Higher homotopy associativity in the Harris decomposition of Lie groups

TL;DR

This paper investigates how the Harris decomposition of certain Lie groups preserves higher homotopy associativity at various primes, extending classical homotopy group relations to a richer algebraic structure.

Contribution

It extends Harris's classical homotopy group results to show preservation of higher homotopy associativity in the p-local decomposition of specific Lie groups.

Findings

Homotopy equivalence $G o H imes G/H$ at primes $p$

Preservation of higher homotopy associativity in the decomposition

Extension of classical Harris results to p-local homotopy theory

Abstract

Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_6,F_4),\,(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\ne 3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G\simeq_{(p)}H\times G/H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.

Equations

$${\pi }_{\ast }(G{)}_{(p)}\cong {\pi }_{\ast }(H{)}_{(p)}\oplus {\pi }_{\ast }(G/H{)}_{(p)}.$$

$$G{\simeq }_{(p)}H\times G/H$$

$$b_k=\max \{(k-1)m+l,kl\}.$$

$$G{\simeq }_{(p)}H\times G/H$$

$$\Sigma X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\hat{P}}^n(X_1,\dots ,X_l)=\bigcup _{i_1+\cdots +i_l=n}{P}^{i_1}{X}_1\times \cdots \times P^{i_l}{X}_l.$$

$$\Sigma X_1\vee \Sigma X_2\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Sigma (X_1\times X_2)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Sigma i_1\vee \Sigma i_2\vee h:\Sigma X_1\vee \Sigma X_2\vee \Sigma (X_1\wedge X_2)\to \Sigma (X_1\times X_2)$$

$$\Sigma (X_1\wedge X_2)\stackrel{h}{\to }\Sigma (X_1\times X_2)\stackrel{r^{\prime }}{\to }\Sigma X_1\vee \Sigma X_2$$

$$\Sigma (X_1\times X_2)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Sigma X_1\vee \cdots \vee \Sigma X_l\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bigcup _{\begin{array}{c}{i}_1+\cdots +i_l=2\\ i_1,\dots ,i_l\le 1\end{array}}{P}^{i_1}{X}_1\times \cdots \times P^{i_l}{X}_l\to BY.$$

$$\Sigma (X_1\times \cdots \times X_l)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Sigma (X_1\times \cdots \times X_l)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$G\simeq S^{2n_1-1}\times \cdots \times S^{2n_r-1}.$$

$$P^l{S}^{2n-1}\simeq S^{2n}\cup e^{4n}\cup \cdots \cup e^{2ln}.$$

$$G/H\simeq S^{2l_1-1}\times \cdots \times S^{2l_t-1}$$

$$X=\bigcup _{\begin{array}{c}i+j=k\\ i\ne k\end{array}}{\hat{P}}^i(S^{2m_1-1},\dots ,S^{2m_s-1})\times {\hat{P}}^j(S^{2l_1-1},\dots ,S^{2l_t-1}).$$

$$Y={\hat{P}}^{k-1}(S^{2m_1-1},\dots ,S^{2m_s-1})\subset X.$$

$$\bigcup _{\begin{array}{c}i+j=k\\ i\ne k\end{array}}{P}^{i}H\times {\hat{P}}^j(S^{2l_1-1},\dots ,S^{2l_t-1})\stackrel{g\circ P^{k-1}h\times 1}{\to }X\to BG$$

$$H\times S^{2l_1-1}\times \cdots \times S^{2l_t-1}\to H.$$

$$\Sigma G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\hat{P}}^k(H,S^{2l_1-1},\dots ,S^{2l_t-1})\to P^{k}H\stackrel{j_k}{\to }BH$$

$$H^{\ast }(BSU(n))=\mathbb{Z}/p[c_2,\dots ,c_n]$$

$$\begin{gathered} {\mathcal{P}}^1{c}_k=\sum _{2i_2+3i_3+\cdots +n{i}_n=k+p-1}(-1{)}^{i_2+\cdots +i_n-1}\frac{(i_2+\cdots +i_n-1)!}{i_2!\cdots i_n!} \\ \times \left(k-1-\frac{{\sum }_{j=2}^{k-1}(k+p-1-j)i_j}{i_2+\cdots +i_n-1}\right)c_2^{i_2}\cdots c_n^{i_n}. \end{gathered}$$

$${\mathcal{P}}^1{c}_2$$

$${\mathcal{P}}^1{c}_4$$

$${\mathcal{P}}^1{c}_2$$

$${\mathcal{P}}^1{c}_4$$

$$H^{\ast }(BSO(2n+1))=\mathbb{Z}/p[p_1,\dots ,p_n]$$

Full text

1. Introduction

Lie groups decompose into products of small spaces when localized at a prime $p$. This is called the mod $p$ decomposition and is fundamental in the homotopy theory of Lie groups. Then it is important to study relations between the mod $p$ decomposition and the group structures (or the loop space structures) of Lie groups, and there are several results on such relations [2, 5, 6, 7, 8, 12, 13, 14, 19, 23]. In this paper, we study a relation between the group structures (or the loop structures) of Lie groups and maps between Lie groups arising from the classical mod $p$ decomposition due to Harris [3, 4].

Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_{6},F_{4})$, $(Spin(8),G_{2})$ and let $p$ be any prime $\geq 5$ for $(G,H)=(E_{6},F_{4})$, any prime $\neq 3$ for $(G,H)=(Spin(8),G_{2})$, and any odd prime otherwise. Then $H$ is a subgroup of $G$ in the obvious way so that there is a fibration $H\to G\to G/H$. Harris [3, 4] showed that the associated homotopy exact sequence splits after localizing at a prime $p$ such that

[TABLE]

The proof of Harris actually implies a stronger result such that there is a $p$-local homotopy equivalence

[TABLE]

which is one of the most classical mod $p$ decomposition of Lie groups. In particular, there is a projection $G_{(p)}\to H_{(p)}$ which has been treated only as a continuous map so far. However, we are interested in relations between the projection and the group structures (or the loop space structures) of $G$ and $H$, and so we naively ask how much this projection respects the group structures of $G$ and $H$. We make this naive question more precise. Stasheff [22] defined $A_{k}$-maps for $k\geq 2$ between loop spaces as H-maps preserving the $(k-2)^{\text{th}}$ higher homotopy associativity. Then $A_{k}$-maps form a gradation between continuous maps and loop maps , and so our question is made precise as follows.

Question 1.1**.**

Given a prime $p$, for which $k$ is the projection $G_{(p)}\to H_{(p)}$ an $A_{k}$-map?

The aim of this paper is to answer this question. Since the case $(G,H)=(SU(2),Sp(1))$ is trivial, we will exclude it throughout.

Theorem 1.2**.**

Let $(G,H),\,a_{k}$ and $p$ be as in the table below. Then for $k\geq 2$ the following statements hold:

1. (1)

for $(G,H)\neq(SO(2n),SO(2n-1))$ the projection $G_{(p)}\to H_{(p)}$ is an $A_{k}$-map if and only if $p\geq a_{k}$; 2. (2)

for $(G,H)=(SO(2n),SO(2n-1))$

1. (a)

if $p\geq a_{k}$ then the projection $G_{(p)}\to H_{(p)}$ is an $A_{k}$-map; 2. (b)

if $p<a_{k}-n+2$ then the projection $G_{(p)}\to H_{(p)}$ is not an $A_{k}$-map.

Harris [4] showed the decomposition (1.1) by constructing a specific map $G/H\to G$, which is a $p$-local section of the projection $G\to G/H$, from a finite order self-map of $G$. But if we use the mod $p$ decomposition of Mimura, Nishida, and Toda [18] and its naturality instead, then we get the decomposition (1.2) for more pairs of Lie groups, e.g. $(SU(n),SU(n-1))$ for $p\geq n$. Our method for showing the projection $G_{(p)}\to H_{(p)}$ is an $A_{k}$-map does not use a specific map $G/H\to G$, and so we get the following general result. We set notation to state it. Suppose that a connected Lie group $G$ has no $p$-torsion in the integral homology for an odd prime $p$. Then its mod $p$ cohomology is an exterior algebra generated by odd degree elements. Suppose further that a pair of connected Lie groups $(G,H)$ admits the decomposition (1.2). Then the mod $p$ cohomology of $H$ and $G/H$ are also exterior algebras generated by odd degree elements. Let $2m-1$ and $2l-1$ be the largest dimensions of the mod $p$ cohomology generators of $H$ and $G/H$, respectively. We define

[TABLE]

Notice that if $(G,H)$ is as in Theorem 1.2 then it satisfies the above conditions. The following table gives a list of $(m,l)$ for $(G,H)$ in Theorem 1.2.

It holds that $a_{k}=b_{k}$ for $(G,H)$ in Theorem 1.2 except for $(G,H)=(E_{6},F_{4})$, in which case $a_{k}=b_{k}-2$.

Theorem 1.3**.**

Let $(G,H)$ be a pair of connected Lie groups, and let $p$ be an odd prime. Suppose $G$ has no $p$-torsion in the integral homology and the decomposition (1.2) holds. Then for $p\geq b_{k}$ the projection $G_{(p)}\to H_{(p)}$ is an $A_{k}$-map.

Theorem 1.3 is a consequence of the following stronger statement (Proposition 2.7).

Theorem 1.4**.**

Under the condition of Theorem 1.3, there is an $A_{k}$-structure on $(G/H)_{(p)}$ such that the decomposition

[TABLE]

is an $A_{k}$-equivalence.

The main technique for proving Theorem 1.3 is a refinement of the reduction of the projective spaces of $p$-regular Lie groups established in the paper [8] on the higher homotopy commutativity of localized Lie groups. Then, indirectly though, our result is connected to higher homotopy commutativity, Sugawara and Williams $C_{k}$-spaces, where we refer to [8] for their definitions. For example, we have the following.

Corollary 1.5**.**

Let $p$ be an odd prime. The following are equivalent:

1. (1)

the projection $SU(2n+1)_{(p)}\to SO(2n+1)_{(p)}$ is an $A_{k}$-map; 2. (2)

$SU(2n+1)_{(p)}$* is a Sugawara $C_{k}$-space;* 3. (3)

$SU(2n+1)_{(p)}$* is a Williams $C_{k}$-space.*

It would be interesting to find a direct connection between an $A_{k}$-structure of the projection $G_{(p)}\to H_{(p)}$ and the higher homotopy commutativity of $G_{(p)}$ and $H_{(p)}$.

Acknowledgement: We are grateful to Mitsunobu Tsutaya for useful advices on $A_{n}$-structures and to the referee for pointing out an ambiguity in the calculation of a Samelson product in $Spin(8)$ in the earlier version and for the suggestion to add Theorem 1.4. The first author was supported in part by JSPS KAKENHI (No. 17K05248).

2. Projective spaces for products

Throughout this section, we assume that spaces are path-connected. This section recalls the reduction of the projective space of a product of $A_{n}$-spaces established in [8] and shows its property that we will use. We refer to [22, 10, 8] for the basics of $A_{n}$-spaces, their projective spaces, and $A_{n}$-maps. Let $X$ be an $A_{n}$-space and $P^{k}X$ be the $k^{\text{th}}$ projective space of $X$ for $k\leq n$. If $X$ is an $A_{\infty}$-space then we write the canonical map $P^{k}X\to BX$ by $j_{k}$. For a map $f\colon X\to Y$ where $Y$ is a topological monoid, let $\bar{f}\colon\Sigma X\to BY$ denote the adjoint of $f$. The following is shown in [10] (cf. [8]).

Lemma 2.1**.**

Let $X$ be an $A_{n}$-space and $Y$ be a topological monoid. A map $f\colon X\to Y$ is an $A_{n}$-map if and only if there is a map $P^{n}X\to BY$ satisfying a homotopy commutative diagram

[TABLE]

Let $X_{1},\ldots,X_{l}$ be $A_{n}$-spaces and let

[TABLE]

We regard $X_{1}\times\cdots\times X_{l}$ as an $A_{n}$-space by the product of multiplications of $X_{1},\ldots,X_{l}$. The following is proved in [8].

Lemma 2.2**.**

Let $X_{i}$ be $A_{n}$-spaces for $i=1,2$. There are maps $\widehat{P}^{i}(X_{1},X_{2})\to P^{i}(X_{1}\times X_{2})$ and $P^{i}(X_{1}\times X_{2})\to\widehat{P}^{i}(X_{1},X_{2})$ for $i=1,2,\ldots,n$ satisfying a homotopy commutative diagram

[TABLE]

where the upper left arrow is the inclusion and $p_{i}\colon X_{1}\times X_{2}\to X_{i}$ is the $i^{\text{th}}$ projection for $i=1,2$.

In order to apply Lemma 2.2 to our case, we need the following simple lemma. Let $X$ be an H-space. Then the projective space $P^{2}X$ is the cofiber of the Hopf construction $H\colon\Sigma X\wedge X\to\Sigma X$, where we write the inclusion $\Sigma X\to P^{2}X$ by $j$.

Lemma 2.3**.**

Let $X_{i}$ be H-spaces for $i=1,2$. Then there is a homotopy commutative diagram

[TABLE]

where $p_{i}\colon X_{1}\times X_{2}\to X_{i}$ is the $i^{\text{th}}$ projection for $i=1,2$ and $\hat{\jmath}$ is the restriction of $j$.

Proof.

Let $h$ be the composite of the inclusion $\Sigma X_{1}\wedge X_{2}\to\Sigma(X_{1}\times X_{2})\wedge(X_{1}\times X_{2})$ and the Hopf construction $H\colon\Sigma(X_{1}\times X2)\wedge(X_{1}\times X_{2})\to\Sigma(X_{1}\times X_{2})$. By the definition of the Hopf construction, $h$ is the right homotopy inverse of the projection $\Sigma(X_{1}\times X_{2})\to\Sigma(X_{1}\wedge X_{2})$, and so the map

[TABLE]

is a homotopy equivalence, where $i_{k}\colon X_{k}\to X_{1}\times X_{2}$ is the inclusion for $k=1,2$. Let $r^{\prime}$ be the composite of the homotopy inverse $(\Sigma i_{1}\vee\Sigma i_{2}\vee h)^{-1}$ and the projection $r\colon\Sigma X_{1}\vee\Sigma X_{2}\vee\Sigma(X_{1}\wedge X_{2})\to\Sigma X_{1}\vee\Sigma X_{2}$. Then there is a homotopy cofibration

[TABLE]

and so one gets a homotopy commutative diagram

[TABLE]

It remains to show $\hat{\jmath}\circ r^{\prime}\simeq\hat{\jmath}\circ(\Sigma p_{1}+\Sigma p_{2})$. Since $\Sigma X_{1}\times\Sigma X_{2}\subset\widehat{P}^{2}(X_{1}\times X_{2})$, it follows from Lemma 2.2 that $\hat{\jmath}$ factors through the inclusion $\Sigma i_{1}\vee\Sigma i_{2}\colon\Sigma X_{1}\vee\Sigma X_{2}\to\Sigma X_{1}\times\Sigma X_{2}$. Then it suffices to show $(\Sigma i_{1}\vee\Sigma i_{2})\circ r^{\prime}\simeq(\Sigma i_{1}\vee\Sigma i_{2})\circ(\Sigma p_{1}+\Sigma p_{2})$, or equivalently, $(\Sigma i_{1}\vee\Sigma i_{2})\circ(\Sigma p_{1}+\Sigma p_{2})\circ h\simeq*$. Now $(\Sigma i_{1}\vee\Sigma i_{2})\circ(\Sigma p_{1}+\Sigma p_{2})\circ h=\Sigma(i_{1}\circ p_{1})\circ h+\Sigma(i_{2}\circ p_{2})\circ h$. On the other hand, $\Sigma p_{k}\circ h\simeq*$ for $k=1,2$ by the definition of the Hopf construction. Thus $(\Sigma i_{1}\vee\Sigma i_{2})\circ(\Sigma p_{1}+\Sigma p_{2})\circ h\simeq*$ as desired. ∎

Proposition 2.4**.**

Let $X_{1},\ldots,X_{l}$ be $A_{n}$-spaces, $Y$ be a topological monoid, and $f\colon X_{1}\times\cdots\times X_{l}\to Y$be a map such that the restriction $f|_{X_{i}}$is an $A_{n}$-map for each $i$. Then $f$ is an $A_{n}$-map if and only if there is a map $\widehat{P}^{n}(X_{1},\ldots,X_{l})\to BY$ satisfying a homotopy commutative diagram

[TABLE]

where $\hat{f}$ is the restriction of $\bar{f}$.

Proof.

We first consider the case $n=2$. The map $f$ is an $A_{2}$-map, that is, an H-map if and only if the Samelson products $\langle f|_{X_{i}},f|_{X_{j}}\rangle$ are trivial for all $i\neq j$. By the adjointness of Samelson products and Whitehead products, this is equivalent to that $\widehat{f}$ extends to a map

[TABLE]

Thus since each $f|_{X_{i}}$ is an H-map, such an extension exists if and only if $\hat{f}$ extends to $\widehat{P}^{2}(X_{1},\ldots,X_{l})\to BY$.

We next consider the case $n\geq 3$. By the $n=2$ case, we may assume that $f$ is an H-map. Then by Lemma 2.3, there is a homotopy commutative diagram

[TABLE]

where the composite of the upper row is $\bar{f}$ and the composite of the lower row is $\hat{f}$. Now suppose that there is a map $\widehat{P}^{n}(X_{1},\ldots,X_{l})\to BY$ extending $\hat{f}$. Then it follows from Lemmas 2.2 and 2.3 together with (2.1) that there is a homotopy commutative diagram

[TABLE]

and thus by Lemma 2.1 $f$ is an $A_{n}$-map.

If $f$ is an $A_{n}$-map then by Lemma 2.1 there is a map $P^{n}(X_{1},\cdots,X_{l})\to BY$ extending $\bar{f}$. Thus by Lemma 2.2 one gets the desired map $\widehat{P}(X_{1},\ldots,X_{l})\to BY$. Therefore the proof is complete. ∎

Hereafter let $p$ be an odd prime and we localize at $p$. Let $G$ be a connected Lie group. By the classical result of Hopf, the rational cohomology of $G$ is an exterior algebra generated by odd degree elements. If generators are in dimensions $2n_{1}-1,\ldots,2n_{r}-1$ for $n_{1}\leq\cdots\leq n_{r}$ then we say that the type of $G$ is $(n_{1},\ldots,n_{r})$. Recall that $G$ is called $p$-regular if it is homotopy equivalent to a product of spheres such that

[TABLE]

It is well known that $G$ is $p$-regular if $p>n_{r}$. On the other hand, it is also well known that any odd sphere is an $A_{p-1}$-space. A cell decomposition of the projective spaces of odd sphres is given in [8] as follows.

Lemma 2.5**.**

For $l\leq p-1$ there is a $p$-local cell decomposition

[TABLE]

The following proposition is proved in [8], which shows an intrinsic feature of the decomposition (2.2) with respect to higher homotopy associativity.

Proposition 2.6**.**

Let $G$ be a connected Lie group of type $(n_{1},\ldots,n_{r})$. If $p>kn_{r}$, implying $G$ is $p$-regular, then the product $A_{k}$-structure on $S^{2n_{1}-1}\times\cdots\times S^{2n_{r}-1}$ and the standard $A_{k}$-structure on $G$ are equivalent.

We slightly improve this proposition in our setting. Let $(G,H)$ and $p$ be as in Theorem 1.3 and suppose $G$ is $p$-regular. Then $G/H$ is a product of odd spheres, and if

[TABLE]

for $l_{1}\leq\cdots\leq l_{t}$ then we say that $G/H$ has type $(l_{1},\ldots,l_{t})$. Hereafter, let $(n_{1},\ldots,n_{r})$, $(m_{1},\ldots,m_{s})$ and $(l_{1},\ldots,l_{t})$ be the types of $G,\,H$ and $G/H$, respectively. Then $(m_{1},\ldots,m_{s})$ and $(l_{1},\ldots,l_{t})$ are subsequences of $(n_{1},\ldots,n_{r})$.

Proposition 2.7**.**

Let $(G,H)$ and $p$ be as in Theorem 1.3. If $p\geq b_{k}$ with $k\geq 2$ then the product $A_{k}$-structure on $H\times S^{2l_{1}-1}\times\cdots\times S^{2l_{t}-1}$ and the standard $A_{k}$-structure on $G$ are equivalent.

Proof.

Note that $G$ is $p$-regular for $p\geq b_{k}$. By Lemma 2.1 and Proposition 2.4 it suffices to show that the natural map $\Sigma(H\vee S^{2l_{1}-1}\vee\cdots\vee S^{2l_{t}-1})\to BG$ extends to a map $\widehat{P}^{k}(H,S^{2l_{1}-1},\ldots,S^{2l_{t}-1})\to BG$. Let

[TABLE]

We first construct an extension $X\to BG$ for $p\geq b_{k}$. Let $b_{k}=\max\{(k-1)m_{s}+l_{t},kl_{t}\}$. By Lemma 2.5, $X$ consists of even dimensional cells and $\dim X\leq 2b_{k}$. Since $G$ is $p$-regular, its homotopy groups can be calculated from those of spheres in [24]. In particular, $\pi_{2i-1}(BG)=0$ for $i\leq p$, and so we get an extension $X\to BG$ for $p\geq b_{k}$.

We next construct an extension $\widehat{P}^{k}(H,S^{2l_{1}-1},\ldots,S^{2l_{t}-1})\to BG$ from an extension $X\to BG$. Let

[TABLE]

By construction, we may assume that the restriction of an extension $X\to BG$ to $Y$ decomposes as $Y\xrightarrow{f}BH\to BG$, where the second map is the induced map from the inclusion $H\to G$. By Proposition 2.6 there is a map $g\colon P^{k-1}H\to Y$ such that the composite $f\circ g\colon P^{k-1}H\to BH$ restricts to the canonical map $\Sigma H\to BH$. Then $f\circ g$ gives the standard $A_{k-1}$-structure of the identity map of $H$, and so by [25] there is an $A_{k-1}$-map $h\colon H\to H$ such that $f\circ g\circ P^{k-1}h$ is homotopic to the canonical map $j_{k-1}\colon P^{k-1}H\to BH$, where $P^{k-1}h\colon P^{k-1}H\to P^{k-1}H$ is the induced map from $h$. Then the composite

[TABLE]

restricts to the canonical map $j_{k-1}\colon P^{k-1}H\to BG$. Since the canonical map $j_{k-1}$ extends to $j_{k}\colon P^{k}H\to BG$, we finally obtain the desired extension $\widehat{P}^{k}(H,S^{2l_{1}-1},\ldots,S^{2l_{t}-1})\to BG$, completing the proof. ∎

Now we prove Theorem 1.3.

Proof of Theorem 1.2.

By Lemma 2.1 and Proposition 2.7 it suffices to show that the adjoint $\bar{q}\colon\Sigma G\to BH$ of the projection $q\colon G\to H$ extends to $\widehat{P}^{k}(H,S^{2l_{1}-1},\ldots,S^{2l_{t}-1})\to BH$. Note that $q\colon G\to H$ is identified with the projection

[TABLE]

Then there is a homotopy commutative diagram

[TABLE]

where the bottom map is the projection. Thus since the composite $\Sigma G\xrightarrow{\Sigma q}\Sigma H\to BH$ is $\bar{q}$, the composite

[TABLE]

is the desired extension of $\bar{q}$. ∎

3. Cohomology calculation

For the rest of the paper, cohomology is assumed to be with mod $p$ coefficients for an odd prime $p$. This section calculates $\mathcal{P}^{1}x$ for some cohomology classes $x$ of the classifying spaces of Lie groups, which we are going to use. We first consider $SU(n)$. The cohomology of $BSU(n)$ is given by

[TABLE]

where $c_{i}$ is the Chern class. In [21], $\mathcal{P}^{i}c_{k}$ is determined, and in particular,

[TABLE]

If a polynomial $P$ includes a monomial $M$ then we write $P\geq M$. By (3.2) one gets:

Lemma 3.1**.**

Let $k\geq 2$.

1. (1)

In $H^{*}(BSU(2n+1))$,

[TABLE] 2. (2)

In $H^{*}(BSU(2n))$ with $k,n\geq 2$,

[TABLE]

We next consider $SO(2n+1)$. The cohomology of $BSO(2n+1)$ is given by

[TABLE]

where $p_{i}$ is the Pontrjagin class. The inclusion $c\colon SO(2n+1)\to SU(2n+1)$ satisfies $c^{*}(c_{2k})=(-1)^{k}p_{k}$ and $c^{*}(c_{2k+1})=0$, and so by (3.2),

[TABLE]

Thus by focusing on $BSO(15)$, one gets the following.

Lemma 3.2**.**

*Let $k\geq 2$. In $H^{*}(BSO(15))$ the following hold: *

[TABLE]

The cohomology of $BSO(2n)$ is given by

[TABLE]

where $p_{i}$ is the Pontrjagin class and $e_{n}$ is the Euler class. Then the inclusion $j\colon SO(2n)\to SO(2n+1)$ satisfies $j^{*}(p_{i})=p_{i}$ for $i=1,\ldots,n-1$ and $j^{*}(p_{n})=e_{n}^{2}$. Thus by (3.4) one gets:

Lemma 3.3**.**

In $H^{*}(BSO(2n))$, for $2(k-2)(n-1)+2\leq p<2(k-1)(n-1)+2$ with $k\geq 3$

[TABLE]

and for $p\leq 2n-1$

[TABLE]

Lemma 3.4**.**

Let $k\geq 2$. In $H^{*}(BSO(8))$ the following hold:

[TABLE]

We finally consider the exceptional Lie groups. Let $p$ be a prime $>5$. The cohomology of $BE_{8}$ is given by

[TABLE]

Let $j_{2}\colon Spin(15)\to E_{8}$ be the canonical inclusion. It is shown in [6] that the generators $x_{i}$ can be chosen such that

[TABLE]

where $BSpin(15)\simeq BSO(15)$ since we are localizing at an odd prime. Then in particular, $x_{60}$ can be chosen such that $j_{2}^{*}(x_{60})$ does not include the monomial $ap_{7}p_{6}p_{2}$ for $a\in\mathbb{Z}/p$. Thus by Lemma 3.2 and a degree reason, one gets:

Lemma 3.5**.**

Let $p$ be a prime $>5$ and $k\geq 2$. In $H^{*}(BE_{8})$,

[TABLE]

There is a commutative square of inclusions

[TABLE]

Let $p$ be a prime $>5$. The cohomology of $BE_{6}$ and $BSO(2m)$ are given by

[TABLE]

It is shown in [6] that $x_{10}$ and $x_{18}$ are chosen as

[TABLE]

and that $i_{2}^{*}(x_{i})=x_{i}$ for $i=4,12,16,24$. Then by the choice of the generators $x_{i}$ of $H^{*}(BE_{8})$ and the commutative diagram (3.5),

[TABLE]

Thus by Lemma 3.5 one finally obtains the following.

Corollary 3.6**.**

Let $p$ be a prime $>5$ and $k\geq 2$. In $H^{*}(BE_{6})$,

[TABLE]

4. Proof of the main theorem

Let $\langle a,b\rangle$ denote the Samelson product of $a,b\in\pi_{*}(X)$ for an H-group $X$. We will need the following lemma.

Lemma 4.1**.**

Let $f\colon X\to Y$ be a map between H-groups. Suppose there are $a,b\in\pi_{*}(X)$ such that $f_{*}(a)=0$ and $f_{*}(\langle a,b\rangle)\neq 0$. Then $f$ is not an H-map.

Proof.

If $f$ is an H-map then by the naturality of Samelson products

[TABLE]

which is a contradiction. Thus $f$ is not an H-map. ∎

All Samelson products that we need to apply Lemma 4.1 are calculated in [1, 5] except for the case $(G,H)=(Spin(8),G_{2})$ and $p=2$. Then we calculate a certain Samelson product in $Spin(8)$ for $p=2$. Recall that the fibration $Spin(7)\to Spin(8)\to S^{7}$ is trivial such that $Spin(8)=Spin(7)\times S^{7}$. Let $\partial\colon\pi_{*+1}(S^{8})\to\pi_{*}(Spin(8))$ be the connecting map of the homotopy exact sequence of a fibration $Spin(8)\to Spin(9)\to S^{8}$. We wll freely use the notation of the homotopy groups of spheres in [24].

Lemma 4.2**.**

Let $p=2$ and $[\iota_{7}]\colon S^{7}\to Spin(7)\times S^{7}=Spin(8)$ be the inclusion. Then

[TABLE]

Proof.

Consider a commutative diagram with fibration rows and columns

[TABLE]

Then in the homotopy exact sequence of the middle row

[TABLE]

one has $\partial(\iota_{8})=[\iota_{7}]$, where $\iota_{8}$ is the identity map of $S^{8}$. Thus by the adjointness of Whitehead products and Samelson products, $\partial([\iota_{8},\iota_{8}])=\langle[\iota_{7}],[\iota_{7}]\rangle$. On the other hand, it is shown in [24, p. 50] that $\pi_{15}(S^{8})=\mathbb{Z}\{\sigma_{8}\}\oplus\mathbb{Z}/8\{\Sigma\sigma^{\prime}\}$ and $[\iota_{8},\iota_{8}]=\pm(2\sigma_{8}-\Sigma\sigma^{\prime})$, where for a cyclic group $A$, $A\{x\}$ is a cyclic group generated by $x$ which is isomorphic to $A$. Thus one gets the desired equality. ∎

There is a commutative diagram of inclusions

[TABLE]

It is shown in [11] that

[TABLE]

Then since $Spin(8)=Spin(7)\times S^{7}$,

[TABLE]

where $\pi_{14}(S^{7})=\mathbb{Z}/8\{\sigma^{\prime}\}$. The following is proved in [11].

Lemma 4.3**.**

For an odd integer $u_{1}$ and an integer $u_{2}$,

[TABLE]

To proceed the calculation, we relate the above generators of $\pi_{14}(Spin(8))$ with $G_{2}$ in $Spin(8)$. As in [16],

[TABLE]

Since $\pi_{6}(G_{2})=0$ as in [16], the fibration $G_{2}\xrightarrow{i}Spin(7)\to S^{7}$ splits such that $Spin(7)\simeq G_{2}\times S^{7}$. Then

[TABLE]

where $[\iota_{7}^{\prime}]\colon S^{7}\to G_{2}\times S^{7}\simeq Spin(7)$ is the inclusion. In particular, for some integers $s_{1},s_{2},s_{3}$,

[TABLE]

Lemma 4.4**.**

We may choose $[\nu_{5}]_{7}\in\pi_{8}(Spin(7))$ and $[\nu_{5}^{2}]\in\pi_{11}(SU(3))$ such that

[TABLE]

Proof.

Consider a commutative diagram with fibration rows

[TABLE]

As in [11], there is $x\in\pi_{8}(Spin(6))$ such that $\pi^{\prime}_{*}(x)=\nu_{5}$, and $[\nu_{5}]_{7}\in\pi_{8}(Spin(7))$ is chosen as $[\nu_{5}]_{7}=j^{\prime}_{*}(x)$, where $\pi_{n+3}(S^{n})=\mathbb{Z}/8\{\nu_{n}\}$ for $n\geq 5$. On the other hand, in [16], $[\nu_{5}^{2}]\in\pi_{11}(SU(3))$ is chosen to be any $y\in\pi_{11}(SU(3))$ satisfying $\bar{\pi}_{*}(y)=\nu_{5}^{2}$. Since $Spin(6)/SU(3)=S^{7}$ and $\pi_{k}(S^{7})=0$ for $k=11,12$, $i^{\prime}$ is an isomorphism in $\pi_{11}$. Then since $\pi^{\prime}_{*}(x\nu_{8})=\nu_{5}^{2}$, we may put $[\nu_{5}^{2}]=(i^{\prime}_{*})^{-1}(x\nu_{8})$. Thus

[TABLE]

as desired. ∎

Lemma 4.5**.**

Let $q\colon Spin(8)\to G_{2}$ be the projection

[TABLE]

Then $q_{*}(\langle[\iota_{7}],[\iota_{7}]\rangle)\neq 0$.

Proof.

Consider a commutative diagram with fibration rows

[TABLE]

In [16, 11], $\langle\bar{\nu}_{6}+\epsilon_{6}\rangle$ and $[\bar{\nu}_{6}+\epsilon_{6}]$ are chosen such that $\pi_{*}(\langle\bar{\nu}_{6}+\epsilon_{6}\rangle)=\pi^{\prime\prime}_{*}([\bar{\nu}_{6}+\epsilon_{6}])=\bar{\nu}_{6}+\epsilon_{6}$, where $\pi_{14}(S^{6})=\mathbb{Z}/8\{\bar{\nu}_{6}\}\oplus\mathbb{Z}/2\{\epsilon_{6}\}$. Then by Lemma 4.4,

[TABLE]

for some integers $t_{1},t_{2}$. Thus by (4.1) and Lemmas 4.2, 4.3, and 4.4,

[TABLE]

Since $q\circ j^{\prime\prime}\circ j\simeq 1$ and $q_{*}(j^{\prime\prime}_{*}([\iota_{7}^{\prime}]\sigma^{\prime}))=0$,

[TABLE]

and so it suffices to show $s_{1}\equiv 4\mod 8$.

By the definition of $[\eta_{5}\epsilon_{6}]_{7}$ in [11], $\pi^{\prime\prime}_{*}([\eta_{5}\epsilon_{6}]_{7})=0$, and so $s_{1}(\bar{\nu}_{6}+\epsilon_{6})+s_{3}\pi^{\prime\prime}_{*}([\iota_{7}^{\prime}]\sigma^{\prime})=0$. Consider a homotopy exact sequence of the lower fibration of (4.2). Since $Spin(6)=SU(4)$ and $\pi_{6}(SU(6))\cong\pi_{6}(SU(\infty))=0$, one gets $\pi_{6}(Spin(6))=\pi_{6}(SU(4))=0$, and so $\pi^{\prime\prime}_{*}\colon\pi_{7}(Spin(7))\to\pi_{7}(S^{6})$ is surjective. By [24] and [16], $\pi_{7}(S^{6})=\mathbb{Z}/2\{\eta_{6}\}$ and $\pi_{7}(Spin(7))=\mathbb{Z}\{[\iota^{\prime}_{7}]\}$, implying $\pi^{\prime\prime}_{*}([\iota^{\prime}_{7}])=\eta_{6}$. Thus $s_{1}(\bar{\nu}_{6}+\epsilon_{6})+s_{3}\eta_{6}\sigma^{\prime}=0$. Now by [24, p. 64], $\eta_{6}\sigma^{\prime}=4\bar{\nu}_{6}$, and so $(s_{1}+4s_{3})\bar{\nu}_{6}+s_{1}\epsilon_{6}=0$. Then $s_{1}$ is even and $s_{1}+4s_{3}\equiv 0\mod 8$. On the other hand, since $[\eta_{5}\epsilon_{6}]_{7}$ has order 8, either $s_{1}$ or $s_{3}$ must be odd. Then $s_{3}$ is odd, implying $s_{1}\equiv 4\mod 8$ as desired. ∎

Proposition 4.6**.**

Let $(G,H),\,a_{k}$ and $p$ be as in Theorem 1.2. If $p<a_{1}$ then the projection $q\colon G\to H$ is not an H-map.

Proof.

Let $(G,H)=(SU(2n+1),SO(2n+1))$. Then $p<2n+1$. Since $SO(2n+1)$ is a direct summand of $SU(2n+1)$, $\pi_{*}(SO(2n+1))$ is a direct summand of $\pi_{*}(SU(2n+1))$ as well. As in [17] $\pi_{4n+2}(SU(2n+1))$ is a cyclic group. Since $SO(2n+1)\simeq Sp(n)$ where we are localizing at an odd prime $p$, $\pi_{4n+2}(SO(2n+1))\neq 0$ as in [17]. Then the projection $q$ induces an isomorphism in $\pi_{4n+2}$. Let $\epsilon_{i}$ be a generator of $\pi_{2i+1}(SU(2n+1))\cong\mathbb{Z}_{(p)}$ for $i=1,\ldots,2n-1$. By [1] that the Samelson product $\langle\epsilon_{2n-p+1},\epsilon_{p-1}\rangle$ in $SU(2n+1)$ is non-trivial. Then in particular, since $q$ is an isomorphism in $\pi_{4n+2}$, $q_{*}(\langle\epsilon_{2n-p+1},\epsilon_{p-1}\rangle)\neq 0$. On the other hand, since $\pi_{2p-1}(SO(2n+1))=0$ for $p<2n+1$, $q_{*}(\epsilon_{p-1})=0$. Thus the proof is done by Lemma 4.1.

The case $(G,H)=(SU(2n),Sp(n))$ follows from the same argument using $\langle\epsilon_{2n-p+1},\epsilon_{p-1}\rangle$ in $SU(2n)$, where $p<2n-1$. There is nothing to do for $(G,H)=(SO(2n),SO(2n-1))$ since $a_{1}=1$.

Let $(G,H)=(E_{6},F_{4})$. Then $p=5$, and so

[TABLE]

as in [17] such that the projection $q\colon E_{6}\to F_{4}$ is identified with the projection $F_{4}\times B(9,17)\to F_{4}$, where $B(9,17)$ is a certain $S^{9}$-bundle over $S^{17}$. Let $\epsilon\colon B(9,17)\to E_{6}$ be the inclusion. In [5] it is shown that the Samelson product $\langle\epsilon,\epsilon\rangle$ is non-trivial, and its proof actually shows that $\langle\epsilon|_{S^{9}},\epsilon|_{S^{9}}\rangle$ is non-trivial, where $\epsilon|_{S^{9}}$ is the restriction of $\epsilon$ to the bottom cell $S^{9}\subset B(9,17)$. The homotopy groups of $B(9,17)$ are calculated in [13] such that $\pi_{18}(B(9,17))=0$. Then $q_{*}(\langle\epsilon|_{S^{9}},\epsilon|_{S^{9}}\rangle)\neq 0$. On the other hand, $q_{*}(\epsilon|_{S^{9}})=0$, and so by Lemma 4.1 $q$ is not an H-map.

Let $(G,H)=(Spin(8),G_{2})$. Then $p=2$. By [16], the inclusion $G_{2}\to Spin(8)$ is injective in $\pi_{14}$. Then by Lemmas 4.1 and 4.5, $q$ is not an H-map. Thus the proof is complete. ∎

We set notation on cohomology. Let $G$ be a connected Lie group whose integral homology has no $p$-torsion. Recall from [9] that there is an isomorphism of unstable algebras

[TABLE]

for some unstable algebra $S$ depending on $G$ and $k$, where the natural map $j_{k}\colon P^{k}G\to BG$ induces the obvious projection in cohomology. Let $(G,H)$ and $p$ be in Theorem 1.2. The cohomology of $BG$ and $BH$ are given by

[TABLE]

where $|x_{i}|=|y_{i}|=2i$.

If $m_{i}$ in $(m_{1},\ldots,m_{s})$ and $n_{j}$ in $(n_{1},\ldots,n_{r})$ are equal as a member of the sequence $(n_{1},\ldots,n_{r})$, then we say that $m_{i}=n_{j}$ in the type of $G$, where $(m_{1},\ldots,m_{s})$ is a subsequence of $(n_{1},\ldots,n_{r})$. Now we state a cohomological criterion for the projection $G\to H$ not being an $A_{l}$-map.

Lemma 4.7**.**

Let $(G,H)$ and $p$ be as in Theorem 1.2, and let $j\colon H\to G$ be the inclusion. Suppose that presentations (4.4) satisfy

[TABLE]

Suppose also that there are an integer $m_{k}$ in the type of $H$ and a monomial $M=ax_{n_{i_{1}}}\cdots x_{n_{i_{l}}}\in H^{*}(BG)$ with $a\neq 0$ and $n_{i_{1}}\leq\cdots\leq n_{i_{l}}$ satisfying the following conditions:

1. (1)

$n_{i_{j}}>m_{k}$* for $j\geq 2$ and $m_{k}-n_{i_{1}}$ is not a sum of integers in the type of $G$;* 2. (2)

for any map $f\colon H^{*}(BH)\to H^{*}(BG)$ satisfying $f(y_{m_{i}})\equiv\pm x_{m_{i}}\mod\widetilde{H}^{*}(BG)^{2}$ for all $i$, there is no polynomial $P\in\mathrm{Im}\,f$ such that

[TABLE]

where $m_{k}=n_{h}$ in the type of $G$ and $I=(x_{n_{1}},\ldots,\widehat{x_{n_{i_{1}}}},\ldots,x_{n_{h}})$.

Then the projection $q\colon G\to H$ is not an $A_{l}$-map.

Proof.

We assume $q$ is an $A_{l}$-map and show a contradiction. By Lemma 2.1 the adjoint $\bar{q}\colon\Sigma G\to BH$ extends to $\bar{q}_{l}\colon P^{l}G\to BH$. Let $S$ be as in (4.3). By the condition (1),

[TABLE]

Then by the Cartan formula,

[TABLE]

Let $f\colon H^{*}(BH)\to H^{*}(BG)$ be the composite of a lift of $\bar{q}_{l}^{*}\colon H^{*}(BH)\to H^{*}(P^{l}G)$ through the projection $H^{*}(BG)\oplus S\to H^{*}(P^{l}G)$ under the isomorphism (4.3) and the projection $H^{*}(BG)\oplus S\to H^{*}(BG)$. Then by (4.5), $f(y_{m_{i}})\equiv\pm x_{m_{i}}\mod\widetilde{H}^{*}(BG)^{2}$, and so by condition (2) there is no $P\in\mathrm{Im}\,f$ such that $P\geq M\mod I$. Thus there is no $Q\in\mathrm{Im}\,\bar{q}_{l}^{*}$ such that $Q\geq M\mod I+S$, which contradicts (4.6). Therefore the proof is complete. ∎

Proposition 4.8**.**

Let $(G,H),\,a_{k}$ and $p$ be as in Theorem 1.2, and let $k\geq 2$. Then the following statements hold:

1. (1)

if $a_{k-1}\leq p<a_{k}$ then the projection $G\to H$ is not an $A_{k}$-map except for $(G,H)=(SO(2n),SO(2n-1))$; 2. (2)

if $a_{k-1}-n+2\leq p<a_{k}-n+2$ then the projection $q\colon SO(2n)\to SO(2n-1)$ is not an $A_{k}$-map.

Proof.

(1) Let $(G,H)=(SU(2n+1),SO(2n+1))$. The inclusion $c\colon SO(2n+1)\to SU(2n+1)$ satisfies $c^{*}(c_{2i})=(-1)^{i}p_{i}$ and $c^{*}(c_{2i+1})=0$. Then the presentations (3.1) and (3.3) satisfy the condition in Lemma 4.7. By a dimensional consideration, we see that the monomial $(-1)^{k-1}c_{2n+1}^{k-1}c_{p-(k-1)(2n+1)+1}$ satisfies the condition (2) of Lemma 4.7. Thus by Lemmas 3.1 and 4.7, the projection $q$ is not an $A_{k}$-map for $a_{k-1}\leq p<a_{k}$.

Let $(G,H)=(SU(2n),Sp(n))$. The cohomology of $BSp(n)$ is given by

[TABLE]

where $q_{i}$ is the symplectic Pontrjagin class. Then the inclusion $c^{\prime}\colon Sp(n)\to SU(2n)$ satisfies $(c^{\prime})^{*}(c_{2i})=(-1)^{i}q_{i}$ and $(c^{\prime})^{*}(c_{2i+1})=0$. Thus by Lemmas Lemmas 3.1 and 4.7, $q$ is not an $A_{k}$-map for $a_{k-1}\leq p<a_{k}$.

Let $(G,H)=(E_{6},F_{4})$. The cohomology of $BF_{4}$ is given by

[TABLE]

In [6], it is shown that we may choose generators of $H^{*}(BF_{4})$ such that the inclusion $j\colon F_{4}\to E_{6}$ satisfies $j^{*}(x_{i})=x_{i}$ for $i=4,12,16,24$ and $j^{*}(x_{i})=0$ for $i=10,18$. Then by Corollary 3.6 and Lemma 4.7, $q$ is not an $A_{k}$-map for $a_{k-1}\leq p<a_{k}$ except for $p=7$ with $k=2$. Let $p=7$. We aim to show that $q$ is not an H-map. As in [17], there is a homotopy equivalence

[TABLE]

Let $\epsilon\colon S^{17}\to E_{6}$ be the inclusion. Then by [5] the Samelson product $\langle\epsilon,\epsilon\rangle$ is non-trivial. By [24], $\pi_{34}(S^{9})=\pi_{34}(S^{17})=0$, and so $q_{*}(\langle\epsilon,\epsilon\rangle)\neq 0$. On the other hand, $q_{*}(\epsilon)=0$. Thus by Lemma 4.1, $q$ is not an H-map.

Let $(G,H)=(Spin(8),G_{2})$ and $j\colon G_{2}\to Spin(8)$ be the inclusion. Since $BSO(8)\simeq BSpin(8)$ at the odd prime $p$,

[TABLE]

The cohomology of $BG_{2}$ is given by

[TABLE]

and so by a degree reason, we may assume

[TABLE]

Then by Lemmas 3.4 and 4.7, $q$ is not an $A_{k}$-map for $a_{k-1}\leq p<a_{k}$.

(2) Let $(G,H)=(SO(2n),SO(2n-1))$. The inclusion $j\colon SO(2n-1)\to SO(2n)$ satisfies $j^{*}(p_{i})=p_{i}$ and $j^{*}(e_{n})=0$. Then by Lemmas 3.3 and 4.7, one gets that $q$ is not an $A_{k}$-map for $a_{k-1}-n+2\leq p<a_{k}-n+2$ unless $p=n$ for $k=2$. Let $p=n$ and suppose $q$ is an H-map. Let $\theta\colon S^{2n-1}\to S^{2n-1}\times SO(2n-1)\simeq SO(2n)$ be the inclusion. In [15] it is shown that the Samelson product $\langle\theta,\theta\rangle$ in $SO(2n)$ is non-trivial. Then since $\pi_{4n-2}(S^{2n-1})=0$ as in [24], $q_{*}(\langle\theta,\theta\rangle)$ is non-trivial. But since $q_{*}(\theta)=0$, we obtain a contradiction by Lemma 4.1. Thus $q$ is not an H-map. ∎

Proof of Theorem 1.2.

The proof is done by Theorem 1.3 and Propositions 4.6 and 4.8 except that $q\colon E_{6}\to F_{4}$ is an $A_{k}$-map for $p=12k-5$. Let $p=12k-5$. From the proofs of Theorem 1.3 and Proposition 2.7, one can see that it suffices to show that the adjoint of the identity map $\Sigma E_{6}\to BE_{6}$ extends to $X\to BE_{6}$, where $X$ is as in the proof of Proposition 2.7. Note that $X=X^{(24k-12)}\cup e^{24k-6}$. Then since $\pi_{2i-1}(BE_{6})=0$ for $i\leq 12k-6$ and $12k-3$ by [24], there is an extension $X\to BE_{6}$ as desired. Thus the proof is complete. ∎

Proof of Corollary 1.5.

The equivalence of (2) and (3) is proved in [8], and Saumell [20] proved that $SU(2n+1)$ is a Williams $C_{k}$-space if and only if $p>k(2n+1)$. Thus the result follows from Theorem 1.2. ∎