is cofibrant and weakly equivalent to ${\mathbf{A}}_{k}$
for all $0\leq k\leq n$.

  2. (b)

For any $J\in\mathcal{I}^{n}$ with $k=\ell(J)$ (so $0\leq k\leq n$) and
$\vec{\eta}:=R(J)$, we have $\widehat{D}(J)=\widehat{D}(J_{k})\Join
L(\vec{\eta})$. Note that this is compatible with (a), since
$R(J_{k})=(0,0,\dotsc,0)$ and ${\mathbf{A}}\Join L(0,\dotsc,0)={\mathbf{A}}$
by the conventon of §2.7.

  3. (c)

The map $\widehat{D}(J_{k})\to\widehat{D}(J_{k+1})$ (necessarily a cofbration,
since $\widehat{D}$ is cofibrant) is weakly equivalent to $f_{k}$ of (2.6).

  4. (d)

The values of $\widehat{D}$ at all morphisms $J\to J^{\prime}$ in
$\widehat{I}^{n}$ with $\ell(J)=\ell(J^{\prime})$ is induced by the inclusion
$L(J)\hookrightarrow L(J^{\prime})$.

Note that the values of $\widehat{D}$ at all objects but $J_{k}$
($k=0,\dotsc,n$) are contractible.

Write
$\vec{\eta}:=R(\widehat{J}^{n}_{\operatorname{fin}}=(1,\dotsc,1)=J^{n-1}_{\operatorname{fin}}$,
and let

(2.11)
${}{\mathbf{C}}_{n-1}~{}:=~{}\widehat{D}(J_{0})\otimes\partial_{-}L(\vec{\eta})/\left(\widehat{D}(J_{0})\otimes\partial_{-}L(\vec{\eta})\cap\widehat{D}(J_{0})\otimes\partial_{+}L(\vec{\eta})\right)~{}.$

Note that if $\mathcal{I}^{\prime}$ is the subcategory of $\mathcal{I}^{n}$
consisting of all objects $J^{\prime}$ with $\ell(J)=0$, then the colimit of
$\widehat{D}\lvert_{\mathcal{I}^{\prime}}$ is precisely
$\widehat{D}(J_{0})\otimes\partial_{-}L(\vec{\eta})$. Thus the maps
$\widehat{D}(J^{\prime})\to\widehat{D}(J^{n})$ determined by $\widehat{D}$
induce a unique map
$F^{\prime}:\widehat{D}(J_{0})\otimes\partial_{-}L(\vec{\eta})\to\widehat{D}(J_{n})$,
and its restriction to $\widehat{D}(J_{0})\otimes\partial_{+}L(\vec{\eta})$ is
trivial by assumption, so it induces a map

(2.12) ${}F:{\mathbf{C}}_{n-1}~{}\to~{}\widehat{D}(J_{n})$

Since ${\mathbf{C}}_{n-1}\simeq\Sigma^{n-2}{\mathbf{A}}_{0}$ by (2.8), and
$\widehat{D}(J_{n})\simeq{\mathbf{A}}_{n}$, we may define the _value_ of the
$(n-2)$-nd order Toda bracket associated to the initial data $\widehat{D}$ to
be the homotopy class of $F$, thought of as a class in
$[\Sigma^{n-2}{\mathbf{A}}_{0},\ {\mathbf{A}}_{n}]$.

Moreover, $F\simeq 0$ if and only if $\widehat{D}$ extends to an enhanced
strictification $\widetilde{D}:\mathcal{I}^{n}\to\mathcal{C}$ of (2.6),
implying that the latter can be strictified, by [BBS2, Lemma 1.13]. Thus the
collection of all values for the $(n-2)$-nd order Toda bracket associated to
(2.6), usually denoted by

(2.13)
${}\langle{f_{0},f_{1},\dotsc,f_{n-1}}\rangle~{}\subseteq~{}[\Sigma^{n-2}{\mathbf{A}}_{0},\
{\mathbf{A}}_{n}]$

is the obstruction to strictification in the sense that the latter is possible
if and only if $0\in\langle{f_{0},f_{1},\dotsc,f_{n-1}}\rangle$.

######  2.14 Example.

For $n=3$ we may depict $\widehat{D}$ as follows, with the associated value of
the ordinary Toda bracket given by $F$:

(2.15)
---
$\textstyle{\stackrel{{\scriptstyle\mbox{${\mathbf{A}}_{0}$}}}{{\mbox{\scriptsize{(000)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{\stackrel{{\scriptstyle\mbox{${\mathbf{A}}_{1}$}}}{{\mbox{\scriptsize{(100)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{\stackrel{{\scriptstyle\mbox{$C{\mathbf{A}}_{0}$}}}{{\mbox{\scriptsize{(010)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle\mbox{${\mathbf{A}}_{2}$}}}{{\mbox{\scriptsize{(110)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{\stackrel{{\scriptstyle\mbox{$C{\mathbf{A}}_{0}$}}}{{\mbox{\scriptsize{(001)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle\mbox{$C{\mathbf{A}}_{1}$}}}{{\mbox{\scriptsize{(100)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{C}}_{2}=\Sigma{\mathbf{A}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\stackrel{{\scriptstyle\mbox{$C\Sigma{\mathbf{A}}_{0}$}}}{{\mbox{\scriptsize{(011)}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\stackrel{{\scriptstyle\mbox{${\mathbf{A}}_{3}$}}}{{\mbox{\scriptsize{(111)}}}}}$

######  2.16 Remark.

Under the assumptions of §2.10, we can reinterpret the $(n-2)$-nd order Toda
bracket defined there as a simplicial higher order operation in the sense of
§1.13, as follows:

First, note that we can associate to each Toda diagram (2.6) of length $n$ in
$\operatorname{ho}\mathcal{C}$ a $(n-1)$-truncated restricted augmented
simplicial object
$\widehat{{\mathbf{X}}}_{\bullet}:\Delta_{\langle{n-1}\rangle}^{\operatorname{op}}\to\operatorname{ho}\mathcal{C}$,
by setting $\widehat{{\mathbf{X}}}_{k}:={\mathbf{A}}_{n-k-1}$ for $-1\leq
k\leq n-1$, with
$d_{0}:\widehat{{\mathbf{X}}}_{k}\to\widehat{{\mathbf{X}}}_{k-1}$ given by
$f_{n-k-1}$ and $d_{i}=0$ for $i\geq 1$, as in (3.8) below.

Now given initial data $\widehat{D}:\widehat{I}^{n}\to\mathcal{C}$ as in
§2.10, we we can associate an $(n-1)$–truncated augmented restricted
simplicial object
${\mathbf{X}}_{\bullet}:\Delta_{\langle{n-1}\rangle}^{\operatorname{op}}\to\mathcal{C}$,
with

(2.17)
${}{\mathbf{X}}_{k}~{}:=~{}\begin{cases}{\mathbf{C}}_{n-1}\vee\bigvee_{\lambda(J)=n-1}\
\widehat{D}(J)&\text{if}\ \ k=1\\\ \bigvee_{\lambda(J)=n-k-1}\
\widehat{D}(J)&\text{otherwise,}\end{cases}$

where ${\mathbf{C}}_{n-1}$ is as in (2.11).

The face map $d_{i}:{\mathbf{X}}_{k}\to{\mathbf{X}}_{k-1}$ takes the wedge
summand $\widehat{D}(J)$ to the summand $\widehat{D}(J^{\prime})$ by the
appropriate map of $\widehat{D}$, where $J^{\prime}$ is obtained from $J$ by
changing the $(i+1)$-st zero to a one. The augmentation
$d_{0}=\varepsilon:{\mathbf{X}}_{0}\to{\mathbf{X}}_{-1}={\mathbf{A}}_{n}$ on
the summand ${\mathbf{C}}_{n-1}$ is the map $F$ of (2.12).

We see that $f$ is nullhomotopic if and only if it extends to the cone on
${\mathbf{C}}_{n-1}$, and by replacing ${\mathbf{C}}_{n-1}$ with this cone in
(2.17) we obtain a strict $(n-1)$-truncated augmented restricted simplicial
object in $\mathcal{C}$ realizing $\widehat{{\mathbf{X}}}_{\bullet}$.

##  3\. The homotopy spectral sequence of a simplicial space

We now show how the above general construction is relates to the homotopy
spectral sequence of a simplicial space ${\mathbf{W}}_{\bullet}$. This was
originally constructed by Quillen in [Q] for bisimplicial groups, with a
version by Bousfield-Friedlander in [BF, Theorem B.5] for simplicial sets.
However, for our purposes it is most convenient to use the approach of Dwyer,
Kan, and Stover in [DKSt]:

######  3.1.

The spiral spectral sequence.  Given a simplicial space
${\mathbf{W}}_{\bullet}\in\mbox{\sf Top}_{0}^{\Delta^{\operatorname{op}}}$,
for each $n\geq 1$ define its $n$-_chains_ object
$C_{n}{\mathbf{W}}_{\bullet}$ to be $\\{w\in{\mathbf{W}}_{n}\,|\ d_{i}w=\ast\
\text{for}\ i=1,\dotsc,n\\}$. Its $n$-_cycles_ object is
$Z_{n}{\mathbf{W}}_{\bullet}=\\{w\in{\mathbf{W}}_{n}\,|\ d_{i}w=\ast\
\text{for}\ i=0,\dotsc,n\\}$ (with
$Z_{0}{\mathbf{W}}_{\bullet}:={\mathbf{W}}_{0}$). We denote the map
$(d_{0})\lvert_{C_{n}{\mathbf{W}}_{\bullet}}:C_{n}{\mathbf{W}}_{\bullet}\to
Z_{n-1}{\mathbf{W}}_{\bullet}$ by $\mathbf{d}_{0}^{W_{n}}$.

When ${\mathbf{W}}_{\bullet}$ is Reedy fibrant (see [Hir, §15.3]), we have a
fibration sequence

(3.2) ${}\dotsb\Omega
Z_{n-1}{\mathbf{W}}_{\bullet}~{}\xrightarrow{\partial_{n-1}}~{}Z_{n}{\mathbf{W}}_{\bullet}~{}\xrightarrow{j_{n}}~{}C_{n}{\mathbf{W}}_{\bullet}~{}\xrightarrow{\mathbf{d}_{0}^{W_{n}}}~{}Z_{n-1}{\mathbf{W}}_{\bullet}~{}.$

Thus the $E_{1}$-exact couple of the spectral sequence may be described as
follows:

(3.3)
$\textstyle{\pi_{i+1}Z_{n}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\partial_{n})_{\\#}}$$\scriptstyle{(j_{n})_{\\#}}$$\textstyle{\pi_{i+1}C_{n}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathbf{d}_{0}^{W_{n}})_{\\#}}$$\textstyle{\pi_{i+1}Z_{n-1}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\partial_{n-1})_{\\#}}$$\scriptstyle{(j_{n-1})_{\\#}}$$\textstyle{\pi_{i+1}C_{n-1}{\mathbf{W}}_{\bullet}}$$\textstyle{\pi_{i}Z_{n+1}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(j_{n+1})_{\\#}}$$\textstyle{\pi_{i}C_{n+1}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathbf{d}_{0}^{W_{n+1}})_{\\#}}$$\textstyle{\pi_{i}Z_{n}{\mathbf{W}}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(j_{n}))_{\\#}}$$\textstyle{\pi_{i}C_{n}{\mathbf{W}}_{\bullet}}$

This agrees from the $E^{2}$-term on with the Bousfield-Friedlander spectral
sequence, having

(3.4)
${}E^{2}_{{n},{i}}~{}\cong~{}\pi_{n}^{h}\pi_{i}^{v}{\mathbf{W}}_{\bullet}~{}\Longrightarrow~{}\pi_{n+i}\|{\mathbf{W}}_{\bullet}\|$

with $d_{r}:E^{r}_{n,i}\to E^{r}_{n-r,i+r-1}$ (see [DKSt, Proposition 8.3]).

This is based on the isomorphism $\pi_{i}C_{n}{\mathbf{W}}_{\bullet}\to
C_{n}\pi_{i}{\mathbf{W}}_{\bullet}$ (see [St, Lemma 2.7]), and the fact that
there are natural maps

(3.5)
${}\pi_{i}Z_{n}{\mathbf{W}}_{\bullet}\to\pi_{0}\operatorname{map}({\mathbf{S}}^{i}\otimes
S^{n},{\mathbf{W}}_{\bullet})\to\pi_{n+i}\|{\mathbf{W}}_{\bullet}\|$

(see [DKSt, §5.2]), where ${\mathbf{S}}^{i}\otimes S^{n}$ is the $n$-skeletal
simplicial space with a single non-degenerate copy of ${\mathbf{S}}^{i}$ in
simplicial dimension $n$ (and $\ast$ in lower dimensions) – in [DKSt, §3.6]
this is written $\Sigma^{n}{\mathbf{S}}^{i}$. As usual,
$\operatorname{map}(-,-)$ is the simplicial enrichment of $\mbox{\sf
Top}_{\ast}^{\Delta^{\operatorname{op}}}$.

######  3.6.

Filtration in the spiral spectral sequence.  A non-zero class
$\gamma\in\pi_{n+i}\|{\mathbf{W}}_{\bullet}\|$ is represented in the spectral
sequence by a permanent cycle, represented in turn by an element $[g]\in
E^{1}_{{n},{i}}=\pi_{i}C_{n}{\mathbf{W}}_{\bullet}$, only if we can choose the
map $g:{\mathbf{S}}^{i}\to C_{n}{\mathbf{W}}_{\bullet}$ so that it lands in
$Z_{n}{\mathbf{W}}_{\bullet}$, in the long exact sequence

(3.7)
${}\dotsc~{}\to~{}\pi_{i}Z_{n}{\mathbf{W}}_{\bullet}~{}\xrightarrow{j_{\\#}}~{}\pi_{i}C_{n}{\mathbf{W}}_{\bullet}~{}\xrightarrow{d_{\\#}}~{}\pi_{i}Z_{n-1}{\mathbf{W}}_{\bullet}~{}\dotsc~{},$

for the fibration sequence (3.2), and then use (3.5).

Now the map $g:{\mathbf{S}}^{i}\to
Z_{n}{\mathbf{W}}_{\bullet}\hookrightarrow{\mathbf{W}}_{n}$ makes the diagram

(3.8)
$\textstyle{{\mathbf{S}}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=g}$$\scriptstyle{d_{1}=\ast}$$\scriptstyle{\vdots}$$\scriptstyle{d_{n}=\ast}$$\scriptstyle{d_{n+1}=\ast}$$\textstyle{{\mathbf{W}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{d_{n}}$$\textstyle{{\mathbf{W}}_{n-1}\
\dotsc\
{\mathbf{W}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\textstyle{{\mathbf{W}}_{0}}$

into a strict $(n+1)$-truncated restricted simplicial object in $\mbox{\sf
Top}_{\ast}$. Furthermore, recall that the $(n+1)$-st _latching object_ of
${\mathbf{W}}_{\bullet}$ is

(3.9)
${}L_{n}G_{\bullet}~{}:=~{}\raisebox{-7.3194pt}{$\stackrel{{\scriptstyle\textstyle\operatorname{colim}}}{{\scriptstyle{\theta^{\operatorname{op}}:[\mathbf{k}]\to[\mathbf{n}+\mathbf{1}]}}}$}\,G_{k}~{},$

where $\theta$ ranges over the surjective maps
$[\mathbf{n}+\mathbf{1}]\to[\mathbf{k}]$ in $\Delta$ (for $k\leq n$). Thus if
we replace ${\mathbf{S}}^{i}$ by ${\mathbf{S}}^{i}\vee
L_{n+1}{\mathbf{W}}_{\bullet}$ in (3.8), then (together with the obvious
degeneracy maps into $L_{n+1}{\mathbf{W}}_{\bullet}$ and face maps out) we
have a full $(n+1)$-truncated simplicial object.

However, since ${\mathbf{W}}_{\bullet}$ is Reedy fibrant, it cannot have an
augmentation ${\bm{\varepsilon}}:{\mathbf{W}}_{\bullet}\to
c({{\mathbf{Y}}})_{\bullet}$ for any space ${\mathbf{Y}}\neq\ast$ (cf. [BK,
Ch. X, §4]), making (3.8) into an $(n+1)$-truncated simplicial space
(restricted or not). However, we can choose
$\varepsilon:{\mathbf{W}}_{0}\to\|{\mathbf{W}}_{\bullet}\|$ making the
extension

(3.10)
$\textstyle{{\mathbf{S}}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}=g}$$\scriptstyle{d_{1}=\ast}$$\scriptstyle{\vdots}$$\scriptstyle{d_{n}=\ast}$$\scriptstyle{d_{n+1}=\ast}$$\textstyle{{\mathbf{W}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{\vdots}$$\scriptstyle{d_{n}}$$\textstyle{{\mathbf{W}}_{n-1}\
\dotsc\
{\mathbf{W}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\textstyle{{\mathbf{W}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\|{\mathbf{W}}_{\bullet}\|}$

of (3.8) commute up to homotopy.

In fact, if we use the standard model

(3.11)
${}\|{\mathbf{W}}_{\bullet}\|~{}:=~{}(\coprod_{n=0}^{\infty}\,{\mathbf{W}}_{n}\ltimes\mathbf{\Delta}^{n})/\sim$

(see [GJ, VII, §3]), we have tautological higher homotopies as in §1.8,
induced by
$\operatorname{Id}:{\mathbf{W}}_{n}\ltimes\mathbf{\Delta}^{n}\to{\mathbf{W}}_{n}\ltimes\mathbf{\Delta}^{n}$,
for all but
${\mathbf{S}}^{i}\ltimes\mathbf{\Delta}^{n+1}\to\|{\mathbf{W}}_{\bullet}\|$.
We call the corresponding element in
$[\Sigma^{n}{\mathbf{S}}^{i},\,\|{\mathbf{W}}_{\bullet}\|]$ a _canonical
value_ of the higher homotopy operation associated to this homotopy-
commutative $(n+1)$-truncated restricted augmented simplicial space, and
deduce:

######  3.12 Proposition.

An element $\gamma\in\pi_{n+i}\|{\mathbf{W}}_{\bullet}\|$ is in filtration $n$
of the spiral spectral sequence for ${\mathbf{W}}_{\bullet}$ if and only if it
is a canonical value for the $(n+1)$-st order simplicial operation associated
to (3.10).

##  4\. Simplicial space approximations

We would like to have a procedure for decomposing elements in the homotopy
groups $\pi_{\ast}{\mathbf{X}}$ of an arbitrary space ${\mathbf{X}}$ in terms
of higher order homotopy operations. By our description of the spiral spectral
sequence, we can do so by providing a suitable simplicial space
${\mathbf{W}}_{\bullet}$ with $\|{\mathbf{W}}_{\bullet}\|\simeq{\mathbf{X}}$.

######  4.1.

Constructing CW approximations.  For simplicity we may assume ${\mathbf{X}}$
is a $(k-1)$-connected pointed space with $k\geq 2$. We define a _sequential
approximation_ to ${\mathbf{X}}$ to be a sequence

(4.2)
${}{\mathbf{W}}_{\bullet}^{[{k}]}~{}\xrightarrow{\iota^{[{k}]}}~{}{\mathbf{W}}_{\bullet}^{[{k+1}]}~{}\xrightarrow{\iota^{[{k+1}]}}~{}{\mathbf{W}}_{\bullet}^{[{k+2}]}~{}\to~{}\dotsc~{}{\mathbf{W}}_{\bullet}^{[{n}]}~{}\xrightarrow{\iota^{[{n}]}}~{}{\mathbf{W}}_{\bullet}^{[{n+1}]}~{}\to~{}\dotsc~{}$

of simplicial spaces with augmentations
${\bm{\varepsilon}}^{[{n}]}:{\mathbf{W}}_{\bullet}^{[{n}]}\to{\mathbf{X}}$
(commuting with the maps $\iota^{[{n}]}$), such that for each $n\geq k$:

  1. (a)

${\mathbf{W}}_{\bullet}^{[{n}]}$ is $(n-k)$-skeletal (in the simplicial
direction).

  2. (b)

For each $i\geq 0$, ${\mathbf{W}}_{i}^{[{n}]}$ is homotopy equivalent to a
wedge of $(k-1)$-connected spheres.

  3. (c)

The map
$\varepsilon^{[{n}]}:\|{\mathbf{W}}_{\bullet}^{[{n}]}\|\to{\mathbf{X}}$
induced by ${\bm{\varepsilon}}^{[{n}]}$ (out of the geometric realization) is
an $n$-equivalence: that is, it induces an isomorphism in $\pi_{i}$ for $i<n$
and an epimorphism for $i=n$). This implies that any $n$-skeleton of
$\|{\mathbf{W}}_{\bullet}^{[{n}]}\|$ is an $n$-skeletal CW approximation for
${\mathbf{X}}$.

######  4.3 Proposition.

Every $(k-1)$-connected pointed space ($k\geq 2$) has a sequential
approximation; if ${\mathbf{X}}$ is of finite type, we may assume each
${\mathbf{W}}_{i}^{[{n}]}$ is homotopy equivalent to a finite wedge of
spheres.

###### Proof.

We construct (4.2) by induction on $n\geq 0$:

Step 1: For $n=k$, start with ${\mathbf{W}}_{0}^{[{k}]}$ a wedge of
$k$-spheres with a map
${\bm{\varepsilon}}^{[{k}]}:{\mathbf{W}}_{0}^{[{k}]}\to{\mathbf{X}}$ which
induces a surjection in $\pi_{k}$. We choose it to be minimal with this
property (that is, no proper sub-wedge has such a surjection). We let
${\mathbf{W}}_{\bullet}^{[{k}]}$ be $c({{\mathbf{W}}_{0}^{[{k}]}})_{\bullet}$.

Step 2: For $n=k+1$, choose $\overline{{\mathbf{W}}}_{1}^{[{k+1}]}$ to be a
(minimal) wedge of $k$-spheres having a map
$d_{0}:\overline{{\mathbf{W}}}_{1}^{[{k+1}]}\to{\mathbf{W}}_{0}^{[{0}]}$
inducing a surjection onto the kernel of
${\bm{\varepsilon}}^{[{k}]}_{\\#}:\pi_{k}{\mathbf{W}}_{0}^{[{k}]}\to\pi_{k}{\mathbf{X}}$.
Let $F:C{\mathbf{W}}_{1}^{[{k+1}]}\to{\mathbf{X}}$ be a nullhomotopy for
${\bm{\varepsilon}}^{[{k}]}\circ d_{0}$, and set
$\widetilde{{\mathbf{W}}}\quad_{0}^{[{k+1}]}:={\mathbf{W}}_{0}^{[{k}]}\vee
C{\mathbf{W}}_{1}^{[{k+1}]}$ and
$\widetilde{{\mathbf{W}}}\quad_{1}^{[{k+1}]}:=\overline{{\mathbf{W}}}_{1}^{[{k+1}]}\vee\widetilde{{\mathbf{W}}}\quad_{0}^{[{k+1}]}$.
We obtain a $1$-skeletal simplicial space
$\widetilde{{\mathbf{W}}}\quad_{\bullet}^{[{k+1}]}$, with an augmentation to
${\mathbf{X}}$ given by ${\bm{\varepsilon}}^{[{k}]}\bot F$. The degeneracy
$s_{0}:\widetilde{{\mathbf{W}}}\quad_{0}^{[{k+1}]}\to\widetilde{{\mathbf{W}}}\quad_{1}^{[{k+1}]}$
is given by the inclusion. The map
$\widetilde{\varepsilon}^{[{k+1}]}:\|\widetilde{{\mathbf{W}}}\quad_{\bullet}^{[{k+1}]}\|\to{\mathbf{X}}$
thus induces an isomorphism in $\pi_{k}$.

Now choose a (minimal) wedge of $(k+1)$-spheres
$\overline{{\mathbf{W}}}_{0}^{[{k+1}]}$ with a map
$e:\overline{{\mathbf{W}}}_{0}^{[{k+1}]}\to{\mathbf{X}}$ mapping onto
$\pi_{k+1}{\mathbf{X}}\setminus\operatorname{Im}(\widetilde{\varepsilon}^{[{k+1}]}_{\\#})$.
Thus if we set
${\mathbf{W}}_{0}^{[{k+1}]}:=\widetilde{{\mathbf{W}}}\quad_{0}^{[{k}]}\vee{\mathbf{W}}^{\prime}$
and
${\mathbf{W}}_{1}^{[{k+1}]}:=\widetilde{{\mathbf{W}}}\quad_{1}^{[{k+1}]}\vee{\mathbf{W}}_{0}^{[{k+1}]}$,
we obtain a $1$-skeletal simplicial space ${\mathbf{W}}_{\bullet}^{[{k+1}]}$
with an augmentation to ${\mathbf{X}}$ such that the induced map
$\varepsilon^{[{k+1}]}:\|{\mathbf{W}}_{\bullet}^{[{k+1}]}\|\to{\mathbf{X}}$
induces a surjection in $\pi_{k+1}$ (and so is a $(k+1)$-equivalence ).

Step 3: Assume given ${\mathbf{W}}_{\bullet}^{[{n-1}]}$ as above, and let
$m\geq n-1$ be maximal such that the map
$\varepsilon^{[{n-1}]}:\|{\mathbf{W}}_{\bullet}^{[{n-1}]}\|\to{\mathbf{X}}$
(induced by
${\bm{\varepsilon}}^{[{n-1}]}:{\mathbf{W}}_{\bullet}^{[{n-1}]}\to{\mathbf{X}}$)
is an $m$-equivalence. If $m\geq n$, we set
${\mathbf{W}}_{\bullet}^{[{n}]}:={\mathbf{W}}_{\bullet}^{[{n-1}]}$; otherwise
$m=n-1$, and we construct ${\mathbf{W}}_{\bullet}^{[{n}]}$ by an inner
induction as follows:

  1. Case 1.

If
$\pi_{n-1}\varepsilon:\pi_{n-1}\|{\mathbf{W}}_{\bullet}^{[{n-1}]}\|\to\pi_{n-1}{\mathbf{X}}$
is injective (and thus an isomorphism), necessarily $\pi_{n}\varepsilon$ is
not surjective. Choose a minimal wedge of $n$-spheres ${\mathbf{W}}^{\prime}$
and a map $\varepsilon^{\prime}:{\mathbf{W}}^{\prime}\to{\mathbf{X}}$ mapping
onto $\pi_{n}{\mathbf{X}}\setminus\operatorname{Im}(\varepsilon_{\\#})$, and
set
${\mathbf{W}}_{0}^{[{n}]}:={\mathbf{W}}_{0}^{[{n-1}]}\vee{\mathbf{W}}^{\prime}$,
with the necessary degeneracies in higher simplicial dimensions. Note that
$\varepsilon^{\prime}$ has non-trivial Hurewicz image.

  2. Case 2.

If $\pi_{n-1}\varepsilon^{[{n-1}]}$ is not injective, let
$K:=\operatorname{Ker}(\pi_{n-1}\varepsilon^{[{n-1}]})$, with $j\geq 0$ the
minimal filtration of elements of $K$, and set
${\mathbf{W}}_{\bullet}^{[{n,i}]}:={\mathbf{W}}_{\bullet}^{[{n-1}]}$ for each
$i\leq j$, and ${\mathbf{W}}_{i}^{[{n,j+1}]}:={\mathbf{W}}_{i}^{[{n}]}$ for
$0\leq i<j$ Next, choose a minimal set of generators $\\{\alpha_{i}\\}_{i\in
I}$ for the elements of $K$ in filtration $j$, represented by maps
$g_{i}:{\mathbf{S}}^{n_{i}}\to Z_{j}{\mathbf{W}}_{j}^{[{n,0}]}$ (see §3.6),
and let
${\mathbf{W}}_{j}^{[{n,j+1}]}:={\mathbf{W}}_{j+1}^{[{n-1}]}\vee\bigvee_{i\in
I}{\mathbf{S}}^{n_{i}}$. If we let $d_{0}=g_{i}$ on ${\mathbf{S}}^{n_{i}}$,
and other face maps vanish there (and add the necessary degeneracies), we
obtain a new simplicial space ${\mathbf{W}}_{\bullet}^{[{n,j+1}]}$ for which
the elements $\\{\alpha_{i}\\}_{i\in I}$ are no longer in
$K:=\operatorname{Ker}(\pi_{n-1}\varepsilon^{[{n-1}]})$. Note that because
$g_{i}$ lands in $Z_{j}{\mathbf{W}}_{\bullet}^{[{n-1}]}$, it represents a
permanent cycle in the spiral spectral sequence of §3.1, so no elements of
$\pi_{n+k-1}\|{\mathbf{W}}_{\bullet}^{[{n-1}]}\|$ have been killed by this
process.

Proceeding in this way – with ${\mathbf{W}}_{\bullet}^{[{n,s+1}]}$ obtained
from ${\mathbf{W}}_{\bullet}^{[{n,s}]}$ in stage $s$ by killing the elements
in $K$ in filtration $s$ – we obtain ${\mathbf{W}}_{\bullet}^{[{n,n}]}$ with
$\pi_{n-1}\widetilde{\varepsilon}^{[{n}]}$ injective (there is no need to
proceed beyond filtration $n$ for dimension reasons).

We then wedge on $n$-spheres in simplicial dimension $0$ as in Case 1 to
obtain ${\mathbf{W}}_{\bullet}^{[{n}]}$.

∎

##  5\. Atomic maps

Now that we have a procedure for generating new elements in
$\pi_{\ast}{\mathbf{X}}$ by higher order operations, we would like to show
which elements are indecomposable with respect to such operations.

######  5.1 Definition.

For a simply-connected space ${\mathbf{X}}$, a homotopy class
$\varphi:{\mathbf{S}}^{n}\to{\mathbf{X}}$ in is called _atomic_ (for a prime
$p$) if it induces a non-zero map
$\varphi_{\ast}:H_{n}({{\mathbf{S}}^{n}};{\mathbb{F}_{p}})\to
H_{n}({{\mathbf{X}}};{\mathbb{F}_{p}})$.

######  5.2 Theorem.

If ${\mathbf{X}}$ is $k$-connected for $k\geq 1$, any homotopy class
$\varphi:{\mathbf{S}}^{n}\to{\mathbf{X}}$ is generated by the atomic maps
under the primary homotopy operations (group operations, composition, and
Whitehead products) and the higher order simplicial operations associated to
the sequential approximations of §3.6.

###### Proof.

We may assume that all spaces are localized at a prime $p$, and consider the
sequential approximations ${\mathbf{W}}_{\bullet}^{[{m}]}$ ($k\leq m\leq n$)
for ${\mathbf{X}}$ constructed in the proof of Proposition 4.3, which we may
also take to be $p$-local and $k$-connected in each simplicial dimension.
Assume that $\varphi$ is not atomic, so that
$\varphi_{\ast}:H_{n}({{\mathbf{S}}^{n}};{\mathbb{F}_{p}})\to
H_{n}({{\mathbf{X}}};{\mathbb{F}_{p}})$ is trivial.

Note that if $\varphi_{\ast}:H_{n}({{\mathbf{S}}^{n}};{\mathbb{Z}_{(p)}})\to
H_{n}({{\mathbf{X}}};{\mathbb{Z}_{(p)}})$ is non-zero, then the Hurewicz image
of $\varphi$ is necessarily divisible by $p^{r}$ for some $r\geq 1$.

Now if
${\mathbf{Z}}\,\xrightarrow{\tau}\,\widehat{{\mathbf{X}}}\,\xrightarrow{i}\,{\mathbf{X}}\,\xrightarrow{q}\,\Sigma{\mathbf{Z}}$
is the cofibration sequence for some CW structure on ${\mathbf{X}}$ with
$n$-skeleton $\widehat{{\mathbf{X}}}$, with
${\mathbf{Z}}:=\bigvee_{i=1}^{m}\,{\mathbf{S}}^{n}_{\langle{i}\rangle}$ (here
$\langle{i}\rangle$ is just an index). We have the associated Hurewicz
diagram:

(5.3)
$\textstyle{\pi_{n}{\mathbf{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\\#}}$$\scriptstyle{h_{n}}$$\scriptstyle{\cong}$$\textstyle{\pi_{n}\widehat{{\mathbf{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\\#}}$$\scriptstyle{h_{n}}$$\textstyle{\pi_{n}{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{n}}$$\textstyle{\pi_{n}\Sigma{\mathbf{Z}}=0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}({{\mathbf{Z}}};{\mathbb{Z}_{(p)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\ast}}$$\textstyle{H_{n}({\widehat{{\mathbf{X}}}};{\mathbb{Z}_{(p)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\ast}}$$\textstyle{H_{n}({{\mathbf{X}}};{\mathbb{Z}_{(p)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}({\Sigma{\mathbf{Z}}};{\mathbb{Z}_{(p)}})=0}$

Since $i_{\\#}$ is surjective, $f$ lifts to
$f^{\prime}:{\mathbf{S}}^{n}\to{\mathbf{X}}$ (cellular approximation), and
since $h_{n}([f])$ is non-trivial by assumption, so is $h_{n}([f^{\prime}])$.
However, we assumed $h_{n}([f])=p^{r}\cdot\hat{y}$ for some $\hat{y}\in
H_{n}({{\mathbf{X}}};{\mathbb{Z}_{(p)}})$ and $r\geq 1$, so there is $y\in
H_{n}({\widehat{{\mathbf{X}}}};{\mathbb{Z}_{(p)}})$ with
$i_{\ast}(y)=\hat{y}$. This means that $p^{r}\cdot
y-h_{n}([f^{\prime}])\in\operatorname{Ker}(i_{\ast})=\operatorname{Im}(\tau_{\ast})$,
so if we write this expression as $\tau_{\ast}(h_{n}(\alpha))$ for some
$\alpha\in\pi_{n}{\mathbf{Z}}$, we see that we may replace our choice
$f^{\prime}$ by $f^{\prime\prime}:=f^{\prime}+\tau_{\\#}(\alpha)$ with
$i_{\\#}(f^{\prime\prime})=f$, but now $h_{n}(f^{\prime\prime})$ itself
divisible by $p^{r}$ in $H_{n}({{\mathbf{X}}};{\mathbb{Z}_{(p)}})$.

In particular, this applies when $\widehat{{\mathbf{X}}}$ is (an $n$-skeleton
of) $\|{\mathbf{W}}_{\bullet}^{[{n}]}\|$, by definition (see §4.1). Thus
$\varphi$ factors through
$f^{(n)}:{\mathbf{S}}^{n}\to\|{\mathbf{W}}_{\bullet}^{[{n}]}\|$, and we may
assume that this is not atomic, either. The class
$[f^{(n)}]\in\pi_{n}\|{\mathbf{W}}_{\bullet}^{[{n}]}\|$ can be written (non-
uniquely) as a sum of elements in various filtrations. All those in positive
filtration are canonical values of higher order simplicial operations, by
Proposition 3.12. The elements in filtration $0$ factor through
${\mathbf{W}}_{0}^{[{n}]}$, which is a wedge of spheres of dimensions $\leq
n$. Thus by Hilton’s Theorem (see [Hil]), each such element is a sum of
compositions of iterated Whitehead products on the fundamental classes of
these spheres. All such summands are decomposable, except possibly for those
which factor through the sub-wedge product ${\mathbf{W}}^{\prime}$ consisting
of the $n$-spheres. However, if
$f^{(n)}_{\ast}:H_{n}({{\mathbf{S}}^{n}};{\mathbb{Z}_{(p)}})\to
H_{n}({\|{\mathbf{W}}_{\bullet}^{[{n}]}\|};{\mathbb{Z}_{(p)}})$ is non-zero,
then (because $f^{(n)}$ is not atomic) its Hurewicz image is $p$-divisible,
and since the Hurewicz map for ${\mathbf{W}}^{\prime}$ is an isomorphism in
dimension $n$, this means the map $g^{(n)}$ is itself $p$-divisible, and thus
decomposable. ∎

##  6\. Atomic maps for spheres

There is little hope of describing all atomic maps for general ${\mathbf{X}}$,
since the Hurewicz image is not known even stably, in general (see [Co, §2]).
However, we can do so for spheres: since the construction of §4.1 is trivial
for ${\mathbf{S}}^{k}$ itself, more precisely, we consider a connected cover.
All spaces in this section are localized at a prime $p$.

######  6.1 Theorem.

For $k\geq 2$, let ${\mathbf{X}}$ be the homotopy fiber of the fundamental
class
$\varepsilon_{k}:{\mathbf{S}}^{k}\to{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})$
(that is, the $(r-1)$-connected cover of the $k$-sphere for $r:=k+2p-3$), and
assume that the lift $f:{\mathbf{S}}^{n}\to{\mathbf{X}}$ of
$\varphi:{\mathbf{S}}^{n}\to{\mathbf{S}}^{k}$ is atomic for $p$. Then the
homotopy cofiber of $\varphi$ supports a non-trivial mod $p$ cohomology
operation.

###### Proof.

Our hypothesis is that the image of $[f]\in\pi_{n}{\mathbf{X}}$ under the
Hurewicz homomorphism $\pi_{n}{\mathbf{X}}\to
H_{n}({{\mathbf{X}}};{\mathbb{F}_{p}})$ is non-trivial. There is thus a class
$\lambda\in H^{n}({{\mathbf{X}}};{\mathbb{F}_{p}})$ with
$f^{\ast}(\lambda)\neq 0$. Such a $\lambda$ is necessarily indecomposable in
the unstable cohomology algebra $H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$
(that is, it does not decompose in terms of any unstable cohomology
operations, including the cup product).

We first observe that it suffices to show that any such indecomposable class
$\lambda$ is transgressive in the $\mathbb{F}_{p}$-cohomology Serre spectral
sequence of the fibration sequence
${\mathbf{X}}\hookrightarrow{\mathbf{S}}^{k}\to{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})$.

If we write $g:{\mathbf{X}}\to{\mathbf{S}}^{k}$ for the covering map, then
$\varphi:=g\circ f$ fits into a commuting diagram with horizontal homotopy
cofibration sequences, as follows:

(6.2)
$\textstyle{{\mathbf{S}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\textstyle{{\mathbf{S}}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{j}$$\textstyle{\operatorname{Cof}(\varphi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\delta}$$\textstyle{{\mathbf{S}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma
f}$$\textstyle{{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{\mathbf{S}}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{t}$$\textstyle{\operatorname{Cof}(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\delta}}$$\scriptstyle{\rho}$$\textstyle{\Sigma{\mathbf{X}}}$$\textstyle{{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})}$

where the map $\rho$ exists by the universal properties of
$\operatorname{Cof}(g)$ and the fibration sequence
${\mathbf{X}}\xrightarrow{g}{\mathbf{S}}^{k}\xrightarrow{t}{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})$.

Since by assumption $f^{\ast}\lambda=\epsilon_{n}$ (the fundamental class in
$H^{n}({{\mathbf{S}}^{n}};{\mathbb{F}_{p}})$), under suspension we have
$\bar{\lambda}\in H^{n+1}({\Sigma{\mathbf{X}}};{\mathbb{F}_{p}})$ with
$(\Sigma f)^{\ast}\bar{\lambda}=\epsilon_{n+1}\in
H^{n+1}({{\mathbf{S}}^{n+1}};{\mathbb{F}_{p}})$.

If $\lambda$ is transgressive, there is a class $c\in
H^{n+1}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})};{\mathbb{F}_{p}})$ with
$\rho^{\ast}c=\bar{\delta}^{\ast}\bar{\lambda}$. But $c$ is necessarily of the
form $\theta(\iota_{k})$ for $\iota_{k}\in
H^{k}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})};{\mathbb{F}_{p}})$ the
fundamental class, and a diagram chase in:

(6.3)
---
$\textstyle{H^{k}{{\mathbf{S}}^{k}}\ni\epsilon_{k}}$$\textstyle{a\in
H^{k}{\operatorname{Cof}(\varphi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\ast}}$$\scriptstyle{\theta}$$\textstyle{H^{n+1}{\operatorname{Cof}(\varphi)}\ni
b}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\epsilon_{n+1}\in
H^{n+1}{{\mathbf{S}}^{n+1}}}$$\scriptstyle{\delta^{\ast}}$$\textstyle{H^{k}{{\mathbf{S}}^{k}}\ni\epsilon_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{H^{n+1}{\operatorname{Cof}(g)}\ni\bar{\delta}^{\ast}(\bar{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{\ast}}$$\textstyle{\bar{\lambda}\in
H^{n+1}{\Sigma{\mathbf{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\delta}^{\ast}}$$\scriptstyle{(\Sigma
f)^{\ast}}$$\textstyle{\iota_{k}\in
H^{k}{{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\ast}}$$\scriptstyle{\theta}$$\textstyle{c\in
H^{n+1}{{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\ast}}$

shows that $b=\theta(a)$, where $a\in
H^{k}({\operatorname{Cof}(\varphi)};{\mathbb{F}_{p}})$ has
$j^{\ast}a=\epsilon_{k}$, and $b\in
H^{n+1}({\operatorname{Cof}(\varphi)};{\mathbb{F}_{p}})$ has
$\delta^{\ast}\epsilon_{n+1}=b$, This $\theta$ is the non-trivial cohomology
operation required by our Theorem. We have thus reduced the proof of the
Theorem to showing that any indecomposable class $\lambda$ is transgressive in
the Serre spectral sequence for
${\mathbf{X}}\to{\mathbf{S}}^{k}\to{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})$.

We compute $H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$, using the associated
fibration sequence
${\mathbf{K}}({\mathbb{Z}_{(p)}},{k-1})\xrightarrow{u}{\mathbf{X}}\to{\mathbf{S}}^{k}$.
Since the base is a sphere, the Serre spectral sequence reduces to the Wang
long exact sequence of [Wan]:

(6.4)
---
$\textstyle{\cdots
H^{n}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k-1})};{\mathbb{F}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{k}}$$\textstyle{H^{n-k+1}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k-1})};{\mathbb{F}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{n+1}({{\mathbf{X}}};{\mathbb{F}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{\ast}}$$\textstyle{H^{n+1}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k-1})};{\mathbb{F}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

We write $E^{0}({p})$ for the image of
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ in
$H^{\ast}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k-1})};{\mathbb{F}_{p}})$ – that
is, $E^{0}({p}):=u^{\ast}H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$.

From here on we distinguish between two cases:

Case I: The prime $2$.

For $p=2$, we have:

(6.5)
${}H^{\ast}({{\mathbf{K}}({\mathbb{Z}_{(2)}},{k})};{\mathbb{F}_{2}})\cong\mathbb{F}_{2}[\operatorname{Sq}^{I}(\iota_{k})~{}|\
I\mbox{ admissible},i_{s}\neq 1,\operatorname{ex}(I)<k].$

(see [Ko, Proposition 3.5.8]). Here the multi-index $I$ is of the form
$(i_{0},\dotsc,i_{s})$ with
$\operatorname{Sq}^{I}:=\operatorname{Sq}^{i_{0}}\dotsc\operatorname{Sq}^{i_{s}}$,
and $\operatorname{ex}(I)$ is the excess.

The exact sequence (6.4) is determined by the fact that $d_{k}$ is an (anti-)
derivation, which sends $\iota_{k-1}$ to $1$ and
$\operatorname{Sq}^{I}(\iota_{k-1})$ to $0$ for $I\neq 0$. It follows that

(6.6) ${}E^{0}({2})~{}=~{}\mathbb{F}_{2}[\iota_{k-1}^{2},\
\operatorname{Sq}^{I}(\iota_{k-1})~{}|\ 0\neq I\mbox{ admissible},\ i_{s}\neq
1,\ \operatorname{ex}(I)<k-1]~{}.$

We choose generators in $H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{2}})$
corresponding to the polynomial algebra generators $\iota_{k-1}^{2}$ and
$\operatorname{Sq}^{I}{\iota_{k-1}}$, and also use the same notation for them.
Thus $H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{2}})$ is generated as an
$E^{0}({2})$-module by $1$ and $\gamma_{2k-1}$ ($=\rho(\iota_{k-1})$), which
lies in $H^{2k-1}({{\mathbf{X}}};{\mathbb{F}_{2}})$ under the exact sequence
(6.4)). Therefore, the indecomposable classes in
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{2}})$ consist of $\gamma_{2k-1}$ and
$\operatorname{Sq}^{j}(\iota_{k-1})$ for $j\leq k-1$.

We now verify that all these classes are transgressive in the
$\mathbb{F}_{2}$-cohomology Serre spectral sequence for
${\mathbf{X}}\to{\mathbf{S}}^{k}\to{\mathbf{K}}({\mathbb{Z}_{(2)}},{k})$,
using the following map $u$ of fibration sequences:

(6.7)
$\textstyle{{\mathbf{K}}({\mathbb{Z}_{(2)}},{k-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathbf{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P{\mathbf{K}}({\mathbb{Z}_{(2)}},{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{S}}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{k}}$$\textstyle{{\mathbf{K}}({\mathbb{Z}_{(2)}},{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{{\mathbf{K}}({\mathbb{Z}_{(2)}},{k})}$$\textstyle{\mathcal{F}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\mathcal{F}^{2}}$

where the left hand side is the path-loop fibration. This induces a map
$u^{\ast}$ of the corresponding Serre spectral sequences, with
$E_{r}(u^{\ast}):E_{r}(\mathcal{F}^{2})\to E_{r}(\mathcal{F}^{1})$.

$0$$1$$\cdot$$\cdot$$\cdot$$\cdot$$k$$k+1$$k+2$$\cdot$$\cdot$$\cdot$$2k$$H^{\ast}({{\mathbf{K}}({\mathbb{Z}},{k})};{\mathbb{F}_{2}})$$0$$1$$\cdot$$\cdot$$\cdot$$k-1$$k$$k+1$$\cdot$$\cdot$$2k-2$$2k-1$$H^{\ast}({{\mathbf{K}}({\mathbb{Z}},{k-1})};{\mathbb{F}_{2}})$$1$$\iota_{k}$$\operatorname{Sq}^{2}\iota_{k}$$\operatorname{Sq}^{k-1}\iota_{k}$$\iota_{k}^{2}$$\iota_{k-1}$$\operatorname{Sq}^{2}\iota_{k-1}$$\iota_{k-1}^{2}$$\iota_{k-1}\iota_{k}$$d_{k}$$d_{k+2}$$d_{k}$$d_{2k-1}$
Figure 6.8. Spectral sequence for the fibration sequence $\mathcal{F}^{1}$.

Observe that the first differential for $\mathcal{F}^{1}$ is $d_{k}$,
determined by $d_{k}(\iota_{k-1})=\iota_{k}$. (See Figure 6.8).

The key step in the argument is the calculation of
${\mathcal{K}}_{r}:=\operatorname{Ker}(E_{r}(u^{\ast}))$. For $j\leq k$,
observe that ${\mathcal{K}}_{r}=\gamma_{2k-1}\cdot E_{r}(\mathcal{F}^{2})$.
For $j=k+1$, multiples of $\iota_{k}$ are zero in $\mathcal{F}^{1}$, so that
${\mathcal{K}}_{k+1}=\gamma_{2k-1}\cdot
E_{k+1}(\mathcal{F}^{2})+\iota_{k}\cdot E_{k+1}(\mathcal{F}^{2})$. Using our
comparison map $u^{\ast}:E_{r}(\mathcal{F}^{2})\to E_{r}(\mathcal{F}^{1})$ and
the fact that $\operatorname{Sq}^{j}(\iota_{k-1})$ transgresses to
$\operatorname{Sq}^{j}(\iota_{k})$ in $\mathcal{F}^{1}$, we deduce that either
the class $\operatorname{Sq}^{j}(\iota_{k-1})$ transgresses to
$\operatorname{Sq}^{j}(\iota_{k})$ in $\mathcal{F}^{2}$, or else some
differential carries $\operatorname{Sq}^{j}(\iota_{k-1})$ to a class in
${\mathcal{K}}_{r}$ for some $r$. Now note that for $2\leq j\leq k-1$,
$k+1\leq|\operatorname{Sq}^{j}(\iota_{k-1})|\leq 2k-2$, and that in these
degrees ${\mathcal{K}}_{r}$ is zero. This forces
$\operatorname{Sq}^{j}(\iota_{k-1})$ to be transgressive.

The class $\gamma_{2k-1}$ lies in ${\mathcal{K}}_{r}$ for all $r$. As the
spectral sequence converges to the cohomology of ${\mathbf{S}}^{k}$, it must
support a differential. Therefore, a differential on it must also lie in
${\mathcal{K}}_{r}$, which leaves the only possibility as
$d_{2k}(\gamma_{2k-1})=\iota_{k}^{2}$, for degree reasons.

$0$$1$$\cdot$$\cdot$$\cdot$$\cdot$$k$$k+1$$k+2$$\cdot$$\cdot$$\cdot$$2k$$H^{\ast}({{\mathbf{K}}({\mathbb{Z}},{k})};{\mathbb{F}_{2}})$$0$$1$$\cdot$$\cdot$$\cdot$$k-1$$k$$k+1$$\cdot$$\cdot$$2k-2$$2k-1$$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{2}})$$1$$\iota_{k}$$\operatorname{Sq}^{2}\iota_{k}$$\operatorname{Sq}^{k-1}\iota_{k}$$\iota_{k}^{2}$$\operatorname{Sq}^{2}\iota_{k-1}$$\iota_{k-1}^{2}$$\gamma_{2k-1}$$d_{k+2}$$d_{2k}$$d_{2k-1}$
Figure 6.9. Spectral sequence for the fibration sequence $\mathcal{F}^{2}$

Case II: odd primes.

For an odd prime $p$, we have:

(6.10)
${}H^{\ast}({{\mathbf{K}}({\mathbb{Z}_{(p)}},{k})};{\mathbb{F}_{p}})\cong
F^{\operatorname{gr}}_{\mathbb{F}_{p}}[\mathcal{P}^{I}(\iota_{k})~{}|\ \mbox{
admissible},\epsilon_{s+1}(I)=0,\operatorname{ex}(I)<k-1]$

(see [Ko, Proposition 3.5.8]), where now
$I:=(\epsilon_{0},i_{0},\cdots,\epsilon_{s},i_{s},\epsilon_{s+1})$ for
$\epsilon_{i}\in\\{0,1\\}$, and
$\mathcal{P}^{I}=\beta^{\epsilon_{0}}\mathcal{P}^{i_{0}}\cdots\mathcal{P}^{i_{s}}\beta^{\epsilon_{s+1}}$.
Here $F^{\operatorname{gr}}_{\mathbb{F}_{p}}$ denotes the free graded
commutative algebra – that is, exterior on the odd degree classes and
polynomial on the even degree classes.

We now distinguish between two cases, depending on the parity of $k$: when $k$
is odd, we find

(6.11)
${}E^{0}({p})=F^{\operatorname{gr}}_{\mathbb{F}_{p}}[\iota_{k-1}^{p},\,\mathcal{P}^{I}(\iota_{k-1})~{}|\
I\neq 0\mbox{ admissible},\epsilon_{s+1}(I)=0,\operatorname{ex}(I)<k]$

Note that $d_{k}(\iota_{k-1}^{p})=0$ in the exact sequence (6.4), while
$d_{k}(\iota_{k-1}^{j})=j\iota_{k-1}^{j-1}$ for $j\leq p-1$. As in the $p=2$
case, we write down the corresponding generators in
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ using the same notation. It
follows that $H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ is generated as an
$E^{0}({p})$-module by $1$ and $\gamma_{(k-1)p+1}$, which is the image of
$\iota_{k-1}^{p-1}$ in $H^{(p-1)(k-1)+k}({{\mathbf{X}}};{\mathbb{F}_{p}})$ in
the exact sequence (6.4). .

Therefore, the indecomposable classes in
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ are $\gamma_{(k-1)p+1}$, and
$\mathcal{P}^{j}(\iota_{k-1})$ for $j\leq\frac{k-1}{2}$ (noting that
$\iota_{k-1}^{p}=\mathcal{P}^{\frac{k-1}{2}}(\iota_{k-1})$). We verify that
all these classes are transgressive in the $\mathbb{F}_{p}$-cohomology Serre
spectral sequence for the fibration
${\mathbf{X}}\to{\mathbf{S}}^{k}\to{\mathbf{K}}({\mathbb{Z}},{k})$.

This follows from a very similar argument to that for the case $p=2$. Note,
however, that for reasons of degree, the classes
$\mathcal{P}^{i}(\iota_{k-1})$ must be transgressive, while the class
$\gamma_{(k-1)p+1}$ must transgress to the class
$\beta\mathcal{P}^{\frac{k-1}{2}}(\iota_{k})$ (which is the differential on
$\iota_{k-1}^{p-1}\cdot\iota_{k}$ in the spectral sequence $\mathcal{F}^{1}$
by the Kudo transgression theorem (see [Ko, Theorem 3.5.3]).

For $k$ even, we have

(6.12)
${}E^{0}({p})=F^{\operatorname{gr}}_{\mathbb{F}_{p}}[\mathcal{P}^{I}(\iota_{k-1})~{}|\
I\neq 0\mbox{ admissible},\epsilon_{s+1}(I)=0,\operatorname{ex}(I)<k-1]~{},$

as $\iota_{k-1}^{2}=0$. It follows that
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ is generated as an
$E^{0}({p})$-module by $1$ and $\gamma_{2k-1}$, which is the image of
$\iota_{k-1}$ in $H^{2k-1}({{\mathbf{X}}};{\mathbb{F}_{p}})$ in the exact
sequence (6.4). Therefore, the indecomposable classes in
$H^{\ast}({{\mathbf{X}}};{\mathbb{F}_{p}})$ are $\gamma_{2k-1}$, and
$\mathcal{P}^{j}(\iota_{k-1})$ for $j<\frac{k-1}{2}$. As above by calculating
${\mathcal{K}}_{r}$ we see that the only class that $d_{r}(\gamma_{2k-1})$ can
be is $\iota_{k}^{2}$, so it is transgressive.

Finally for the classes $\mathcal{P}^{j}(\iota_{k-1})$, observe that if these
classes are _not_ transgressive, the differential on these classes must lie in
${\mathcal{K}}_{r}$ for some $r$ (as they are transgressive in the spectral
sequence for $\mathcal{F}^{1}$). Observe that
${\mathcal{K}}_{r}=\\{\gamma_{2k-1}E_{r}(\mathcal{F}^{2})$ for $r\leq k$,
while ${\mathcal{K}}_{k+1}=\gamma_{2k-1}\cdot
E_{k+1}(\mathcal{F}^{2})+\iota_{k}\cdot E_{k+1}(\mathcal{F}^{2})$.

If $d_{r}(\mathcal{P}^{j}(\iota_{k-1})\neq 0$ for some $r<2k$, we must have
$r>k$ and $d_{r}(\mathcal{P}^{j}(\iota_{k-1}))=\gamma_{2k-1}q$. Note that $q$
must be in the $r$-th column, and hence is of the form
$\mathcal{P}^{L}(\iota_{k})q^{\prime}$ with $\operatorname{ex}(L)<k-1$, where
$q^{\prime}$ is in the $0$-th-column, since $r<2k$. Thus
$d_{r}(\mathcal{P}^{L}(\iota_{k-1})q^{\prime}\gamma_{2k-1})=\gamma_{2k-1}q$.

Observe that the classes $\mathcal{P}^{j}(\iota_{k-1})$ in the cohomology of
$X$ are defined only up to a multiple of $\gamma_{2k-1}$, so we may change the
representative in such a way that the differential $d_{r}$ vanishes on it.
Finally, note that ${\mathcal{K}}_{2k+1}$ is simply
$\mathbb{F}_{p}\\{\iota_{k}\\}$, so from this page on the differentials are
determined by those of the spectral sequence for $\mathcal{F}^{1}$. The result
follows. ∎

######  6.13 Corollary.

All homotopy classes of maps between wedges of spheres are generated under
group operations, Whitehead products, compositions and higher order simplicial
operations by the fundamental class $\iota_{k}\in\pi_{k}{\mathbf{S}}^{k}$, the
Hopf maps $\eta_{k}\in\pi_{k+1}{\mathbf{S}}^{k}$,
$\nu_{k}\in\pi_{k+3}{\mathbf{S}}^{k}$, and
$\sigma_{k}\in\pi_{k+7}{\mathbf{S}}^{k}$, and the maps
$\alpha_{1}(p)\in\pi_{k+2p-3}{\mathbf{S}}^{k}$, for odd primes $p$.

###### Proof.

This follows from the stable analysis in [Ada] and [Liu]. ∎

##  7\. Suspensions

So far we have only considered the way elements in the homotopy groups of a
single space ${\mathbf{Y}}$ can be decomposed. However, our methods also allow
us to describe the elements generated by suspending ${\mathbf{Y}}$.

######  7.1.

The suspension spectral sequence.  Given a connected pointed space
${\mathbf{Y}}$, as in§3.1 we denote by ${\mathbf{Y}}\otimes S^{n}$ the
$n$-skeletal simplicial space with a single non-degenerate copy of
${\mathbf{Y}}$ in simplicial dimension $n$, so that its geometric realization
is weakly equivalent to $\Sigma^{n}{\mathbf{Y}}$.

In particular, the homotopy spectral sequence for
${\mathbf{W}}_{\bullet}:={\mathbf{Y}}\otimes S^{1}$ will be called the
_suspension spectral sequence_ for ${\mathbf{Y}}$. This simplicial space may
be described combinatorially in terms of the degeneracies of ${\mathbf{Y}}$:

(7.2) $\textstyle{\dotsc s_{1}s_{0}{\mathbf{Y}}\vee s_{2}s_{0}{\mathbf{Y}}\vee
s_{2}s_{1}{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{3}}$$\textstyle{s_{0}{\mathbf{Y}}\vee
s_{1}{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\textstyle{{\mathbf{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\textstyle{\ast}$$\textstyle{\dotsc{\mathbf{W}}_{3}}$$\textstyle{{\mathbf{W}}_{2}}$$\textstyle{{\mathbf{W}}_{1}}$$\textstyle{{\mathbf{W}}_{0}}$

with the degeneracies as indicated, and all face maps determined by the
simplicial identities.

Since $\pi_{\ast}{\mathbf{W}}_{0}=0$, $\pi_{\ast}\Sigma{\mathbf{Y}}$ has no
elements in filtration $0$ in this spectral sequence, so all its elements are
formally exhibited as values of higher simplicial homotopy operations.

######  7.3 Remark.

The elements of $\pi_{\ast}\Sigma{\mathbf{Y}}$ in filtration $1$ are evidently
just the image of the suspension homomorphism
$E:\pi_{\ast}{\mathbf{Y}}\to\pi_{\ast+1}\Sigma{\mathbf{Y}}$. Note that it is
natural to think of suspensions as values of a secondary homotopy operation,
since any map $f:{\mathbf{S}}^{n}\to{\mathbf{Y}}$ is null homotopic in
$\Sigma{\mathbf{Y}}$ for two different reasons – namely, the two cones on the
“equatorial” copy of ${\mathbf{Y}}$.

######  7.4 Example.

In the spectral sequence for ${\mathbf{W}}_{\bullet}:={\mathbf{S}}^{2}\otimes
S^{1}$, the $1$-cycles $\iota_{2}\in\pi_{2}{\mathbf{W}}_{1}$,
$\eta_{2}\in\pi_{3}{\mathbf{W}}_{1}$, and
$\eta_{2}\eta_{3}\in\pi_{4}{\mathbf{W}}_{1}$ represent the suspension classes
$\iota_{3}\in\pi_{3}{\mathbf{S}}^{3}$, $\eta_{3}\in\pi_{4}{\mathbf{S}}^{3}$,
and $\eta_{3}\eta_{4}\in\pi_{5}{\mathbf{S}}^{3}$, respectively.

The $2$-chain $[s_{0}\iota_{2},s_{1}\iota_{2}]\in\pi_{3}{\mathbf{W}}_{2}$ is
not a cycle, since $[\iota_{2},\iota_{2}]\neq 0$ in $\pi_{3}{\mathbf{W}}_{1}$.
However,
$[s_{0}\iota_{2},s_{1}\iota_{2}]\circ\eta_{3}\in\pi_{4}{\mathbf{W}}_{2}$ is a
cycle, since $[\iota_{2},\iota_{2}]\circ\eta_{3}=(2\eta_{2})\circ\eta_{3}=0$
in $\pi_{4}{\mathbf{W}}_{1}$. It has order $2$, and is not a boundary, so it
must represent $\pm\nu^{\prime}\in\pi_{6}{\mathbf{S}}^{3}$, with
$2\nu^{\prime}=\eta_{3}\eta_{4}\eta_{5}$ (a suspension, and thus in filtration
$1$).

######  7.5 Remark.

More generally, we could start with any simplicial resolution
${\mathbf{W}}_{\bullet}$ of ${\mathbf{Y}}$, and by letting
${\mathbf{V}}_{\bullet}:={\mathbf{W}}_{\bullet}\otimes S^{k}$ we obtain a
resolution of $\Sigma^{k}{\mathbf{Y}}$.

######  7.6.

Whitehead products.  To identify certain terms in higher filtration in the
suspension spectral sequence, we consider the special case where
${\mathbf{Y}}={\mathbf{S}}^{p}\vee{\mathbf{S}}^{q}$. Note that we have two
split inclusions $i_{p}:{\mathbf{S}}^{p}\hookrightarrow{\mathbf{Y}}$ and
$i_{q}:{\mathbf{S}}^{q}\hookrightarrow{\mathbf{Y}}$, so the spectral sequences
for each summand spit off from that for ${\mathbf{Y}}$; we are only interested
in the remaining cross-term part.

First assume that $p,q\geq 2$: by Hilton’s Theorem, we know that the lowest
dimensional non-trivial cross-term cycle in the simplicial abelian group
$\pi_{i}{\mathbf{W}}_{\bullet}$ in filtration (=simplicial dimension) $2$
necessarily has the form:

(7.7)
${}w_{2}~{}:=~{}[s_{0}i_{p},\,s_{1}i_{q}]~{}-~{}[s_{1}i_{p},\,s_{0}i_{q}]~{}\in~{}\pi_{p+q-1}{\mathbf{W}}_{2}$

(see (7.2)). For dimensional reasons this cycle cannot be hit by any
differentials, and the class it represents in
$\pi_{p+q+1}\|{\mathbf{W}}_{\bullet}\|=\pi_{p+q+1}({\mathbf{S}}^{p+1}\vee{\mathbf{S}}^{q+1})$
is therefore the generator $[i_{p+1},\,i_{q+1}]$ (up to sign) – which thus can
be expressed as the value of a third order homotopy operation.

Likewise, the lowest dimensional cycles in filtration $3$ will have the form

(7.8)
${}\begin{split}&~{}[s_{1}s_{0}i_{p},[s_{2}s_{0}i_{p},\,s_{2}s_{1}i_{q}]]-[s_{1}s_{0}i_{p},[s_{2}s_{1}i_{p},\,s_{2}s_{0}i_{q}]]+[s_{2}s_{0}i_{p},[s_{2}s_{1}i_{p},\,s_{1}s_{0}i_{q}]]\\\
&-[s_{2}s_{0}i_{p},[s_{1}s_{0}i_{p},\,s_{2}s_{1}i_{q}]]+[s_{2}s_{1}i_{p},[s_{1}s_{0}i_{p},\,s_{2}s_{0}i_{q}]]-[s_{2}s_{1}i_{p},[s_{2}s_{0}i_{p},\,s_{1}s_{0}i_{q}]]\end{split}$

representing $[i_{p+1},[i_{p+1},i_{q+1}]]$ in
$\pi_{2p+q+1}\in{\mathbf{S}}^{p+1}\vee{\mathbf{S}}^{q+1}$ (and similarly for
the other Hall basis elements). Note that the $2$-cycle
$[s_{0}i_{p},[_{0}i_{p},\,s_{1}i_{q}]]-[s_{1}i_{p},[s_{0}i_{p},\,s_{1}i_{q}]]$
in the cross-term of $\pi_{2p+q-2}{\mathbf{W}}_{2}$

When $p=q=1$, $\pi_{\ast}{\mathbf{W}}_{\bullet}$ is concentrated in dimension
$1$, and the Whitehead products are commutators, so we cannot use a dimension
argument to identify the cycles. In this case it is convenient to work with a
simplicial group model for ${\mathbf{S}}^{1}\vee{\mathbf{S}}^{1}$, namely,
$F(S^{0}\vee S^{0})$ – a constant simplicial group. Thus
${\mathbf{W}}_{\bullet}$ is a bisimplicial group which is equivalent to its
own diagonal, which is thus a simplicial group model $F(S^{1}\vee S^{1})$ for
$S^{2}\vee S^{2}$, and the spectral sequence collapses at the $E^{2}$-term.

In this case we see that the $1$-dimensional analogue of (7.7) is the product
of two commutators:

(7.9) ${}\omega~{}:=~{}\left(s_{0}\alpha s_{1}\beta
s_{0}\alpha^{-1}s_{1}\beta^{-1}\right)\cdot\left(s_{1}\alpha s_{0}\beta
s_{1}\alpha^{-1}s_{0}\beta^{-1}\right)~{},$

where $\alpha\in\pi_{1}{\mathbf{S}}^{1}_{(1)}$ and
$\beta\in\pi_{1}{\mathbf{S}}^{1}_{(2)}$ are the fundamental classes of the two
wedge summands in
${\mathbf{Y}}={\mathbf{S}}^{1}_{(1)}\vee{\mathbf{S}}^{1}_{(2)}$. This is known
to represent the Whitehead product in $S^{2}\vee S^{2}$ (see [Cu, §11.11]). A
similar statement holds for the case $p=1<q$.

Since ${\mathbf{S}}^{k}\vee{\mathbf{S}}^{k}$ is the universal space for
Whitehead products of $k$-dimensional classes, we see that whenever
${\mathbf{Y}}$ is $(k-1)$-connected and $\alpha,\beta\in\pi_{k}{\mathbf{Y}}$,
the Whitehead product $[E\alpha,E\beta]$ in $\pi_{2k+1}\Sigma{\mathbf{Y}}$ is
represented in the spectral sequence by

(7.10)
${}\gamma~{}:=~{}[s_{0}\alpha,\,s_{1}\beta]-[s_{1}\alpha,\,s_{0}\beta]~{},$

modulo suspensions.

We can use the suspension spectral sequence to provide an elementary proof of
the following well-known facts (cf. [HW, (3.5)]):

######  7.11 Lemma.

Let ${\mathbf{Y}}$ be $(k-1)$-connected ($k\geq 1$), and
$\alpha,\beta\in\pi_{k}{\mathbf{Y}}$:

  1. (a)

If $[\alpha,\beta]=0$ in $\pi_{2k-1}{\mathbf{Y}}$, then $[E\alpha,E\beta]$ is
divisible by $2$ in $\pi_{2k+1}\Sigma{\mathbf{Y}}$, modulo elements in the
image of the suspension
$E:\pi_{\ast}{\mathbf{Y}}\to\pi_{\ast+1}\Sigma{\mathbf{Y}}$.

  2. (b)

If $\alpha=\beta$ and $k$ is even, then $[E\alpha,E\beta]$ is a suspension.

###### Proof.

By the universal example (7.7), the class
$\gamma\in\pi_{2}\pi_{2k-1}{\mathbf{W}}_{\bullet}$ of (7.10) is a cycle
representing $[E\alpha,E\beta]$ (modulo lower filtration, which is
$\operatorname{Im}(E)$).

  1. (a)

If $[\alpha,\beta]=0$, then $\delta:=[s_{0}\alpha,\,s_{1}\beta]$ is another
cycle which survives to $\pi_{\ast}\Sigma{\mathbf{Y}}$ whose double is
$\gamma$.

  2. (b)

When $k$ is even, $\gamma=0$, so $[E\alpha,E\beta]$ vanishes modulo
suspensions.

∎

######  7.12 Examples.

  1. (a)

When ${\mathbf{Y}}={\mathbf{S}}^{1}$ and $\alpha=\beta=\iota_{1}$, then
$[\alpha,\beta]=0$ and the Proposition yields the Hopf map
$\eta_{2}\in\pi_{3}{\mathbf{S}}^{2}$, with $2\eta_{2}=[\iota_{2},\iota_{2}]$
(since there are no elements in filtration $0$ in this dimension).

  2. (b)

When ${\mathbf{Y}}={\mathbf{S}}^{3}$ and $\alpha=\beta=\iota_{3}$, again
$[\alpha,\beta]=0$ and the Proposition yields
$\nu_{4}\in\pi_{7}{\mathbf{S}}^{4}$, and in fact

(7.13) ${}2\nu_{4}=[\iota_{4},\iota_{4}]+E\nu^{\prime}$

by [T2, (5.8)], although we cannot determine the extension using the spectral
sequence alone.

  3. (c)

When ${\mathbf{Y}}={\mathbf{S}}^{4}$ and $\alpha=\beta=\iota_{4}$, we have
$[\iota_{5},\iota_{5}]=\nu_{5}\circ\eta_{8}=E(\nu_{4}\circ\eta_{7})$, where
the first equality is [T2, (5.10)].

  4. (d)

When ${\mathbf{Y}}={\mathbf{S}}^{6}$ and $\alpha=\beta=\iota_{6}$, we have
$[\iota_{7},\iota_{7}]=0$.

  5. (e)

When ${\mathbf{Y}}={\mathbf{S}}^{7}$ and $\alpha=\beta=\iota_{7}$, we have
$[\alpha,\beta]=0$ and the Proposition yields
$\sigma_{8}\in\pi_{15}{\mathbf{S}}^{8}$, with

(7.14) ${}2\sigma_{8}=[\iota_{8},\iota_{8}]+E\sigma^{\prime}$

by [T2, (5.16)].

  6. (f)

When ${\mathbf{Y}}={\mathbf{S}}^{8}$ and $\alpha=\beta=\iota_{8}$, we have

$[\iota_{9},\iota_{9}]=\sigma_{9}\circ\eta_{16}+\bar{\nu}_{9}+\epsilon_{9}=E(\sigma_{8}\circ\eta_{15}+\bar{\nu}_{8}+\epsilon_{8})~{},$

where the first equality is [T2, (7.1)].

######  7.15 Remark.

As noted above, in the spectral sequence for
${\mathbf{W}}_{\bullet}:={\mathbf{S}}^{4}\otimes S^{1}$, the $1$-chain
$\delta_{4}:=[s_{0}\iota_{4},\,s_{1}\iota_{4}]\in\pi_{7}{\mathbf{W}}_{2}$ is
not a cycle, since
$d_{0}\delta_{4}=[\iota_{4},\iota_{4}]=2\nu_{4}-E\nu^{\prime}$ by (7.13).
Since $E\nu^{\prime}\circ\eta_{7}\neq 0$ and
$E\nu^{\prime}\circ\eta_{7}\circ\eta_{8}\neq 0$, also
$\delta_{4}\circ\eta_{7}$ and $\delta_{4}\circ\eta_{7}\circ\eta_{8}$ are not
cycles. Similarly,

$(2\nu_{4}-E\nu^{\prime})\circ(2\nu_{7})=4\nu_{4}\circ\nu_{7}-(2E\nu^{\prime})\circ\nu_{7}=4\nu_{4}\circ\nu_{7}-\eta^{3}\circ\nu_{7}=4\nu_{4}\circ\nu_{7}\neq
0~{},$

since $2\nu^{\prime}=\eta^{3}$ and already $\eta_{5}\circ\nu_{6}=0$, while
$\nu_{4}\circ\nu_{7}$ has order $8$ in $\pi_{10}{\mathbf{S}}^{4}$. This
implies that $2\delta_{4}\circ\nu_{4}$ and thus $\delta_{4}\circ\nu_{4}$ are
not cycles, but $\delta_{4}\circ\eta^{3}$ is a cycle in
$\pi_{10}{\mathbf{W}}_{2}$. Since it cannot bound anything, it must therefore
represent the generator $\sigma^{\prime\prime\prime}$ of
$\pi_{12}{\mathbf{S}}^{5}_{(2)}\cong\mathbb{Z}/2$.

Similarly, in the spectral sequence for
${\mathbf{W}}_{\bullet}:={\mathbf{S}}^{6}\otimes S^{1}$,
$\delta_{6}:=[s_{0}\iota_{6},\,s_{1}\iota_{6}]\in\pi_{11}{\mathbf{W}}_{2}$ is
not a cycle, since
$d_{0}\delta_{6}~{}=~{}[\iota_{6},\iota_{6}]\in\pi_{13}{\mathbf{S}}^{6}$ has
infinite order. However, $\delta_{6}\circ\eta_{11}$ is the only possible
representative for $\sigma^{\prime}\in\pi_{14}{\mathbf{S}}^{7}$, of order $8$,
since it is not a suspension (though
$2\sigma^{\prime}=E\sigma^{\prime\prime}$). Incidentally, this shows that
$[\iota_{6},\iota_{6}]\circ\eta_{11}=0$ in $\pi_{12}{\mathbf{S}}^{6}$.

We conclude this section by showing

######  7.16 Theorem.

All elements in the homotopy groups of a finite wedge of simply-connected
spheres are generated under simplicial homotopy operations by the group
operation and Whitehead products.

###### Proof.

Since any suspension is the value of a secondary homotopy operation (see
§7.3), by Corollary 6.13, it suffices to show this for
$\eta_{2}\in\pi_{3}{\mathbf{S}}^{2}$, $\nu_{4}\in\pi_{7}{\mathbf{S}}^{4}$, and
$\sigma_{k}\in\pi_{15}{\mathbf{S}}^{8}$, and the maps
$\alpha_{1}(p)\in\pi_{2p}{\mathbf{S}}^{3}$ for odd primes $p$.

Indeed, from (7.9) we see that by Lemma 7.11(a):

  1. (a)

$\eta_{2}$ is represented by either half of $\omega$ – e.g.,
$h:=s_{0}\iota_{1}\cdot s_{1}\iota_{1}\cdot
s_{0}\iota_{1}\cdot^{-1}s_{1}\iota_{1}^{-1}$ in $\pi_{1}{\mathbf{W}}_{2}$ for
${\mathbf{W}}_{\bullet}={\mathbf{S}}^{1}\otimes S^{1}$;

  2. (b)

$\nu_{4}$ is represented by
$[s_{0}i_{3},\,s_{1}i_{3}]\in\pi_{5}{\mathbf{W}}_{2}$ for
${\mathbf{W}}_{\bullet}={\mathbf{S}}^{3}\otimes S^{1}$;

  3. (c)

$\sigma_{8}$ is represented by
$[s_{0}i_{7},\,s_{1}i_{7}]\in\pi_{13}{\mathbf{W}}_{2}$ for
${\mathbf{W}}_{\bullet}={\mathbf{S}}^{7}\otimes S^{1}$.

Now for any odd prime $p$ we have

(7.17)
${}\pi_{i}{\mathbf{S}}^{2}_{(p)}~{}:=~{}\begin{cases}\mathbb{Z}_{(p)}\langle{\iota_{2}}\rangle&\text{for}\
\ i=2\\\ \mathbb{Z}_{(p)}\langle{[\iota_{2},\iota_{2}]}\rangle&\text{for}\ \
i=3\\\ 0&\text{for}\ 4\leq i<2p\end{cases}$

with $[[\iota_{2},\iota_{2}],\iota_{2}]=0$. Since
$\alpha_{1}(p)\in\pi_{2p}{\mathbf{S}}^{3}$ is not a suspension, it must appear
in filtration $k\geq 2$ and in dimension $i=2p-k$ in the spectral sequence for
${\mathbf{W}}_{\bullet}={\mathbf{S}}^{2}_{(p)}\otimes S^{1}$, and all classes
in $\pi_{i}{\mathbf{W}}_{k}$ are iterated Whitehead products, by (7.17) and
Hilton’s Theorem. ∎

######  7.18 Example.

A Hall basis for the lowest dimension $2$-cycles for
${\mathbf{W}}_{\bullet}={\mathbf{S}}^{2}_{(3)}\otimes S^{1}$ is given by
$\varepsilon_{1}:=[[s_{0}\iota_{2},s_{1}\iota_{2}],\,s_{0}\iota_{2}]$ and
$\varepsilon_{2}:=[[s_{0}\iota_{2},s_{1}\iota_{2}],\,s_{1}\iota_{2}]$ in
$\pi_{4}{\mathbf{W}}_{2}$; a Hall basis for the non-degenerate classes in
$\pi_{4}{\mathbf{W}}_{3}$ is given by
$\theta_{1}:=[[s_{2}s_{0}\iota_{2},s_{1}s_{0}\iota_{2}],\,s_{2}s_{1}\iota_{2}]$
and
$\theta_{2}:=[[s_{2}s_{1}\iota_{2},s_{1}s_{0}\iota_{2}],\,s_{2}s_{0}\iota_{2}]$.

We have $d_{0}\theta_{i}=0$ and $d_{3}\theta_{i}=0$ for $i=1,2$. Since
$[s_{1}\iota_{2},s_{1}\iota_{2}]=2(s_{1}\iota_{2}\circ\eta_{2})$ and

(7.19)
${}[s_{0}\iota_{2},s_{1}\iota_{2}\eta_{2}]~{}=~{}[s_{0}\iota_{2},s_{1}\iota_{2}]\circ\eta_{3}~{}-~{}[[s_{0}\iota_{2},s_{1}\iota_{2}],\,s_{1}\iota_{2}]~{}\equiv~{}-\varepsilon_{2}\pmod{3}$

by [BH, Theorem 3.24], we see that $d_{1}\theta_{1}=\varepsilon_{2}$, and
$d_{12}\theta_{1}=-2\varepsilon_{2}$, so $[\varepsilon_{2}]$ has order $3$ in
$E^{2}_{2,4}=\pi_{2}\pi_{4}{\mathbf{W}}_{\bullet}$. Similarly,
$d_{1}\theta_{2}=\varepsilon_{2}$, and $d_{2}\theta_{2}=\varepsilon_{1}$, so
$[\varepsilon_{1}]=[\varepsilon_{2}]$. Thus the generator $\alpha_{1}(3)$ for
$\pi_{6}{\mathbf{S}}^{3}_{(3)}\cong\mathbb{Z}/3$ is represented by the either
of these two classes.

##  8\. Complex projective spaces

For each $n\geq 1$, the complex projective space $\mathbb{C}\mathbf{P}^{n}$
fits into a fibration sequence

(8.1)
${}{\mathbf{S}}^{1}~{}\hookrightarrow~{}{\mathbf{S}}^{2n+1}~{}\xrightarrow{g_{n}}~{}\mathbb{C}\mathbf{P}^{n}~{},$

with $g_{n}$ a Hopf map, as well as a homotopy cofibration sequence

(8.2)
${}{\mathbf{S}}^{2n+1}~{}\xrightarrow{g_{n}}~{}\mathbb{C}\mathbf{P}^{n}~{}\xrightarrow{j_{n}}~{}\mathbb{C}\mathbf{P}^{n+1}~{},$

starting with $\mathbb{C}\mathbf{P}^{1}={\mathbf{S}}^{2}$ and
$g_{2}=\eta_{2}$. Thus $g_{n}$ determines an isomorphism
$\pi_{i}{\mathbf{S}}^{2n+1}\cong\pi_{i}\mathbb{C}\mathbf{P}^{n}$ for $i\neq
2$, with $\pi_{2}\mathbb{C}\mathbf{P}^{n}\cong\mathbb{Z}$ generated by
$\iota_{2}=j_{n-1}\circ\dotsc\circ j_{1}$.

It is nevertheless illuminating to analyze how all elements in
$\pi_{\ast}\mathbb{C}\mathbf{P}^{n}$ are generated by the unique
indecomposable $\iota_{2}$ using (higher) homotopy operations, as follows:

######  8.3.

The four dimensional complex projective space.  Our approach is necessarily
inductive, starting with $\mathbb{C}\mathbf{P}^{2}$. From (8.2) we see that a
(minimal) simplicial resolution ${\mathbf{W}}_{\bullet}$ for
$\mathbb{C}\mathbf{P}^{2}$ as in Section 4 must start with
${\mathbf{W}}_{0}={\mathbf{S}}^{2}$ and ${\mathbf{W}}_{1}={\mathbf{S}}^{3}\vee
s_{0}{\mathbf{S}}^{2}$, with $d_{0}$ given by the Hopf map
$\eta_{2}=g_{1}:{\mathbf{S}}^{3}\to\mathbb{C}\mathbf{P}^{1}={\mathbf{S}}^{2}$
on the summand ${\mathbf{S}}^{3}$.

As in §7.6, the lowest dimensional $1$-cycle in
$\pi_{i}{\mathbf{W}}_{\bullet}$ (for $i=4$) is given by the Whitehead product
$\gamma_{2}=[\iota_{3},s_{0}\iota_{2}]$. We see that indeed

(8.4)
${}d_{0}(\gamma_{2})~{}=~{}[\eta_{2},\iota_{2}]~{}=~{}[\iota_{2},\iota_{2}]\circ\eta_{3}~{}=~{}(2\eta_{2})\circ\eta_{3}~{}=~{}0~{},$

and since this is a permanent cycle which cannot be hit by a differential (for
dimension reasons), it must represent
$g_{2}:{\mathbf{S}}^{5}\to\mathbb{C}\mathbf{P}^{2}$.

Note that (8.4) exhibits $g_{2}$ as a canonical value of a secondary operation
in the sense of Proposition 3.12 (though not as a Toda bracket in the usual
sense). Moreover, this also defines a map $\widehat{g}$ from the simplicial
space ${\mathbf{V}}_{\bullet}:={\mathbf{S}}^{4}\otimes S^{1}$ to the
simplicial $1$-skeleton of ${\mathbf{W}}_{\bullet}$ described above (and thus
to ${\mathbf{W}}_{\bullet}$ itself), realized by $g_{2}$:

(8.5) |  |  |
---|---|---|---
$\textstyle{\dotsc}$$\textstyle{{\mathbf{V}}_{2}}$$\textstyle{{\mathbf{V}}_{1}}$$\textstyle{{\mathbf{V}}_{0}}$$\textstyle{\dotsc}$$\textstyle{s_{0}{\mathbf{S}}^{4}\vee
s_{1}{\mathbf{S}}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{\widehat{g}_{2}=}$$\scriptstyle{[s_{0}\iota_{3},s_{1}s_{0}\iota_{2}]\bot[s_{1}\iota_{3},s_{1}s_{0}\iota_{2}]}$$\textstyle{{\mathbf{S}}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}_{1}=}$$\scriptstyle{[\iota_{3},s_{0}\iota_{2}]}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}_{0}}$$\textstyle{\dotsc}$$\textstyle{s_{0}{\mathbf{S}}^{3}\vee
s_{1}{\mathbf{S}}^{3}\vee
s_{1}s_{0}{\mathbf{S}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\textstyle{{\mathbf{S}}^{3}\vee
s_{1}s_{0}{\mathbf{S}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\textstyle{{\mathbf{S}}^{2}}$$\textstyle{\dotsc}$$\textstyle{{\mathbf{W}}_{2}}$$\textstyle{{\mathbf{W}}_{1}}$$\textstyle{{\mathbf{W}}_{0}}$

From the description in §7.1 we see that for any map
$f:{\mathbf{S}}^{N}\to{\mathbf{S}}^{4}$, precomposition with $f\otimes
S^{1}:{\mathbf{S}}^{N}\otimes S^{1}\to{\mathbf{V}}_{\bullet}$ induces
precomposition with $\Sigma f$ in $\pi_{5}\mathbb{C}\mathbf{P}^{2}$. Thus
$[\gamma_{2}\circ f]\in\pi_{N}{\mathbf{W}}_{1}$ represents $[g_{2}\circ\Sigma
f]\in\pi_{N+1}\mathbb{C}\mathbf{P}^{2}$.

######  8.6 Remark.

Any class
$\alpha\in\pi_{k}{\mathbf{S}}^{5}\cong\pi_{k}\mathbb{C}\mathbf{P}^{2}$ ($k>5$)
which is not a suspension is represented by an element in higher filtration in
the spectral sequence for ${\mathbf{V}}_{\bullet}$. The map of simplicial
spaces $\widehat{g}:{\mathbf{V}}_{\bullet}\to{\mathbf{W}}_{\bullet}$ then
induces a map of spectral sequences $\widehat{g}_{\ast}$, yielding a
representative for the composite $g_{2}\circ\alpha$ (which in this case is
always non-trivial since $(g_{2})_{\\#}$ is an isomorphism).

######  8.7 Example.

The first such class is
$\sigma^{\prime\prime\prime}\in\pi_{12}{\mathbf{S}}^{5}$, of order $2$. By
§7.15, this is represented by the element
$[s_{0}\iota_{4},\,s_{1}\iota_{4}]\circ\eta^{3}$ in
$E^{1}_{2,10}=\pi_{10}{\mathbf{V}}_{2}$, which maps under
$(\widehat{g}_{2})_{\ast}$ to the iterated Whitehead product

$\begin{split}&[[s_{0}\iota_{3},s_{1}s_{0}\iota_{2}],\
[s_{1}\iota_{3},s_{1}s_{0}\iota_{2}]]~{}\circ\eta^{3}\\\
\quad&~{}~{}~{}~{}~{}~{}~{}~{}=~{}[[[s_{1}\iota_{3},s_{1}s_{0}\iota_{2}],\
s_{1}s_{0}\iota_{2}],\
s_{0}\iota_{3}]~{}\circ\eta^{3}~{}+~{}[[[s_{1}\iota_{3},s_{1}s_{0}\iota_{2}],\
s_{0}\iota_{3}],\ s_{1}s_{0}\iota_{2}]~{}\circ\eta^{3}\end{split}$

in $\pi_{10}{\mathbf{W}}_{2}$.

######  8.8.

Higher projective spaces.  A simplicial resolution for
$\mathbb{C}\mathbf{P}^{3}$ will start with ${\mathbf{W}}_{0}={\mathbf{S}}^{2}$
and ${\mathbf{W}}_{1}={\mathbf{S}}^{3}\vee s_{0}{\mathbf{S}}^{2}$, as for
$\mathbb{C}\mathbf{P}^{2}$, but now we set
${\mathbf{W}}_{2}:={\mathbf{S}}^{4}\vee s_{0}{\mathbf{S}}^{3}\vee
s_{1}{\mathbf{S}}^{3}\vee s_{1}s_{0}{\mathbf{S}}^{2}$, with $d_{0}$ given by
the representative for $g_{2}:{\mathbf{S}}^{5}\to\mathbb{C}\mathbf{P}^{2}$ –
that is, by $\gamma_{2}=[\iota_{3},s_{0}\iota_{2}]$ – on the summand
${\mathbf{S}}^{4}$.

As before, the lowest dimensional $2$-cycle in $\pi_{i}{\mathbf{W}}_{\bullet}$
is given by

(8.9)
${}\gamma_{3}~{}:=~{}[\iota_{4},s_{1}s_{0}\iota_{2}]~{}-~{}[s_{0}\iota_{3},s_{1}\iota_{3}]~{}\in~{}\pi_{5}{\mathbf{W}}_{2}~{}.$

By the Jacobi identity

$[[\iota_{3},s_{0}\iota_{2}],s_{0}\iota_{2}]~{}+~{}[[s_{0}\iota_{2},s_{0}\iota_{2}],\iota_{3}]~{}+~{}[[s_{0}\iota_{2},\iota_{3}],s_{0}\iota_{2}]~{}=~{}0~{},$

so

$2[[\iota_{3},s_{0}\iota_{2}],s_{0}\iota_{2}]~{}=~{}-[[s_{0}\iota_{2},s_{0}\iota_{2}],\iota_{3}]~{}=~{}-2[s_{0}\iota_{2}\circ\eta_{2},\iota_{3}]~{}=~{}2[\iota_{3},s_{0}\iota_{2}\circ\eta_{2}]~{}.$

Therefore,

$d_{0}(\gamma_{3})~{}=~{}[[\iota_{3},s_{0}\iota_{2}],s_{0}\iota_{2}]-[\iota_{3},s_{0}\iota_{2}\circ\eta_{2}]~{}=~{}[\iota_{3},s_{0}\iota_{2}\circ\eta_{2}]-[\iota_{3},s_{0}\iota_{2}\circ\eta_{2}]=0~{},$

Since $d_{1}\gamma_{3}=[\iota_{3},\iota_{3}]=0$, and $d_{2}\gamma_{3}=0$, too,
this is a permanent cycle. It cannot be hit by a differential (for dimension
reasons), so it must represent
$g_{3}:{\mathbf{S}}^{7}\to\mathbb{C}\mathbf{P}^{3}$ as a third-order
simplicial operation.

It also defines a map $\widehat{g}_{3}$ from the simplicial space
${\mathbf{U}}_{\bullet}:={\mathbf{S}}^{5}\otimes S^{2}$ to
${\mathbf{W}}_{\bullet}$, realized by $g_{3}$. Once more, for any map
$f:{\mathbf{S}}^{N}\to{\mathbf{S}}^{5}$, precomposition with $f\otimes
S^{2}:{\mathbf{S}}^{N}\otimes S^{2}\to{\mathbf{U}}_{\bullet}$ shows that
$[\gamma_{3}\circ f]\in\pi_{N}{\mathbf{W}}_{2}$ represents
$[g_{3}\circ\Sigma^{2}f]\in\pi_{N+2}\mathbb{C}\mathbf{P}^{3}$.

Any class in $\pi_{k}{\mathbf{S}}^{7}\cong\pi_{k}\mathbb{C}\mathbf{P}^{3}$
($k>7$) which is not a double suspension is represented by an element in
higher filtration in the spectral sequence for ${\mathbf{V}}_{\bullet}$. The
first such class is $\sigma^{\prime}\in\pi_{14}{\mathbf{S}}^{7}$ – in fact,
$\sigma^{\prime}$ is not a suspension, and
$2\sigma^{\prime}=E\sigma^{\prime\prime}$ is not a double suspension, but
$4\sigma^{\prime}=E^{2}\sigma^{\prime\prime\prime}$.

######  8.10.

Some combinatorics.

In order to describe the general case, we require some combinatorial notions:

######  8.11 Definition.

Given a set $I$ of $k$ non-negative integers, we denote by
$\langle{I}\rangle=(i_{1}<\dotsc<i_{k})$ the elements of $I$ arranged in
ascending order, and let $s_{I}=s_{i_{k}}s_{i_{k-1}}\dotsc s_{i_{2}}s_{i_{1}}$
the corresponding iterated degeneracy map (so that $\forall x\exists y\
s_{I}x=s_{j}y$ if and only if $j\in I$).

We now define the notion of a _sign_ for two finite sets $I$ and $J$ of non-
negative integers:

  1. (i)

If $I$ and $J$ form a partition of $\\{1,2,\dotsc,n\\}$, we write
$\sigma_{(I,J)}\in S_{n}$ for the permutation obtained by concatenating
$\langle{I}\rangle$ with $\langle{J}\rangle$, and let
$\operatorname{sgn}(I,J):=\operatorname{sgn}(\sigma_{(I,J)})$.

  2. (ii)

If $I$ and $J$ are disjoint subsets of $\\{0,1,2,\dotsc,N\\}$, let
$\pi:I\sqcup J\to\\{1,2,\dotsc,n\\}$ be an order-preserving isomorphism, with
$\operatorname{sgn}(I,J):=\operatorname{sgn}(\pi[I],\pi[J])$.

  3. (iii)

If $I$ and $J$ are any two finite sets $I$ and $J$ of non-negative integers,
let $I^{\prime}:=I\setminus J$ and $J^{\prime}:=J\setminus I$, and set
$\operatorname{sgn}(I,J):=\operatorname{sgn}(I^{\prime},J^{\prime})$.

######  8.12 Example.

For $I=\\{2,4\\}$ and $J=\\{1,3,5\\}$, we have $\sigma_{(I,J)}=\binom{1\ 2\ 3\
4\ 5}{2\ 4\ 1\ 3\ 5}$, so $\operatorname{sgn}(I,J)=-1$.

For $I=\\{1,5\\}$ and $J=\\{0,2,7\\}$, the map $\pi$ is given by $\binom{0\ 1\
2\ 5\ 7}{1\ 2\ 3\ 4\ 5}$ and so $\pi[I]=\\{2,4\\}$ and $\pi[J]=\\{1,3,5\\}$,
and thus $\operatorname{sgn}(I,J)=-1$.

For $I=\\{1,3,5,6\\}$ and $J=\\{0,2,3,6,7\\}$, we have $I^{\prime}=\\{1,5\\}$
and $J^{\prime}=\\{0,2,7\\}$, so again $\operatorname{sgn}(I,J)=-1$.

######  8.13 Notation.

Given $n\geq 2$, for each $0\leq k<n-1$, let $\mathcal{I}_{k}^{n}$ denote the
collection of all sets of $k$ integers in $\\{0,\dotsc,n-2\\}$. For each
$I\in\mathcal{I}_{k}^{n}$, we denote by $\Phi^{n}(I)$ its complement
$J:=\\{0,\dotsc,n-2\\}\setminus I$ (so also $I=\Phi^{n}(J)$). When $2k=n-1$,
we include in $\mathcal{I}_{k}^{n}$ only sets with $0\in I$ (to avoid double
counting).

######  8.14 Remark.

If $(I^{\prime},J^{\prime})$ is obtained from $(I,J)$ by switching a pair of
indices between $I$ and $J$, while maintaining descending order in
$\langle{I^{\prime}}\rangle$ and $\langle{J^{\prime}}\rangle$, then
$\operatorname{sgn}(I^{\prime},J^{\prime})=-\operatorname{sgn}(I,J)$.

Note also that if $I\setminus J$ has cardinality $k$ and $J\setminus I$ has
cardinality $\ell$, then

(8.15)
${}\operatorname{sgn}(I,J)~{}=~{}(-1)^{k\cdot\ell}\operatorname{sgn}(J,I)$

Finally, if $I=I^{\prime}\sqcup I^{\prime\prime}$ is a disjoint union, then
for any $J$ we have

(8.16)
${}\operatorname{sgn}(I,J)~{}=~{}\operatorname{sgn}(I^{\prime},J)\cdot\operatorname{sgn}(I^{\prime\prime},J)~{}.$

We can now state our main result for this section:

######  8.17 Proposition.

For each $n\geq 1$, there is a sequential approximation
${\mathbf{W}}_{\bullet}$ for $\mathbb{C}\mathbf{P}^{n}$ with a single non-
degenerate $(k+2)$-sphere in ${\mathbf{W}}_{k}$ for each $0\leq k<n$. The Hopf
map $g_{n}:{\mathbf{S}}^{2n+1}\to\mathbb{C}\mathbf{P}^{n}$ is represented in
the homotopy spectral sequence for ${\mathbf{W}}_{\bullet}$ by the
$(n-1)$-cycle

(8.18) ${}\gamma_{n}~{}:=~{}\sum_{j=2}^{\lfloor\frac{n+3}{2}\rfloor}\
\sum_{I\in\mathcal{I}_{j-2}^{n}}\ (-1)^{n\cdot
j}\operatorname{sgn}(I,\Phi^{n}(I))\cdot[s_{I}\iota_{n-j+3},\
s_{\Phi^{n}(I)}\iota_{j}]$

in $\pi_{n+2}{\mathbf{W}}_{n-1}$. This also serves as the face map $d_{0}$ on
$\iota_{n+1}\in\pi_{n+1}{\mathbf{S}}^{n+1}$ in simplicial dimension $n-1$ in
the sequential approximation for $\mathbb{C}\mathbf{P}^{n+1}$.

###### Proof.

First, note that all iterated Whitehead products in $\pi_{i}{\mathbf{W}}_{j}$
are degenerate, if $i\leq j>k$. Thus it suffices to show that $\gamma_{n}$ is
a cycle, since it is the first non-trivial class that does not come from
$\mathbb{C}\mathbf{P}^{n-1}$.

Step 1: If $J:=\Phi(I)$ and $r\geq 1$, then
$d_{r}([s_{I}\iota_{n-j+3},s_{J}\iota_{j}]=0$ unless one of $\\{r,r+1\\}$ is
in $I$ and one is in $J$. In this case, if $(I^{\prime},J^{\prime})$ is
obtained form $(I,J)$ by interchanging $r$ and $r+1$, by Remark 8.14 we have
$\operatorname{sgn}(I^{\prime},J^{\prime})=-\operatorname{sgn}(I,J)$ but
$d_{r}[s_{I}\iota_{n-j+3},s_{J}\iota_{j}]=d_{r}[s_{I^{\prime}}\iota_{n-j+3},s_{J^{\prime}}\iota_{j}]$.
Thus we see that $d_{r}\gamma_{n}=0$ for $1\leq r\leq n$.

Step 2: By the Jacobi identity for any $n\geq 2$ and iterated degeneracy
$s_{I}$ we have
$[[\iota_{n},s_{I}\iota_{2}],s_{I}\iota_{2}]+[[s_{I}\iota_{2},s_{I}\iota_{2}],\iota_{n}]+[[s_{I}\iota_{2},\iota_{n}],s_{I}\iota_{2}]=0$,
so

$2[[\iota_{n},s_{I}\iota_{2}],s_{I}\iota_{2}]~{}=~{}-[s_{I}[\iota_{2},\iota_{2}],\iota_{n}]~{}=~{}-2[s_{I}\eta_{2}^{\\#}\iota_{2},\iota_{n}]~{}=~{}(-1)^{n+1}2[\iota_{n},s_{I}\eta_{2}^{\\#}\iota_{2}]$

and thus

(8.19)
${}[[\iota_{n},s_{I}\iota_{2}],s_{I}\iota_{2}]~{}=~{}(-1)^{n+1}[\iota_{n},s_{I}\eta_{2}^{\\#}\iota_{2}]$

(since all summands in Hilton’s Theorem in this dimension are of infinite
order). This implies that the only summand of $d_{0}\gamma_{n}$ which is not a
three-fold Whitehead product - namely,

(8.20)
${}d_{0}([s_{0}\iota_{n},s_{J}\iota_{3}])=[\iota_{n},s_{J^{\prime}}\gamma_{1}]=[\iota_{n},s_{J^{\prime}}\eta_{2}^{\\#}\iota_{2}]$

(where $J=\\{1,\dotsc,n-2)$ and $J^{\prime}=\\{0,\dotsc,n-3\\}$) – is equal to
the first summand of
$d_{0}([\iota_{n+1},s_{J^{\prime\prime}}\iota_{2}])=[\gamma_{n-1},s_{J^{\prime}}\iota_{2}]$
for $J^{\prime\prime}=J^{\prime}\cup\\{n-2\\}$ – which is
$[\iota_{n},s_{J^{\prime}}\iota_{2}],s_{J^{\prime}}\iota_{2}]$ but with the
opposite sign (due to the sign in (8.19) and the coefficient $(-1)^{n\cdot j}$
in (8.18), where $j=3$ in (8.20)).

Step 3: Thus it remains to deal with the summands of $d_{0}\gamma_{n}$ which
are triple Whitehead products of the form

(8.21) ${}A~{}=~{}[[s_{I}\iota_{p},\,s_{J}\iota_{q}],\,s_{K}\iota_{r}]]~{},$

with $p+q+r=n+4$ and $I$, $J$, $K$ non-empty subsets of $\\{0,\dotsc,n\\}$ (of
cardinality $n-p$, $n-q$, and $n-r$, respectively) with no element common to
all three. We may assume without loss of generality that $p\geq q$, and
further assume for now that $q>r$.

Associated to $A$ we have $B=[[s_{J}\iota_{q},s_{K}\iota_{r}],s_{I}\iota_{p}]$
and $C=[[s_{K}\iota_{r},s_{I}\iota_{p}],s_{J}\iota_{q}]$, with

(8.22) ${}(-1)^{pr}A~{}+~{}(-1)^{qp}B~{}+~{}(-1)^{rq}C~{}=~{}0$

by the Jacobi identity (using the sign conventions of [Wh, X §7]).

Thus to show that $d_{0}\gamma_{n}=0$, it suffices to show that each of $A$,
$B$, and $C$ appears exactly once in (the expansion of) $d_{0}\gamma_{n}$,
with the appropriate sign.

By inspection of (8.18) we see that $A$ can only appear as a summand in
$[s_{L}\gamma_{p+q-3},\,s_{K}\iota_{r}]=d_{0}([s_{\widehat{L}}\iota_{p+q-1},\,s_{\widehat{K}}\iota_{r}])$
corresponding to the summand $\widehat{A}=[s_{N}\iota_{p},\,s_{M}\iota_{q}]$
in $\gamma_{p+q-3}$. This exists because we assumed $q>r$, so $p+q-1\geq r$,
and we must have $0\in\widehat{K}$, so $\widehat{K}$ is obtained from $K$ by
adding one to each index and then adjoining the index $0$, while $\widehat{L}$
is obtained from $L$ just by adding one to each index. Thus

(8.23)
${}\operatorname{sgn}(\widehat{K},\widehat{L})~{}=~{}\operatorname{sgn}(K,L)~{},$

and so by (8.15):

(8.24)
${}\begin{split}\operatorname{sgn}(\widehat{L},\widehat{K})~{}=&~{}(-1)^{(r-2)(n-p+1)}\operatorname{sgn}(\widehat{K},\widehat{L})~{}=~{}(-1)^{r(n-p+1)}\operatorname{sgn}(K,L)\\\
=&~{}(-1)^{r(n-p+1)}(-1)^{(r-2)(n-p)}\operatorname{sgn}(L,K)~{}=~{}(-1)^{r}\operatorname{sgn}(L,K)~{}.\end{split}$

Note that we have partitions $I=L\sqcup N$, $J=L\sqcup M$, $K=M\sqcup N$, and
$\\{0,1,\dotsc,n-2\\}=L\sqcup M\sqcup N$, with cardinalities $|N|=q-2$,
$|M|=p-2$, and $|L|=r-2$. If $p=q$, we shall assume that $\min(N)<\min(M)$. By
the remark in the beginning of §8.11, $s_{I}=s_{L}\circ s_{N}$ and
$s_{J}=s_{L}\circ s_{M}$, and by Definition 8.11(ii)-(iii) we have:

(8.25)
${}\begin{cases}\operatorname{sgn}(I,J)~{}=&~{}\operatorname{sgn}(N,M)\\\
\operatorname{sgn}(J,K)~{}=&~{}\operatorname{sgn}(L,N)\\\
\operatorname{sgn}(I,K)~{}=&~{}\operatorname{sgn}(L,M)\end{cases}$

Finally, by (8.16) we have

(8.26)
${}\begin{cases}\operatorname{sgn}(L,K)~{}=&~{}\operatorname{sgn}(L,N)\cdot\operatorname{sgn}(L,M)\\\
\operatorname{sgn}(M,I)~{}=&~{}\operatorname{sgn}(M,L)\cdot\operatorname{sgn}(M,N)\\\
\operatorname{sgn}(N,J)~{}=&~{}\operatorname{sgn}(N,M)\cdot\operatorname{sgn}(N,L)\end{cases}$

Step 4: We see that by (8.24), the sign of
$[s_{\widehat{L}}\iota_{p+q-1},s_{\widehat{K}}\iota_{r}]$ in $\gamma_{n}$ is

$\begin{split}(-1)^{nr}\operatorname{sgn}(\widehat{L},\widehat{K})~{}=&~{}(-1)^{nr+r}\operatorname{sgn}(L,K)~{}=~{}(-1)^{(p+q)r}\operatorname{sgn}(L,N)\cdot\operatorname{sgn}(L,M)\\\
=&~{}(-1)^{pr}\operatorname{sgn}(N,L)\cdot\operatorname{sgn}(L,M)\\\
,\end{split}$

while the sign of $\widehat{A}$ in $\gamma_{p+q-3}$ is

$(-1)^{(p+q-3)q}\operatorname{sgn}(N,M)=(-1)^{pq}\operatorname{sgn}(N,M)=\operatorname{sgn}(M,N)~{}.$

Therefore, if we set

(8.27)
${}\Theta~{}:=~{}\operatorname{sgn}(N,L)\cdot\operatorname{sgn}(L,M)\cdot\operatorname{sgn}(M,N)~{},$

the sign of $A$ in $d_{0}(\gamma_{n})$ is $(-1)^{pr}\,\Theta$, so
multiplication by $(-1)^{pr}$ yields $\Theta$ as the “corrected” coefficient
of $A$ for (8.22).

Step 5: If we further assume that $q+r-1>p$ (or $q+r-1=p$ and $0\not\in I$),
the sign of $B$ in $d_{0}(\gamma_{n})$ is given analogously by

$\begin{split}&(-1)^{(q+r)p+(q+r-3)r}\,\operatorname{sgn}(M,L)\cdot\operatorname{sgn}(M,N)\cdot\operatorname{sgn}(L,N)\\\
&~{}=~{}(-1)^{pq+pr+qr}(-1)^{pr+pq}\,\operatorname{sgn}(L,M)\cdot\operatorname{sgn}(M,N)\cdot\operatorname{sgn}(N,L)~{}=~{}(-1)^{pq}\cdot\Theta~{},\end{split}$

after applying (8.15), so multiplication by $(-1)^{qp}$ yields $\Theta$ again
as the corrected coefficient of $B$.

Step 6: Finally, $C$ appears in
$[s_{N}\gamma_{p+r-3},s_{J}\iota_{q}]=d_{0}([s_{\widehat{N}}\iota_{p+r-1},s_{\widehat{J}}\iota_{q}])$,
with $0\in\widehat{J}$, but since we assumed $r\leq p$, the corresponding
summand in $\gamma_{p+r-3}$ is
$\widehat{C}=[s_{L}\iota_{p},\,s_{M}\iota_{r}]=(-1)^{pr}[s_{M}\iota_{r},s_{L}\iota_{p}]$,
with sign $(-1)^{(r+p-1)p}\operatorname{sgn}(M,L)=\operatorname{sgn}(L,M)$. On
the other hand, by (8.24) the sign of
$[s_{\widehat{N}}\iota_{p+r-1},s_{\widehat{J}}\iota_{q}])$ is

$\begin{split}(-1)^{nq}\operatorname{sgn}(\widehat{N},\widehat{J})~{}=&~{}(-1)^{pq+q+qr}(-1)^{q}\operatorname{sgn}(N,J)=(-1)^{pq+qr}\operatorname{sgn}(N,M)\operatorname{sgn}(N,L)\\\
~{}=&~{}(-1)^{qr}\,\operatorname{sgn}(M.N)\operatorname{sgn}(L,N)\end{split}$

Thus the sign of $C$ in $d_{0}(\gamma_{n})$ is $(-1)^{qr}\cdot\Theta$, so
multiplication by $(-1)^{qr}$ again yields $\Theta$ as the corrected
coefficient of $C$. This shows that (8.22) indeed holds, so these three terms
in the boundary sum to $0$.

Remark: There are a number of other possible orderings of $p+q-1$, $q+r-1$,
and so on, with respect to $p\geq q$ and $r$, with subcases when we have
equalities (modulo permutations of $\\{p,q,r\\}$, of course). We shall
consider the ordering $r\geq p+q-1$:

(i) In this case we have the summand
$A^{\prime}=[s_{K}\iota_{r},\,[s_{I}\iota_{p},\,s_{J}\iota_{q}]]=(-1)^{r(p+q-1)}A$
of $d_{0}\gamma_{n}$ in
$[s_{K}\iota_{r},\,s_{L}\gamma_{p+q-3}]=d_{0}([s_{\widehat{K}}\iota_{r},\,s_{\widehat{L}}\iota_{p+q-1}])$,
with $0\in\widehat{K}$ (indeed, by the convention of §8.13 this must happen
when $r=p+q-1$). It corresponds as in Step 3 to
$\widehat{A}=[s_{N}\iota_{p},\,s_{M}\iota_{q}]$ in $\gamma_{p+q-3}$, with sign
$(-1)^{pq}\operatorname{sgn}(N,M)=\operatorname{sgn}(M,N)$.

The sign of $[s_{\widehat{K}}\iota_{r},\,s_{\widehat{L}}\iota_{p+q-1}]$ in
$\gamma_{n}$ is

$(-1)^{n(p+q-1)}\operatorname{sgn}(\widehat{K},\widehat{L})=(-1)^{pr+qr+r}\operatorname{sgn}(K,L)=(-1)^{pr+r}\operatorname{sgn}(L,N)\operatorname{sgn}(M,L)$

Thus the sign of $A^{\prime}$ in $d_{0}(\gamma_{n})$ is $(-1)^{pr+r}\Theta$,
for $\Theta$ as in (8.27). Since $A^{\prime}=(-1)^{r(p+q-1)}A$, after also
multiplying by $(-1)^{pr}$ as in (8.22) we find the corrected coefficient of
$A$ is $(-1)^{pr+qr}\Theta$.

(ii) Since $r>q$, we will have
$B^{\prime}=[[s_{K}\iota_{r},s_{J}\iota_{q}],s_{I}\iota_{p}]=(-1)^{qr}B$ as a
summand of
$[s_{M}\gamma_{r+q-3},\,s_{I}\iota_{p}]=d_{0}([s_{\widehat{M}}\iota_{r+q-1},\,s_{\widehat{I}}\iota_{p}])$
instead of $B$, with $0\in\widehat{I}$, corresponding to
$\widehat{B}^{\prime}=[s_{N}\iota_{r},\,s_{L}\iota_{q}]$ with sign
$(-1)^{(r+q-3)q}\operatorname{sgn}(N,L)=\operatorname{sgn}(L,N)$ in
$\gamma_{r+q-3}$.

Since
$\operatorname{sgn}(\widehat{M},\widehat{I})=(-1)^{p}\operatorname{sgn}(M,I)$,
the sign of $[s_{\widehat{M}}\iota_{r+q-1},\,s_{\widehat{I}}\iota_{p}]$ in
$\gamma_{n}$ is

$(-1)^{np}\operatorname{sgn}(\widehat{M},\widehat{I})~{}=~{}(-1)^{np+p}\operatorname{sgn}(M,I)~{}=~{}(-1)^{pq+pr}\operatorname{sgn}(M,L)\operatorname{sgn}(M,N)~{}.$

Thus the sign of $B^{\prime}$ in $d_{0}(\gamma_{n})$ is $(-1)^{pq+pr}\Theta$.
Since $B^{\prime}=(-1)^{qr}B$, multiplying by $(-1)^{pq}$ yields
$(-1)^{pr+qr}\,\Theta$ as the corrected coefficient of $B$ in (8.22).

(iii) Finally, $C=[[s_{K}\iota_{r},s_{I}\iota_{p}],s_{J}\iota_{q}]$ appears in
$[s_{N}\gamma_{p+r-3},s_{J}\iota_{q}]=d_{0}([s_{\widehat{N}}\iota_{p+r-1},s_{\widehat{J}}\iota_{q}])$
(with $0\in\widehat{J}$), corresponding to
$\widehat{C}=[s_{M}\iota_{r},s_{L}\iota_{p}]$ with sign
$(-1)^{pr}\operatorname{sgn}(M,L)$, and the sign of
$[s_{\widehat{N}}\iota_{p+r-1},s_{\widehat{J}}\iota_{q}]$ is

$(-1)^{nq}\operatorname{sgn}(\widehat{N},\widehat{J})=(-1)^{q}\operatorname{sgn}(N,J)=(-1)^{pq+qr}\operatorname{sgn}(L,N)\operatorname{sgn}(M,N)~{},$

as in (8.24). Thus the sign of $C$ is $(-1)^{qr+pq+pr}\Theta$, and the
corrected coefficient of $C$ is therefore $(-1)^{pq+pr}\,\Theta$.

The remaining cases are treated similarly. ∎

######  8.28 Example.

The first three cycles, after $\gamma_{3}$ of (8.9), are:

$\gamma_{4}~{}:=~{}[\iota_{5},s_{2}s_{1}s_{0}\iota_{2}]~{}+~{}[s_{0}\iota_{4},s_{2}s_{1}\iota_{3}]~{}-~{}[s_{1}\iota_{4},s_{2}s_{0}\iota_{3}]~{}+~{}[s_{2}\iota_{4},s_{1}s_{0}\iota_{3}]$

in $\pi_{6}{\mathbf{W}}_{3}$, with

$\begin{split}d_{0}(\gamma_{4})=&~{}[[\iota_{4},s_{1}s_{0}\iota_{2}],s_{1}s_{0}\iota_{2}]-[[s_{0}\iota_{3},s_{1}\iota_{3}],s_{1}s_{0}\iota_{2}]+[\iota_{4},s_{1}s_{0}\eta_{2}^{\\#}\iota_{2}]\\\
&-[[s_{0}\iota_{3},s_{1}s_{0}\iota_{2}],s_{1}\iota_{3}]+[[s_{1}\iota_{3},s_{1}s_{0}\iota_{2}],s_{0}\iota_{3}]~{},\end{split}$

which vanishes by combining (8.19) with (8.22). Next,

$\begin{split}\gamma_{5}~{}:=&~{}[\iota_{6},s_{3}s_{2}s_{1}s_{0}\iota_{2}]-[s_{0}\iota_{5},s_{3}s_{2}s_{1}\iota_{3}]+[s_{1}\iota_{5},s_{3}s_{2}s_{0}\iota_{3}]-[s_{2}\iota_{5},s_{3}s_{1}s_{0}\iota_{3}]\\\
&+[s_{3}\iota_{5},s_{2}s_{1}s_{0}\iota_{3}]+[s_{1}s_{0}\iota_{4},s_{3}s_{2}\iota_{4}]-[s_{2}s_{0}\iota_{4},s_{3}s_{1}\iota_{4}]+[s_{3}s_{0}\iota_{4},s_{2}s_{1}\iota_{4}]\end{split}$

in $\pi_{7}{\mathbf{W}}_{4}$. Finally,

$\begin{split}\gamma_{6}~{}:=&~{}[\iota_{7},s_{4}s_{3}s_{2}s_{1}s_{0}\iota_{2}]+[s_{0}\iota_{6},s_{4}s_{3}s_{2}s_{1}\iota_{3}]-[s_{1}\iota_{6},s_{4}s_{3}s_{2}s_{0}\iota_{3}]+[s_{2}\iota_{6},s_{4}s_{3}s_{1}s_{0}\iota_{3}]\\\
&-[s_{3}\iota_{6},s_{4}s_{2}s_{1}s_{0}\iota_{3}]+[s_{4}\iota_{6},s_{3}s_{2}s_{1}s_{0}\iota_{3}]+[s_{1}s_{0}\iota_{5},s_{4}s_{3}s_{2}\iota_{4}]-[s_{2}s_{0}\iota_{5},s_{4}s_{3}s_{1}\iota_{4}]\\\
&+[s_{3}s_{0}\iota_{5},s_{4}s_{2}s_{1}\iota_{4}]-[s_{4}s_{0}\iota_{5},s_{3}s_{2}s_{1}\iota_{4}]+[s_{2}s_{1}\iota_{5},s_{4}s_{3}s_{0}\iota_{4}]-[s_{3}s_{1}\iota_{5},s_{4}s_{2}s_{0}\iota_{4}]\\\
&+[s_{4}s_{1}\iota_{5},s_{3}s_{2}s_{0}\iota_{4}]+[s_{3}s_{2}\iota_{5},s_{4}s_{1}s_{0}\iota_{4}]-[s_{4}s_{2}\iota_{5},s_{3}s_{1}s_{0}\iota_{4}]+[s_{4}s_{3}\iota_{5},s_{2}s_{1}s_{0}\iota_{4}]\end{split}$

in $\pi_{8}{\mathbf{W}}_{5}$.

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