A formula on Stirling numbers of the second kind and its application to the unstable $K$-theory of stunted complex projective spaces

TL;DR

This paper proves a new formula for Stirling numbers of the second kind and applies it to derive algebraic topology results related to the unstable K-theory of stunted complex projective spaces.

Contribution

It introduces a novel formula for Stirling numbers and applies it to establish isomorphisms in unstable K-theory of stunted complex projective spaces.

Findings

Proved a new formula for Stirling numbers of the second kind.

Showed divisibility properties of $k!S(n, k)$ for odd $n$ and even $k$.

Derived isomorphisms of unstable $K^1$-groups for certain spaces.

Abstract

A formula on Stirling numbers of the second kind $S(n, k)$ is proved. As a corollary, for odd $n$ and even $k$, it is shown that $k!S(n, k)$ is a positive multiple of the greatest common divisor of $j!S(n, j)$ for $k+1\leq j\leq n$. Also, as an application to algebraic topology, some isomorphisms of unstable $K^1$-groups of stunted complex projective spaces are deduced.

Equations

$$R(m)=\frac{2(2^{2m+2}-1)B_{2m+2}}{m+1}$$

$$\tilde{S}(n,k)=\frac{1}{2}\sum _{h=0}^{(n-k-1)/2}R(h)\left(\genfrac{}{}{0ex}{}{k+2h}{k-1}\right)\tilde{S}(n,k+2h+1)$$

$$\tilde{S}(7,4)$$

$$=\frac{1}{2}\left(1\times 4\times 16800-\frac{1}{2}\times 20\times 5040\right)=8400.$$

$$R(m)$$

$$=\sum _{j=0}^m{\left(-\frac{1}{2}\right)}^j\sum _{(I_0,I_1,\cdots ,I_j)\in {\tilde{O}}_{m,j}}\left(\genfrac{}{}{0ex}{}{2I_1-1}{2I_0-1}\right)\left(\genfrac{}{}{0ex}{}{2I_2-1}{2I_1-1}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j-1}{2I_{j-1}-1}\right)$$

$${\tilde{E}}_{m,j}$$

$${\tilde{O}}_{m,j}$$

$$\left(\genfrac{}{}{0ex}{}{2I_2}{2I_1}\right)\left(\genfrac{}{}{0ex}{}{2I_3}{2I_2}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j}{2I_{j-1}}\right),\left(\genfrac{}{}{0ex}{}{2I_1-1}{2I_0-1}\right)\left(\genfrac{}{}{0ex}{}{2I_2-1}{2I_1-1}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j-1}{2I_{j-1}-1}\right)$$

$$\tilde{S}(n,k)=\sum _{j=k}^n(-1{)}^{n-j}(\genfrac{}{}{0ex}{}{j-1}{k-1})\tilde{S}(n,j).$$

$$\tilde{S}(n,k)=\frac{1}{2}\sum _{j=k+1}^n(-1{)}^{n-j}(\genfrac{}{}{0ex}{}{j-1}{k-1})\tilde{S}(n,j).$$

$${\Delta }_{n,k}={\Delta }_{n,k+1}$$

$${\nu }_p(\tilde{S}(n,k))\ge {\nu }_p({\Delta }_{n,k+1})=\min \{{\nu }_p(\tilde{S}(n,j))|j=k+1,k+2,\cdots ,n\}$$

$$B_{2m+2}^{\prime }=\prod _{p:\text{prime},\frac{2m+2}{p-1}\in \mathbb{Z}}p$$

$$U_n(\mathbb{C}{P}_n^n)\stackrel{q^{\ast }}{\to }{U}_n(\mathbb{C}{P}_{n-1}^n)\stackrel{q^{\ast }}{\to }\cdots \stackrel{q^{\ast }}{\to }{U}_n(\mathbb{C}{P}_3^n)\stackrel{q^{\ast }}{\to }{U}_n(\mathbb{C}{P}_2^n)\stackrel{q^{\ast }}{\to }{U}_n(\mathbb{C}{P}_1^n)=0$$

$$q^{\ast }:U_n(\mathbb{C}{P}_{k+1}^n)\to U_n(\mathbb{C}{P}_k^n)$$

$$(e^z+e^{-z}-2{)}^j=\sum _{m=1}^{\infty }\frac{(2j)!}{(2m)!}M(m,j)z^{2m}$$

$$e^z+e^{-z}-2=2\sum _{m=1}^{\infty }\frac{z^{2m}}{(2m)!}=2\left(\frac{z^2}{2!}+\frac{z^4}{4!}+\cdots \right),$$

$$M(m,j)$$

$$=\frac{2^j}{(2j)!}\sum _{(I_1,\cdots ,I_j)\in {\tilde{E}}_{m,j}}\left(\genfrac{}{}{0ex}{}{2I_2}{2I_1}\right)\left(\genfrac{}{}{0ex}{}{2I_3}{2I_2}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j}{2I_{j-1}}\right)$$

$$E_{m,j}$$

$${\tilde{E}}_{m,j}$$

$$\left(\genfrac{}{}{0ex}{}{2I_2}{2I_1}\right)\left(\genfrac{}{}{0ex}{}{2I_3}{2I_2}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j}{2I_{j-1}}\right)$$

$$R_E(m)$$

$$=(2m)!\sum _{j=1}^m{\left(-\frac{1}{2}\right)}^j\sum _{(i_1,\cdots ,i_j)\in E_{m,j}}\frac{1}{(2i_1)!(2i_2)!\cdots (2i_j)!}$$

$$=\sum _{j=1}^m{\left(-\frac{1}{2}\right)}^j\sum _{(I_1,\cdots ,I_j)\in {\tilde{E}}_{m,j}}\left(\genfrac{}{}{0ex}{}{2I_2}{2I_1}\right)\left(\genfrac{}{}{0ex}{}{2I_3}{2I_2}\right)\cdots \left(\genfrac{}{}{0ex}{}{2I_j}{2I_{j-1}}\right)\in \mathbb{Q}$$

$$R_E(m)=\frac{2(2^{2m+2}-1)B_{2m+2}}{m+1}.$$

$$\sum _{j=1}^{\infty }{\left(-\frac{1}{4}\right)}^j(e^z+e^{-z}-2{)}^j=\sum _{j=1}^{\infty }{\left(-\frac{1}{2}\right)}^j{\left(\frac{z^2}{2!}+\frac{z^4}{4!}+\cdots \right)}^j$$

$$D=\{z\in \mathbb{C}|\left|\frac{1}{4}(e^z+e^{-z}-2)\right|=\left|{\sinh }^2\frac{z}{2}\right|<1\},$$

$$\frac{d}{dz}(\tanh z)=1-{\tanh }^{2}z$$

$$\frac{d}{dz}\left(\tanh \frac{z}{2}\right)=\frac{1}{2}\left(1-{\tanh }^2\frac{z}{2}\right).$$

$$\tanh z=\sum _{m=1}^{\infty }\frac{2^{2m}(2^{2m}-1)B_{2m}}{(2m)!}{z}^{2m-1}$$

$$\tanh \frac{z}{2}=\sum _{m=1}^{\infty }\frac{2(2^{2m}-1)B_{2m}}{(2m)!}{z}^{2m-1}.$$

$$\sum _{j=1}^{\infty }{\left(-\frac{1}{4}\right)}^j(e^z+e^{-z}-2{)}^j$$

Full text

1 Introduction

Let $S(n,k)$ be the Stirling number of the second kind, which is the number of partitions of a set consisting of $n$ elements into $k$ pairwise disjoint non-empty subsets, and let $\tilde{S}(n,k)=k!S(n,k)$. Let $R$ be the function defined on non-negative integers by

[TABLE]

where $B_{s}$ is the $s$th Bernoulli number. The main purpose of this paper is to show the following theorem.

Theorem 1.1**.**

For any odd integer $n\geq 3$ and any positive even integer $k$ such that $k<n$, the equation

[TABLE]

holds.

For example, we have

[TABLE]

For this purpose, first we investigate properties of the function $R$, which we call the recurrence weight (of Stirling numbers of the second kind) in this paper. It has representations

[TABLE]

for any positive integer $m$ where

[TABLE]

and

[TABLE]

are to be interpreted as $1$ for $j=1$ and for $j=0$ respectively. Thus, the recurrence weight $R$ also has combinatorial nature, from which some simple recurrence formulae are derived. We also describe recurrence formulae for Bernoulli numbers $B_{2m+2}$ which are derived from those for the recurrence weight $R$, because they may be of independent interest.

Next, we consider the identity

[TABLE]

(See Quaintance and Gould [20, (9.18)].) In particular, if $n$ and $k$ are as in Theorem 1.1, we have

[TABLE]

Then, we can show Theorem 1.1 by induction and by properties of the recurrence weight $R$.

Let $\Delta_{n,k}$ be the greatest common divisor of $\tilde{S}(n,j)$ for $j=k,k+1,\cdots,n$. (See Lundell [19] for $\Delta_{n,k}$ where $k\leq 12$.) Then, we have the following corollary of Theorem 1.1.

Corollary 1.2**.**

For any odd integer $n\geq 3$ and any positive even integer $k$ such that $k<n$, the equation

[TABLE]

holds. In other words, $\tilde{S}(n,k)$ is a positive multiple of $\Delta_{n,k+1}$.

To prove Corollary 1.2, it suffices to show that

[TABLE]

for any prime $p$, where $\nu_{p}(i)$ denotes the power of $p$ in the factorization of a non-zero integer $i$ into prime powers. By (1), for any odd prime $p$, it is immediate that $\nu_{p}(\tilde{S}(n,k))\geq\nu_{p}(\Delta_{n,k+1})$. For $p=2$, we invoke the theorem of Staudt [21] and Clausen [6] which states that

[TABLE]

where $B^{\prime}_{s}$ is the denominator of $B_{s}$ when the fraction is expressed in its lowest terms. Then, by Theorem 1.1, we can estimate $\nu_{2}(\tilde{S}(n,k))$ so that $\nu_{2}(\tilde{S}(n,k))\geq\nu_{2}(\Delta_{n,k+1})$. (See section 4 for details.)

It is known that the numbers $\Delta_{n,k}$ are important in algebraic topology. (See for example [4, 7, 8, 9, 10, 18, 19].) We have an application of Corollary 1.2 which concerns the unstable $K$-theory of stunted complex projective spaces and which is the motivation of this paper. Let $\mathbb{C}P^{n}$ be the complex projective space of complex lines in $\mathbb{C}^{n+1}$. We abbreviate the stunted complex projective space $\mathbb{C}P^{n}/\mathbb{C}P^{k-1}$ by $\mathbb{C}P^{n}_{k}$. The set $U_{n}(X)=[X,U(n)]_{*}$ of pointed homotopy classes of pointed maps from a topological space $X$ to the unitary group $U(n)$ of rank $n$ has a group structure induced by the canonical multiplication of $U(n)$. (See Hamanaka and Kono [12].) Following [12], we call $U_{n}(X)$ the unstable $K^{1}$-group of $X$. It is known that $U_{n}(X)$ is useful for studying homotopy types of certain gauge groups. (See for example Hamanaka and Kono [13, 14].) Using the result of [12], we can see that $U_{n}(\mathbb{C}P^{n}_{k})$ is a cyclic group of order $\Delta_{n,k}$ and that we have a sequence of epimorphisms of groups

[TABLE]

where $q=q_{j,j+1}\colon\mathbb{C}P^{n}_{j}\to\mathbb{C}P^{n}_{j+1}$ is the natural projection for $j=1,2,\cdots,n-1$ and $q^{*}$ is the homomorphism induced by $q$. (See Proposition 5.1.) Then, by Corollary 1.2, we have the following theorem.

Theorem 1.3**.**

For any odd integer $n\geq 3$ and any positive even integer $k$ such that $k<n$,

[TABLE]

is an isomorphism of groups.

This paper is organized as follows. In section 2, we prepare for the recurrence weight $R$ stated above. In section 3, we prove Theorem 1.1. In section 4, we prove Corollary 1.2. In section 5, we prove Theorem 1.3 and give some corollaries and examples. Moreover, in terms of the sequence (2), we give a characterization of the complex James number $b_{k}$ of Stiefel manifolds (see James [15] and Atiyah [2]) which is the same as the Atiyah-Todd number $M_{k}$ (see Atiyah and Todd [3]) by the result of Adams and Walker [1]. (See Proposition 5.5.)

2 Definition and some properties of the recurrence weight $R$

In this section, we define a function $R$, which we call the recurrence weight, and investigate its some properties.

For any positive integers $m$ and $j$, let $M(m,j)$ be the central factorial number of the second kind which is defined by

[TABLE]

for $z\in\mathbb{C}$. Since

[TABLE]

$M(m,j)$ is explicitly given by

[TABLE]

where

[TABLE]

and

[TABLE]

is to be interpreted as $1$ for $j=1$.

Let $R_{E}$ be the function defined on non-negative integers by

[TABLE]

for any positive integer $m$ and by $R_{E}(0)=1$.

Lemma 2.1**.**

We have

[TABLE]

Proof.

By (3), we have

[TABLE]

which is analytic on

[TABLE]

a neighborhood of [math]. Here recall that

[TABLE]

and hence,

[TABLE]

Also recall that the Maclaurin series of $\tanh z$ is given by

[TABLE]

and hence,

[TABLE]

Then, on one hand, we have

[TABLE]

On the other hand, we have

[TABLE]

Hence, we have

[TABLE]

as desired. ∎

Similarly, let $R_{O}$ be the function defined on non-negative integers by

[TABLE]

for any non-negative integer $m$ where

[TABLE]

and

[TABLE]

is to be interpreted as $1$ for $j=0$.

Lemma 2.2**.**

We have

[TABLE]

Proof.

By

[TABLE]

and

[TABLE]

for $z\in\mathbb{C}$, we have

[TABLE]

which is analytic on $D$. Then, on one hand, we have

[TABLE]

On the other hand, we have

[TABLE]

Hence, we have

[TABLE]

as desired. ∎

By Lemma 2.1 and Lemma 2.2, we have

Corollary 2.3**.**

The functions $R_{E}$ and $R_{O}$ coincide. In fact,

[TABLE]

In the following, we abbreviate $R=R_{E}=R_{O}$. As stated in the introduction, we call $R$ the recurrence weight. Here we give two fundamental recurrence formulae for $R$, of which one is derived from the definition of $R_{E}$ and the other from the definition of $R_{O}$.

Proposition 2.4**.**

The recurrence weight $R=R_{E}$ satisfies the relation

[TABLE]

for any non-negative integer $m$.

Proof.

We have

[TABLE]

Here note that $\tilde{E}_{m+1,j+1}$ is the disjoint union of

[TABLE]

for $h=j,j+1,\cdots,m$. Then, we have

[TABLE]

as desired. ∎

Proposition 2.5**.**

The recurrence weight $R=R_{O}$ satisfies the relation

[TABLE]

for any non-negative integer $m$.

Proof.

We have

[TABLE]

Here note that $\tilde{O}_{m+1,j+1}$ is the disjoint union of

[TABLE]

for $h=j,j+1,\cdots,m$. Then, we have

[TABLE]

as desired. ∎

Thus, we can compute the recurrence weight $R$ recursively as

[TABLE]

or as

[TABLE]

In computing $R$ recursively by hand, it seems to be easier to use Proposition 2.4 than to use Proposition 2.5, though we use the latter to prove Theorem 1.1 later.

Corollary 2.6**.**

We have

[TABLE]

and

[TABLE]

for any positive integer $m$.

Proof.

In fact, by Corollary 2.3 and Proposition 2.4, we have

[TABLE]

and hence,

[TABLE]

Thus, the former follows. Similarly, the latter follows from Corollary 2.3 and Proposition 2.5. ∎

The author cannot find these recurrence formulae in the literature. We can deduce other recurrence formulae for the recurrence weight $R$ and then we can transform them to ones for Bernoulli numbers by Corollary 2.3. In the following, for example, we give two other simple recurrence formulae for the recurrence weight $R$.

Proposition 2.7**.**

The recurrence weight $R=R_{E}=R_{O}$ satisfies the relation

[TABLE]

and hence, it satisfies the relation

[TABLE]

for any positive integer $m$.

Proof.

Since we have

[TABLE]

for any non-negative integers $a,b,c$ such that $c\leq b\leq a$, we can see that for $(I_{0},I_{1},\cdots,I_{j})\in\tilde{O}_{m,j}$,

[TABLE]

Thus, we have

[TABLE]

Here let

[TABLE]

Note that as $(I_{0},I_{1},\cdots,I_{j})$ runs over $\tilde{O}_{m,j}$ with $1\leq j\leq m$, $(I^{\prime}_{1},I^{\prime}_{2},\cdots,I^{\prime}_{j})$ bijectively runs over $\tilde{E}_{h,j}$ with $1\leq j\leq h\leq m$. Then, it follows that the above sum is equal to

[TABLE]

Thus we have

[TABLE]

as desired. ∎

Corollary 2.8**.**

We have

[TABLE]

for any positive integer $m$.

Proof.

This follows from Corollary 2.3 and the previous proposition. ∎

Proposition 2.9**.**

The recurrence weight $R$ satisfies the relation

[TABLE]

for any non-negative integer $m$.

Proof.

By Lemma 2.2 and its proof,

[TABLE]

has an analytic continuation

[TABLE]

on

[TABLE]

Then, by

[TABLE]

on $\tilde{D}$, we have

[TABLE]

and hence

[TABLE]

It follows that

[TABLE]

as desired. ∎

The recurrence formula of Bernoulli numbers which is derived from Corollary 2.3 and this proposition is of course merely the one derived from the Maclaurin series (4) of $\tanh\frac{z}{2}$, those of $\cosh\frac{z}{2}$ and $\sinh\frac{z}{2}$, and the relation (6).

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

Recall that Stirling numbers of the second kind $S(n,k)$ satisfy the conditions that

[TABLE]

for any positive integers $n$ and $k$ such that $n<k$, and also satisfy the recurrence formula

[TABLE]

for any positive integers $n$ and $k$ such that $k\leq n$. It is known that Stirling numbers of the second kind $S(n,k)$ satisfy many combinatorial identities which are derived from (7). (See Quaintance and Gould [20].) Here, we use the identity

[TABLE]

for any positive integers $n$ and $k$ such that $k\leq n$. (See Quaintance and Gould [20, (9.18)].) In particular, for any odd integer $n\geq 3$ and any positive even integer $k$ such that $k<n$, we have

[TABLE]

and hence

[TABLE]

Now we prove Theorem 1.1. In the following, we prove the equation

[TABLE]

for any non-negative integer $m$ and any positive even integer $k$, which is a paraphrase of Theorem 1.1, by induction on $m$.

For $m=0$, the assertion is obvious by (8). Suppose that the assertion is proved for $m\leq a$ where $a$ is a non-negative integer. In the following, we abbreviate $\tilde{S}(k+2(a+1)+1,j)=S^{\prime}(j)$. Then, by (8), we have

[TABLE]

Here, we have

[TABLE]

Hence, we have

[TABLE]

that is,

[TABLE]

Thus, by induction on $m$, Theorem 1.1 follows.

4 Proof of Corollary 1.2

In this section, we prove Corollary 1.2.

Let

[TABLE]

As stated in the introduction, to prove Corollary 1.2, it suffices to show that $\nu_{p}(\tilde{S}(n,k))\geq t$ for any odd integer $n\geq 3$, any positive even integer $k$ such that $k<n$, and any prime $p$. In other words, it suffices to show that $\tilde{S}(n,k)$ is divisible by $p^{t}$.

First let $p$ be an odd prime. Consider the equation (8):

[TABLE]

Then, since the right hand side is of course an integer and is divisible by $p^{t}$, so is the left hand side.

In the rest of this section, let $p=2$. Consider the equation of Theorem 1.1:

[TABLE]

Since $\tilde{S}(n,k+2h+1)$ is divisible by $2^{t}$, it suffices to show that

[TABLE]

for $h=0,1,\cdots,(n-k-1)/2$ where $\nu_{2}(q_{1}/q_{2})=\nu_{2}(q_{1})-\nu_{2}(q_{2})$ for a non-zero rational number $q_{1}/q_{2}\in\mathbb{Q},\ 0\neq q_{1}\in\mathbb{Z},\ 0\neq q_{2}\in\mathbb{Z}$. As stated in the introduction, we invoke the theorem of Staudt [21] and Clausen [6] which states that

[TABLE]

Hence $\nu_{2}(B_{2h+2})=-1$ and

[TABLE]

It follows that

[TABLE]

which completes the proof.

5 Proof of Theorem 1.3

In this section, we recall some facts on the unstable $K$-theory from Hamanaka and Kono [12] and then, we prove Theorem 1.3. Also we give some examples of Theorem 1.3 and a characterization of $b_{k}=M_{k}$ in terms of the sequence (2).

As stated in the introduction, let $\mathbb{C}P^{n}$ be the complex projective space and $\mathbb{C}P^{n}_{k}=\mathbb{C}P^{n}/\mathbb{C}P^{k-1}.$ Let $U(n)$ be the unitary group and $U_{n}(X)=[X,U(n)]_{*}$. According to Hamanaka and Kono [12], for a CW-complex $X$ of dimension less than or equal to $2n$, the group $U_{n}(X)$ fits into an exact sequence

[TABLE]

of groups. Here $H^{\ell}(X)$ and $\tilde{K}^{\ell}(X)$ are the $\ell$th integral cohomology group of $X$ and the $\ell$th reduced complex $K$ cohomology group of $X$ respectively, and the homomorphism $\Theta$ is given as follows.

Let $T$ be a maximal torus of $U(n)$, $i\colon T\to U(n)$ the inclusion, $BT$ and $BU(n)$ the classifying spaces of $T$ and $U(n)$ respectively, and $Bi\colon BT\to BU(n)$ the map induced by $i$. Recall that

[TABLE]

the polynomial algebra generated by $t_{1},t_{2},\cdots,t_{n}\in H^{2}(BT)$. Let $W(U(n))$ be the Weyl group of $U(n)$ which is isomorphic to the symmetric group of $n$ letters and which acts on $H^{*}(BT)$ by permuting $t_{1},t_{2},\cdots,t_{n}$. Let $H^{*}(BT)^{W(U(n))}$ be the algebra which consists of the polynomials in $H^{*}(BT)$ invariant under the action of $W(U(n))$. Then it is well known that

[TABLE]

is an isomorphism of algebras where $c_{r}\in H^{2r}(BU(n))$ is the $r$th universal Chern class which corresponds to the $r$th elementary symmetric function of $t_{1},t_{2},\cdots,t_{n}$ for $r\leq n$. Let $s_{r}\in H^{2r}(BU(n))$ be the element which corresponds to $t_{1}^{r}+t_{2}^{r}+\cdots+t_{n}^{r}\in H^{2r}(BT)^{W(U(n))}$ through $(Bi)^{*}$ and take the polynomial function $f_{r}$ of $r$ variables such that $s_{r}=f_{r}(c_{1},c_{2},\cdots,c_{r})$ for $r\leq n$.

Let $U$ be the infinite unitary group, $BU$ the classifying space of $U$, $i_{n}\colon U(n)\to U$ the inclusion, and $Bi_{n}\colon BU(n)\to BU$ the map induced by $i_{n}$. Recall that

[TABLE]

where $c_{r}\in H^{2r}(BU)$ is the $r$th universal Chern class which corresponds to $c_{r}\in H^{2r}(BU(n))$ if $r\leq n$ through the homomorphism $(Bi_{n})^{*}\colon H^{2r}(BU)\to H^{2r}(BU(n))$ induced by $Bi_{n}$. Denote the element $f_{r}(c_{1},c_{2},\cdots,c_{r})$ for $c_{1}$, $c_{2}$, $\cdots$, $c_{r}\in H^{*}(BU)$ by the same letter $s_{r}\in H^{2r}(BU)$ as $s_{r}\in H^{2r}(BU(n))$ for $r\leq n$.

For a CW-complex $X$ and for $\theta\in\tilde{K}^{0}(X)=[X,BU]_{*}$, let $s_{r}(\theta)=\theta^{*}(s_{r})$, the image of $s_{r}\in H^{2r}(BU)$ through the homomorphism $\theta^{*}\colon H^{2r}(BU)\to H^{2r}(X)$ induced by $\theta$. Then, $\Theta(\theta)$ is given by

[TABLE]

Moreover, recall that for a CW-complex $X$ of dimension less than or equal to $2n$ and for $\theta\in\tilde{K}^{0}(X)=[X,BU]_{*}$,

[TABLE]

where

[TABLE]

is the Chern character and $\iota\colon H^{*}(BU)\to H^{*}(BU;\mathbb{Q})$ is induced by the inclusion $\mathbb{Z}\to\mathbb{Q}$. Thus, if $H^{2n}(X)$ is free, then $\iota\colon H^{2n}(X)\to H^{2n}(X;\mathbb{Q})$ is monomorphic and hence $\Theta(\theta)$ is determined by

[TABLE]

where $\mathop{\rm ch}\nolimits_{n}$ is the component of $\mathop{\rm ch}\nolimits$ in $H^{2n}(X;\mathbb{Q})$.

Now we consider the case $X=\mathbb{C}P^{n}_{k}$. (See Atiyah and Todd [3] for $\tilde{K}^{j}(\mathbb{C}P^{n}_{k})$.) Let $\gamma$ be the canonical complex line bundle over $\mathbb{C}P^{n}=\mathbb{C}P_{1}^{n}$. Let $\mathbb{Z}[a]/(a^{r})$ denote the truncated polynomial algebra generated by an element $a$. Then it is well known that

[TABLE]

and $\tilde{K}^{-1}(\mathbb{C}P^{n})=0$ where $t\in H^{2}(\mathbb{C}P^{n})$ and $z=\gamma-1\in\tilde{K}^{0}(\mathbb{C}P^{n})$. Also it is well known that

[TABLE]

for the suitably chosen generator $t\in H^{2}(\mathbb{C}P^{n})\subset H^{2}(\mathbb{C}P^{n};\mathbb{Q})$. Then we have

[TABLE]

(see Quaintance and Gould [20, (9.59)]) and hence we have

[TABLE]

for $1\leq j\leq n$.

Next, let $q=q_{j,\ell}\colon\mathbb{C}P^{n}_{j}\to\mathbb{C}P^{n}_{\ell}$ be the natural projection for $j\leq\ell$. Let $\mathbb{Z}\{a_{1},a_{2},\cdots\}$ denote the free additive group generated by elements $a_{1},a_{2},\cdots$. Then, it is well known that

[TABLE]

and $\tilde{K}^{-1}(\mathbb{C}P_{k}^{n})=0$. Here the elements which correspond to $t^{j}\in H^{2j}(\mathbb{C}P^{n})$ and $z^{j}\in\tilde{K}^{0}(\mathbb{C}P^{n})$ through $q_{1,k}^{*}\colon H^{2j}(\mathbb{C}P_{k}^{n})\to H^{2j}(\mathbb{C}P^{n})$ and $q_{1,k}^{*}\colon\tilde{K}^{0}(\mathbb{C}P_{k}^{n})$ $\to\tilde{K}^{0}(\mathbb{C}P^{n})$ are denoted by the same letters $t^{j}\in H^{2j}(\mathbb{C}P_{k}^{n})$ and $z^{j}\in\tilde{K}^{0}(\mathbb{C}P_{k}^{n})$ respectively for $k\leq j\leq n$.

Then, by (9), we have an exact sequence

[TABLE]

of groups and by the naturality of $\mathop{\rm ch}\nolimits$, we have

[TABLE]

for $k\leq j\leq n$. It follows that $\mathrm{Im}\Theta$ is the subgroup generated by $\tilde{S}(n,j)t^{n}$ for $j=k,k+1,\cdots,n$, that is, the subgroup generated by $\Delta_{n,k}t^{n}$ in $H^{2n}(\mathbb{C}P^{n}_{k})=\mathbb{Z}\{t^{n}\}$. Thus, we have the following proposition.

Proposition 5.1**.**

Let $n$ and $k$ be positive integers such that $k\leq n$. Then, the group $U_{n}(\mathbb{C}P^{n}_{k})$ is isomorphic to a cyclic group of order $\Delta_{n,k}$. Moreover, there exists a sequence of epimorphisms of groups

[TABLE]

Hence, Theorem 1.3 follows from this proposition and Corollary 1.2.

As an example of Theorem 1.3, for an odd integer $n$ such that $n\geq 3$,

[TABLE]

by the equality $\Delta_{n,n}=n!$ or by the theorem of Borel and Hirzebruch [5] for the homotopy group $\pi_{2n}(U(n))$. As another example, for an odd integer $n$ such that $n\geq 3$,

[TABLE]

by the theorem of Komatsu, Luca, and Ruiz [17] which states that $\Delta_{n,2}=B^{\prime}_{n-1}$.

Corollary 5.2**.**

Let $n$ be an integer which is a power of two. Then, the $2$-primary component of $U_{n}(\mathbb{C}P^{n}_{k})$ is a cyclic group of order $2^{k-1}$.

Proof.

In Wannemacker [23], it is shown that $\nu_{2}(\tilde{S}(n,k))=k-1$ if $n$ is a power of two. Then, the corollary follows from Proposition 5.1. ∎

Let $\Sigma X$ denote the reduced suspension of a topological space $X$. Let $b\colon S^{2k}\to\mathbb{C}P^{n}_{k}$ be the inclusion of the bottom cell of $\mathbb{C}P^{n}_{k}$ and put

[TABLE]

for positive integers $n$ and $k$ such that $k<n$.

Proposition 5.3**.**

Let $n$ and $k$ be positive integers such that $k<n$. Then, the group $U_{n}(\Sigma\mathbb{C}P^{n}_{k})$ is isomorphic to a free aditive group of rank $n-k$. Moreover, there exists a sequence of monomorphisms of groups

[TABLE]

and the image of

[TABLE]

is $\kappa_{n,k}U_{n}(S^{2k+1})$.

Proof.

We have an exact sequence of groups

[TABLE]

by Theorem 1.2 and Theorem 1.3 of Hamanaka [11], by $H^{2n}(\Sigma\mathbb{C}P^{n}_{k})=0$, and by $Sq^{2}\rho H^{2n-1}(\Sigma\mathbb{C}P^{n}_{k})=H^{2n+1}(\Sigma\mathbb{C}P^{n}_{k};\mathbb{Z}/2\mathbb{Z})$ if $n$ is even where

[TABLE]

is the mod $2$ reduction and

[TABLE]

is the Steenrod squaring operation. Moreover, the homomorphism

[TABLE]

is identified with

[TABLE]

through the suspension isomorphism

[TABLE]

and the isomorphism

[TABLE]

where $\beta\colon BU\to\Omega U$ is the Bott map. It follows that $U_{n}(\Sigma\mathbb{C}P^{n}_{k})\cong\textrm{Ker}\ T$ is isomorphic to a free additive group of rank $n-k$.

Next, for $k\leq n-2$, considering the following homotopy cofibration sequence

[TABLE]

and applying the functor $U_{n}(-)$, we have the following exact sequence of groups

[TABLE]

Thus, by Proposition 5.1, the proposition follows. ∎

Remark.  As is well known, $U_{n}(\Sigma\mathbb{C}P^{n}_{n})=\pi_{2n+1}(U(n))$ is trivial if $n$ is odd and is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ if $n$ is even. See Toda [22].

Corollary 5.4**.**

For any odd integer $n\geq 3$ and any positive even integer $k$ such that $k<n$, the sequence of groups

[TABLE]

is split exact.

Proof.

This follows from Proposition 5.3 and Corollary 1.2. ∎

Remark.  We can give generators of

[TABLE]

using arithmetic relations of $\tilde{S}(n,k)$. As an example, for $n=7$, we have

[TABLE]

Hence, we have

[TABLE]

and

[TABLE]

Define elements $\alpha_{6},\cdots,\alpha_{1}$ by

[TABLE]

Then, for $k=6,5,\cdots,1$, we can easily see that $\alpha_{6},\cdots,\alpha_{k}$ generates

[TABLE]

which is a subgroup of

[TABLE]

Finally, we give a characterization of $b_{k}=M_{k}$ in terms of the sequence (11). Recall that for a positive integer $k$, there exists a positive integer $b_{k}$, the complex James number of Stiefel manifolds, such that the following conditions are equivalent:

(A)

the integer $n$ is a positive multiple of $b_{k}$

(B)

the fibration

[TABLE]

of complex Stiefel manifolds admits a cross-section.

(See James [15] and Atiyah [2].) Also recall that for a positive integer $k$, the Atiyah-Todd number $M_{k}$ is defined by

[TABLE]

(See Atiyah and Todd [3].) For example, $M_{1}=1,\ M_{2}=2,\ M_{3}=M_{4}=24,\ M_{5}=M_{6}=2880,\cdots$. It is obvious that $M_{k}$ divides $M_{k+1}$ and it is shown in [3] that $M_{k}=M_{k+1}$ for any odd integer $k\geq 3$. In [3], it is shown that $b_{k}$ is a positive multiple of $M_{k}$ for any positive integer $k$. Later, Adams and Walker show that $b_{k}=M_{k}$ for any positive integer $k$ by examining the $J$-group of $\mathbb{C}P^{k-1}$. (See [1]. Also see [2].) Also it is known that there are several conditions which are equivalent to (A) and (B). For example, in terms of stable homotopy theory, the condition that

(C)

the stunted complex projective space $\mathbb{C}P^{n-1}_{n-k}$ is $S$-reducible, that is, the stable homotopy class of the attaching map of the top cell of $\mathbb{C}P^{n-1}_{n-k}$ is inessential

is equivalent to (A) and (B). (See James [16].)

Now, by the result of Atiyah and Todd [3, Theorem (1.7)], the condition that $n$ is a positive multiple of $M_{k}$ is equivalent to that the coefficient of $t^{n-1}$ in $(e^{t}-1)^{j}$ is an integer for $j=n-k,n-k+1,\cdots,n-1$. By (10), this condition is equivalent to that $\tilde{S}(n-1,j)$ is a positive multiple of $(n-1)!$ for $j=n-k,n-k+1,\cdots,n-1$ and hence, is equivalent to that $\Delta_{n-1,n-k}=(n-1)!$. Thus we have another condition which is equivalent to (A), (B), and (C) as in the following proposition.

Proposition 5.5**.**

The integer $n$ is a positive multiple of the complex James number and the Atiyah-Todd number $b_{k}=M_{k}$ if and only if we have isomorphisms of groups

[TABLE]

For example, since $24$ is a positive multiple of $b_{4}=M_{4}=24$, we have

[TABLE]

and

[TABLE]