1. Prerequisites

All spaces and maps in this note are assumed to be connected, based and of the homotopy type of $CW$-complexes. We also do not distinguish notationally between a continuous map and its homotopy class. We write $\Omega (X)$ (respectively $E(X)$) for the loop (respectively suspension) space on a space $X$, $\simeq$ for the homotopy relation and $[Y,X]$ for the set of homotopy classes of maps $Y\to X$.

Given a space $X$, we use the customary notation $X\vee X$ and $X\wedge X$ for the wedge and the smash product of $X$, respectively.

Recall that an $H$-space is a pair $(X,\mu )$, where $X$ is a space and $\mu : X \times X \to X$ is a map such that the diagram

commutes up to homotopy, where $\nabla : X\vee X\to X$ is the codiagonal map.

We call $\mu$ a multiplication or an $H$-structure on $X$. Two examples of $H$-spaces come in mind: topological groups and the spaces $\Omega (X)$ of loops on $X$. In the sequel, we identify an $H$-space $(X,\mu )$ with the space $X$.

An $H$-space $X$ is called a group-like space if $X$ satisfies all the axioms of groups up to homotopy. Recall that a homotopy associative an $H$-space always has a homotopy inverse. More precisely, according to [Reference Zabrodsky22, Corollary 1.3.2] (see also [Reference Arkowitz3, Proposition 8.4.4]), we have the following result.

If $X$ is a homotopy associative $H$-space, then the functor $[ -, X]$ takes its values in the category of groups. One may then ask when this functor takes its values in various subcategories of groups.

For example, $X$ is homotopy commutative if and only if $[Y, X]$ is abelian for all $Y$.

For an integer $n\ge 1$, let $X^{\times n}$ and $X^{\wedge n}$ be the $n$-fold Cartesian and smash power of $X$, respectively. Write $q_{X,n} : X^{\times n}\to X^{\wedge n}$ for the quotient map. Given a group-like space $X$, we write $\gamma _{X,1}=\mbox {id}_X : X \to X$ and $\gamma _{X,2} : X \times X \to X$ for the commutator map of $X$. Since the restriction ${\gamma _{X,2}}_{\mid X\vee X}\simeq \ast$, we get a map $\bar {\gamma }_{X,2} : X \wedge X \to X$ with $\bar {\gamma }_{X,2}q_{X,2}\simeq \gamma _{X,2}$. Next, define inductively the maps

\[ \gamma_{X,n+1} : X^{{\times}(n+1)}\to X\quad \mbox{and}\quad \bar{\gamma}_{X,n+1} : X^{{\wedge}(n+1)}\to X \]

by $\gamma _{X,n+1}=\gamma _{X,2}\circ (\gamma _{X,1} \times \gamma _{X,n})$ and $\bar {\gamma }_{X,n+1}=\bar {\gamma }_{X,2}\circ (\gamma _{X,1} \wedge \bar {\gamma }_{X,n})$ for $n\ge 2$, respectively. Then, the diagram

commutes up to homotopy for $n\ge 2$.

One might ask if there is an upper bound for the nilpotency class of $[Y, X]$ that is independent of $Y$. The homotopy nilpotency class of $X$ defined by Berstein–Ganea [Reference Berstein and Ganea5] is the least $n$ such that $\gamma _{X,n+1}\simeq \ast$ and $\gamma _{X,n}\not \simeq \ast$. Equivalently, the homotopy nilpotency class of $X$ is the least $n$ such that $\bar {\gamma }_{X,n+1}\simeq \ast$ and $\bar {\gamma }_{X,n}\not \simeq \ast$. In this case, we write $\mbox {nil}\ X =n$ and call the homotopy associative $H$-space $X$ homotopy nilpotent. If no such integer exists, we put $\mbox {nil}\ X=\infty$.

Note that $\mbox {nil}\ X=1$ if and only if $X$ is homotopy commutative. Given a space $X$, the number $\mbox {nil}\ \Omega (X)$ (if any) is called the homotopy nilpotency class of $X$.

Now, let $MU$ be the complex Thom spectrum, $BP^{\ast }(-)$ the Brown–Peterson cohomology with coefficients $BP^{\ast } = \mathbb {Z}_{(p)}[v_1,\ldots ]$ and $K(n)$ the $n$th Morava $K$-theory at a prime $p$. Thus, $K(n)_\ast (pt) = \mathbb {Z}/p[v_n, v_n^{-1}]$ with $|v_n|=2p^{n}-2$. Hopkins [Reference Hopkins10] described a cohomological criteria for the homotopy nilpotence of finite connected associative $H$-spaces.

We recall Rao's formulation [Reference Rao15, Theorem 0.2] of Hopkins’ result [Reference Hopkins10, Theorem 2.1] needed in the sequel.

Then, in [Reference Hopkins10, Corollary 2.2], it was deduced the following homological criterion for the homotopy nilpotency.

Furthermore, we derive the following result the proof of which is essentially a small modification of Hopkins’ argument [Reference Hopkins10, Corollary 2.2].

Proof. Note that $H\mathbb {Q}^{\ast }(\bar {\gamma }_n) = 0$ for the field of rationals $\mathbb {Q}$ sufficiently large $n$ since $X_{(p)}^{\wedge n}$ is at least $(n-1)$-connected. By assumption, the canonical map $MU^{\ast }(X_{(p)}^{\wedge n})\to MU^{\ast }(X_{(p)}^{\wedge n})\otimes \mathbb {Q}$ is injective, so by Theorem 1.2, it suffices to show that $MU^{\ast }(\bar {\gamma }_n)\otimes \mathbb {Q}=0$. But, for a finite $CW$-complex $X$, there is a natural isomorphism

\begin{align*} MU^{{\ast}}(X^{{\wedge} n}_{(p)})\otimes\mathbb{Q}&\approx MU^{{\ast}}(X^{{\wedge} n})\otimes\mathbb{Q}\approx MU^{{\ast}}(\mbox{pt})\otimes H\mathbb{Q}^{{\ast}}(X^{{\wedge} n})\\ &\approx MU^{{\ast}}(\mbox{pt})\otimes H\mathbb{Q}^{{\ast}}(X^{{\wedge} n}_{(p)}), \end{align*}

so the map $MU^{\ast }(\bar {\gamma }_n)\otimes \mathbb {Q}= 0$ as soon as $H\mathbb {Q}^{\ast }(\bar {\gamma }_n)=0$. This completes the proof.

Since the homology $H_\ast (\mathbb {S}^{2m-1},\mathbb {Z})$ are torsion free and $\mathbb {S}^{2m-1}$ is a finite $CW$-complex, Corollary 1.4 yields the result on the homotopy nilpotency of $\mathbb {S}^{2m-1}_{(p)}$.

Theorem 1.5 If $m\ge 2$ and $p>3$ is a prime then

\[ \mbox{nil}\ \mathbb{S}^{2m-1}_{(p)}<\infty \]

with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$.

In the sequel, we make use of the following. Let $f: X\to Y$ be an $H$-map of homotopy associative $H$-spaces. Recall from [Reference Zabrodsky22, Chapter II] that:

1. (1) it is said $\mbox {nil}\ f \le n$ if $f\bar {\gamma }_{X,n}\simeq \ast$;

2. (2) $f$ is called central if $\bar {\gamma }_{Y,2}(f \wedge \mbox {id}_Y)\simeq \ast$.

Notice that $\mbox {nil}\ f\le \min \{\mbox {nil}\ X,\mbox {nil}\ Y\}$.

Then, in view of [Reference Zabrodsky22, Lemma 2.6.6], we have the following techniques for the study of the homotopy nilpotency.

2. $A_m$-spaces

Recall that by Stasheff [Reference Stasheff18], an $A_m$-structure on a space $X$ consists on $m$-tuples

such that ${\frak q}_{n\ast } : \pi _k(\mathcal{E}_n(X),X)\to \pi _k(\mathcal {B}_n({X}))$ is an isomorphism for all $k\ge 1$, together with a contracting homotopy $h: C\mathcal {E}_{n-1}(X)\to \mathcal {E}_n(X)$ such that $h(C\mathcal {E}_{n-1}(X))\subseteq \mathcal {E}_n(X)$ for $n=2,\ldots ,m$. For the purposes of homotopy theory, in the light of [Reference Stasheff18, Proposition 2], we can think of $X\to \mathcal {E}_n(X)\stackrel {\frak {q}_n}{\to }\mathcal {B}_n(X)$, as a fibration.

An $A_m$-space for $m=0,1\ldots ,\infty$ is a space $X$ with a multiplication $\mu : X\times X\to X$ that is associative up to higher homotopies involving up to $n$ variables. Further, an $A_\infty$-space has all coherent higher associativity homotopies and is equivalent to a loop space $\Omega (Y)$ for a space $Y$ called the classifying space of $X$.

By [Reference Stasheff18, Theorem 5], classes of spaces with $A_m$-structures and $A_m$-spaces coincide.

The $X$-projective $n$-space $XP(n)$ for $n\le m$, associated with an $A_m$-space X is the base space $\mathcal {B}_{n+1}(X)$ of the derived $A_m$-structure. The space $\mathcal {B}_1(X)$ is a point and $\mathcal {B}_2(X)$ can be recognized as the suspension $E(X)$. Notice that $\mathcal {B}_{m+1}(X)$ can be defined even when $\frak {p}_{m+1}$ cannot; it has the homotopy type of the mapping cone $C\mathcal {E}_m(X)\cup _{\frak {q}_m} \mathcal {B}_m(X)$. By means of [Reference Stasheff18, Theorem 11, Theorem 12], the spaces $\mathcal {E}_n(X)$ and $\mathcal {B}_{n+1}(X)$ have the homotopy types of the $n$th join $X^{\ast ^{n}}$ and $C\mathcal {E}_n(X)\cup _{\frak {p}_n} \mathcal {B}_n(X)$ for $n\le m$, respectively provided $X$ is path-connected. Because of a homotopy equivalence $X^{\ast ^{n}}\simeq E^{n-1}(X^{\wedge n})$ for the $(n-1)$th suspension $E^{n-1}$, we deduce that the fibration $X\to \mathcal {E}_n(X)\stackrel {\frak {q}_n}{\to }\mathcal {B}_n(X)$ is homotopy equivalent to

(2.2)\begin{equation} X\to E^{n-1}X^{{\wedge} n}\stackrel{q_n}{\longrightarrow} XP(n-1). \end{equation}

3. Localized spheres $\mathbb {S}^{2m-1}_{(p)}$ and $\mathbb {S}^{2m-1}_{(p)}$-projective spaces $\mathbb {S}^{2m-1}_{(p)}P(n-1)$

Let $\mathbb {S}^{2m-1}_{(p)}$ be the $p$-localization of the sphere $\mathbb {S}^{2m-1}$ at a prime $p$. It is known by [Reference James11, Theorem 1.4] that $\mathbb {S}^{2m-1}_{(2)}$ does not admit a homotopy associative multiplication if $m\not =1,2$. The sole obstruction to putting an $H$-structure on $\mathbb {S}^{2m-1}$ is the Whitehead square $[\iota _{2m-1},\iota _{2m-1}]$ of a generator $\iota _{2m-1}\in \pi _{2m-1}(\mathbb {S}^{2m-1})$. Since the order of $[\iota _{2m-1},\iota _{2m-1}]$ is $\le 2$, it follows that, if $p$ is an odd prime, $\mathbb {S}^{2m-1}_{(p)}$ admits an $H$-space structure. Which $p$-localized spheres $\mathbb {S}^{2m-1}_{(p)}$ with $p>2$ have an $H$-structures or loop structures is known by Adams [Reference Adams1]. More precisely, in view of [Reference Adams2] (see also [Reference Neisendorfer14, Proposition 11.2.2]), we have the following $H$-structure on $\mathbb {S}_{(p)}^{2m-1}$.

Loosely speaking, via the double suspension map $E^{2} : \mathbb {S}_{(p)}^{2m-1}\to \Omega ^{2}(\mathbb {S}_{(p)}^{2m+1})$, the multiplication on the double loop space $\Omega ^{2}\mathbb {S}_{(p)}^{2m+1}$ restricts to the multiplication $\mu _A$ on the bottom cell $\mathbb {S}_{(p)}^{2m-1}$. Next, by Mimura et al. [Reference Mimura, Nishida and Toda13, Proposition 6.8], Stasheff [Reference Stasheff18] and Sullivan [Reference Sullivan19], we have the result on $H$-structures on $\mathbb {S}_{(p)}^{2m-1}$.

This implies that $\mathbb {S}^{2m-1}_{(p)}$ does not admit an $A_{p}$-structure provided $m\nmid p-1$. We also point out that an $A_{p-1}$-structure on $\mathbb {S}^{2m-1}_{(p)}$ is induced from $\Omega ^{2}(\mathbb {S}_{(p)}^{2m+1})$ which is of course an $A_\infty$-space, via the double suspension map $E^{2} : \mathbb {S}_{(p)}^{2m-1}\to \Omega ^{2}(\mathbb {S}_{(p)}^{2m+1})$. Further, in view of Proposition 3.2, the sphere $\mathbb {S}_{(p)}^{2m-1}$ admits an $A_{p}$-structure if and only if it admits a classifying space.

Now, we show the nilpotency of $\mathbb {S}^{2m-1}_{(p)}$ provided $m\mid p-1$.

Proof. Sullivan [Reference Sullivan19], to construct a classifying space for $\mathbb {S}_{(p)}^{2m-1}$ with $m\mid p-1$, considered the space $K(\mathbb {Z}_p, 2)$, where $\hat {\mathbb {Z}}_p$ is the $p$-adic integers and the cyclic subgroup $\Gamma <\mathbb {Z}_{p-1}<\hat {\mathbb {Z}}_p^{\ast }$ (the $p$-adic units) of order $m$. Then $\Gamma$ acts freely on a model of $K(\hat {\mathbb {Z}}_p,2)$ and $X = K(\hat {\mathbb {Z}}_p,2)/\Gamma$ has cohomology $H^{\ast }(X,\mathbb {Z}/p\mathbb {Z}) = S(x,2m)$, the graded symmetric algebra generated by $x$ with degree $|x|=2m$ and $\pi _1(X) = \Gamma$. After completing $X$ at $p$ to $\hat {X}_p$, we have a space with $\pi _1(\hat {X}_p) = 0$ and $H^{\ast }(\hat {X}_p, \mathbb {Z}/p\mathbb {Z}) = S(x,2m)$. The map $\mathbb {S}^{2m-1}\to \Omega (\hat {X}_p)$ defines a homotopy equivalence $\mathbb {S}^{2m-1}_{(p)}\stackrel {\simeq }{\to } \Omega (\hat {X}_p)$ and $\hat {X}_p$ is a classifying space for $\mathbb {S}_{(p)}^{2m-1}$.

But, by [Reference Neisendorfer14, Chapter 2], the $p$-completion preserves a fibration of simply-connected spaces. Hence, the $p$-completion of the fibration $\Gamma \to K(\hat {\mathbb {Z}}_p,2)\to X$ leads to the fibration $\hat {\Gamma }_p\to K(\hat {\mathbb {Z}}_p,2)\to \hat {X}_p$. Consequently, we get the $H$-fibration

\[ \Omega( K(\hat{\mathbb{Z}}_p,2))= K(\hat{\mathbb{Z}}_p,1)\longrightarrow \Omega(X)\longrightarrow \Gamma. \]

Since the space $X = K(\hat {\mathbb {Z}}_p,2)/\Gamma$ is simply-connected, we have $\widehat {(\Omega (X))}_p=\Omega (\hat {X}_p)$. Then, by means of the $p$-completeness of $K(\hat {\mathbb {Z}}_p,1)$, the $p$-completion of the fibration above yields the $H$-fibration

\[ K(\hat{\mathbb{Z}}_p,1)\longrightarrow \Omega(\hat{X}_p)\longrightarrow \hat{\Gamma}_p \]

determined by the canonical $H$-map $\hat {\Gamma }_p\to K(\hat {\mathbb {Z}}_p,2)$.

Thus, by means of Proposition 1.6(2), we derive that the $H$-map $\Omega ( K(\hat {\mathbb {Z}}_p,2))= K(\hat {\mathbb {Z}}_p,1)\to \Omega (X)$ is central and so Proposition 1.6(1) yields

\[ \mbox{nil}\ \mathbb{S}^{2m-1}_{(p)}\le 2. \]

This completes the proof.

Then, Arkowitz, Ewing and Schiffman [Reference Arkowitz, Ewing and Schiffman4, Theorem 0.1] have proved the following result on $H$-structures on $\mathbb {S}^{2m-1}_{(p)}$.

Thus, the above and Theorem 3.4(2) yield the conclusion.

Corollary 3.5 If $m\mid p-1$ and $p>3$ then

\[ \mbox{nil}\ \mathbb{S}^{2m-1}_{(p)}= 2 \]

with respect to all $(p-1)$ loop $H$-structures on $\mathbb {S}^{2m-1}_{(p)}$.

We point out that Proposition 3.3 has been already shown by Meier [Reference Meier12] in the special case when $m = p-1$ using the result [Reference Toda20, Theorem 13.4].

Theorem 3.6 Let $p$ be an odd prime.

\[ \pi_{2m-1+k}(\mathbb{S}^{2m-1)}_{(p)}) \approx\begin{cases} \mathbb{Z}/p\mathbb{Z}\ \text{for}\ k = 2i(p-1)-1,\quad i = 1,\ldots, p-1,\ m\ge 2;\\ \mathbb{Z}/p\mathbb{Z}\ \text{for}\ k = 2i(p-1)-2,\quad i=m,\ldots,p-1;\\ 0\ \text{ortherwise}\ \text{for}\ 1\le k< 2p(p-1)-2. \end{cases} \]

Given a pointed connected topological space $X$ and a prime $p$, write $\pi _m(X;p)$ for the $p$-primary component of its $m$th homotopy group $\pi _m(X)$ for $m\ge 1$. Recall that by [Reference Copeland6], the set of homotopy classes of possible $H$-structures on $\mathbb {S}^{2m-1}_{(p)}$ is in one-to-one correspondence with $[\mathbb {S}^{2m-1}_{(p)}\wedge \mathbb {S}^{2m-1}_{(p)},\mathbb {S}^{2m-1}_{(p)}]=\pi _{4m-2}(\mathbb {S}^{2m-1},p)$. Consequently, if $p>3$ then we may study the homotopy nilpotency of $\mathbb {S}^{2m-1}_{(p)}$.

Nevertheless, in some particular cases, an estimation for $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}$ might be stated. First, notice that Theorem 3.6 implies that

(3.7)\begin{equation} \pi_{2m-1+k}(\mathbb{S}^{2m-1}_{(p)})=0 \end{equation}

provided $k<2p(p-1)-2$, $k\not =2i(p-1)-1$ for $i = 1,\ldots , p-1$ and $k\not =2i(p-1)-2$ for $i = m,\ldots , p-1$.

Certainly, the homotopy group $\pi _{4m-2}(\mathbb {S}^{2m-1})$ is finite and write $\sharp \pi _{4m-2}(\mathbb {S}^{2m-1})$ for its order. Then, for $p_m=\max \{p;\, p \,\mbox {is a prime with}\ p\mid \sharp \pi _{4m-2}(\mathbb {S}^{2m-1})\}$, we apply Theorem 3.6 to state the result on an $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$.

Proof. If $m< p-1$ then $2m-1<2p-3$ and Equation (3.7) implies that

\[ [(\mathbb{S}^{2m-1}_{(p)})^{{\wedge} 2},\mathbb{S}^{2m-1}_{(p)}]=\pi_{2(2m-1)}(\mathbb{S}^{2m-1}_{(p)})=0. \]

If $p>\max \{3,p_m\}$ then $\pi _{2(2m-1)}(\mathbb {S}^{2m-1}_{(p)})=0$ as well. Then, Proposition 3.1 provides an existence of a unique homotopy associative and commutative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$ and the proof follows.

Now, we apply the results above to $\mathbb {S}^{2m-1}_{(p)}$-projective spaces $\mathbb {S}^{2m-1}_{(p)}P(n-1)$. Write $J_k(\mathbb {S}^{2n})$ for the $k$th stage of the James construction on the sphere $\mathbb {S}^{2m}$.

Since $\mathbb {S}^{2m-1}_{(p)}P(1)\simeq \mathbb {S}^{2m}_{(p)}$ and $\mathbb {S}^{2m-1}_{(p)}P(n-1)\simeq C\mathbb {S}^{2(n-1)m-1}_{(p)} \cup _{q_{n-1}}\mathbb {S}^{2m-1}_{(p)}P(n-2)$ for the fibration (2.2)

\[ q_{n-1} : \mathbb{S}^{2(n-1)m-1}_{(p)}\to \mathbb{S}^{2m-1}_{(p)}P(n-2) \]

with $X=\mathbb {S}^{2m-1}_{(p)}$, we can define inductively a map

\[ \mathbb{S}^{2m-1}_{(p)}P(n-1)\longrightarrow J_{n-1}(\mathbb{S}^{2m}_{(p)}). \]

for $p>3$ provided $n-1< p$ with $m\nmid p-1$ or any $n\ge 1$ with $m\mid p-1$.

Furthermore, one can state the result on some $\mathbb {S}^{2m-1}_{(p)}P(n-1)$.

Proposition 3.9 The canonical map

\[ \mathbb{S}^{2m-1}_{(p)}P(n-1)\longrightarrow J_{n-1}(\mathbb{S}^{2m}_{(p)}) \]

is an integral homology isomorphism for $p>3$ provided $n-1< p$ with $m\nmid p-1$ or any $n\ge 1$ with $m\mid p-1$.

Consequently, by means of the Whitehead Theorem, we get a homotopy equivalence $\mathbb {S}^{2m-1}_{(p)}P(n-1)\stackrel {\simeq }{\longrightarrow }J_{n-1}(\mathbb {S}^{2m}_{(p)})$ which yields an $H$-homotopy equivalence

\[ \Omega(\mathbb{S}^{2m-1}_{(p)}P(n-1))\stackrel{\simeq}{\longrightarrow}\Omega(J_{n-1}(\mathbb{S}^{2m}_{(p)})) \]

for $p>3$ provided $n-1< p$ with $m\nmid p-1$ or any $n\ge 1$ with $m\mid p-1$. But, Gray showed [Reference Gray9, Theorem 1 and the footnote on p. 182] that $\Omega (J_{jp^{s}-1}(\mathbb {S}^{2m}))$ with $p\ge 3$ is universal in the category of homotopy associative commutative $H$-spaces, with its generating subspace being the $(2mp-2)$-skeleton provided $p\ge 3$ with $s>0$ and an odd $j\le p$. Hence, Proposition 3.9 yields the conclusion on the $H$-structure on $\Omega (\mathbb {S}^{2m-1}_{(p)}P(jp^{s}-1))$.

For further studies of the homotopy nilpotency of $\Omega (\mathbb {S}^{2m-1}_{(p)}P(n-1))$, we need to show an existence of some $H$-fibration.

Lemma 3.11 If $p> 3$ is a prime, $m\ge 2$ and $m\mid p-1$ then for a fixed $A_\infty$-structure on $\mathbb {S}^{2m-1}_{(p)}$ and $n\ge 2$ then there is an $H$-fibration

\[ \Omega (\mathbb{S}^{2mn-1}_{(p)})\longrightarrow \Omega (\mathbb{S}^{2m-1}_{(p)}P(n-1))\longrightarrow\mathbb{S}^{2m-1}_{(p)} \]

with the central map $\Omega (\mathbb {S}^{2mn-1}_{(p)})\longrightarrow \Omega (\mathbb {S}^{2m-1}_{(p)}P(n-1))$.

Proof. Recall that by Proposition 3.2(3) the space $\mathbb {S}^{2m-1}_{(p)}$ admits an $A_\infty$-structure provided $m\mid p-1$. Furthermore, for such the space $\mathbb {S}^{2m-1}_{(p)}$, Sullivan [Reference Sullivan19] constructed a classifying space denoted in the proof of Proposition 3.3 by $\hat {X}_p$.

Next, write $i_n : \mathbb {S}^{2mn-1}_{(p)}\hookrightarrow \mathbb {S}^{2m-1}_{(p)}P(n-1)$ and $j_n : \mathbb {S}^{2m-1}_{(p)}P(n-1)\hookrightarrow \hat {X}_p$ for the canonical inclusion maps and notice that $E^{n-1}(\mathbb {S}^{2m-1}_{(p)})^{\wedge n}=\mathbb {S}^{2mn-1}_{(p)}$. Since $\Omega (\hat {X}_p)\simeq \mathbb {S}^{2m-1}_{(p)}$, we get the Puppe fibration sequence

\[ \cdots\to \Omega(\mathbb{S}^{2mn-1}_{(p)})\stackrel{\Omega(i_n)}{\longrightarrow}\Omega(\mathbb{S}^{2m-1}_{(p)}P(n-1)) \stackrel{\Omega(j_n)}{\longrightarrow}\mathbb{S}^{2m-1}_{(p)}\stackrel{\partial_n}{\to} \mathbb{S}^{2mn-1}_{(p)}\stackrel{i_n}{\hookrightarrow}\mathbb{S}^{2m-1}_{(p)}P(n-1)\stackrel{j_n}{\hookrightarrow}\hat{X}_p. \]

But, the $H$-deviation [Reference Zabrodsky22, Definition 1.4.1.] of $\partial _n$ is a map $\mathbb {S}^{2m-1}_{(p)}\wedge \mathbb {S}^{2m-1}_{(p)}\to \mathbb {S}^{2mn-1}_{(p)}$ which is null homotopic for dimension and connectivity reasons if $n\ge 2$. Then, by Zabrodsky [Reference Zabrodsky22, Proposition 1.5.1.], $\partial _n$ is an $H$-map. Hence, Proposition 1.6(2) implies that $\Omega (i_n) : \Omega (\mathbb {S}^{2mn-1}_{(p)})\to \Omega (\mathbb {S}^{2m-1}_{(p)}P(n-1))$ is central in the $H$-fibration

\[ \Omega(\mathbb{S}^{2mn-1}_{(p)})\stackrel{\Omega(i_n)}{\longrightarrow}\Omega(\mathbb{S}^{2m-1}_{(p)}P(n-1)) \stackrel{\Omega(j_n)}{\longrightarrow}\mathbb{S}^{2m-1}_{(p)} \]

and this completes the proof.

Thus, Propositions 1.6(1), 3.3, Corollary 3.10 and Lemma 3.11 yield the result on the homotopy nilpotency of $\Omega (\mathbb {S}^{2m-1}_{(p)}P(n-1))$ under some conditions.

To conclude, we point out that Theorem 3.12 applies to more cases than Meier's result [Reference Meier12, Theorem 5.4].

We close the paper with the following conjecture.

Conjecture 3.13 If $p> 3$ is a prime and $m,n\ge 2$ then

\[ \mbox{nil}\ \Omega (\mathbb{S}^{2m-1}_{(p)}P(n-1))<\infty. \]