Stable unextendibility of vector bundles over the quaternionic projective spaces

Stable unextendibility of vector bundles over the quaternionic projective spaces

Mitsunori Imaoka

(Received December 12, 2002) (Revised April 17, 2003)

Abstract. We study the stable unextendibility of vector bundles over the quaternionic projective spaceHPⁿby making use of combinatorial properties of the Stiefel-Whitney classes and the Pontrjagin classes. First, we show that the tangent bundle of HPⁿ is not stably extendible toHP^nþ1 fornb2, and also induce such a result for the normal bundle associated to an immersion ofHPⁿ intoR^4nþk. Secondly, we show a su‰cient condition for a quaternionic r-dimensional vector bundle over HPⁿ not to be stably extendible toHP^nþl forranandl>0, which is also a necessary condition whenr¼n and l¼1.

1. Introduction and results

Let F be the real field R, the complex field C or the quaternionic skew field H. Then, an F-vector bundle V of dimension k over a base space B is called extendible to a spaceB⁰withBHB⁰if there exists anF-vector bundleW of dimension k over B⁰ whose restriction to B is isomorphic to V as F-vector bundles. That is,iWGV for the inclusion mapi:B!B⁰. IfiW is stably equivalent to V, namely iWþmFGVþmF for a trivial F-vector bundle mF

of dimension mb0, then V is called stably extendible to B⁰ ([6]).

It is an interesting problem to determine when given vector bundles are stably extendible or not, which is related to some stable properties of vector bundles like geometrical dimensions or decompositions to line bundles (cf. [12], [2], [11], [9]). In this paper, we study the stable unextendibility of some vector bundles over the quaternionic projective space HPⁿ.

Schwarzenberger ([4, Appendix 1]) has shown, as an application of the Riemann-Roch theorem, that the tangent bundle TðCPⁿÞ of the complex pro- jective space CPⁿ for nb2 is not extendible to CP^nþ1 as C-vector bundle.

Kobayashi-Maki-Yoshida [7] has also shown that the tangent bundle TðRPⁿÞ (resp.TðLⁿðpÞÞ) of the real projective space RPⁿ (resp. the lens spaceLⁿðpÞfor

2000 Mathematics Subject Classification. Primary 55R50; secondary 55R40.

Key words and phrases. vector bundle, extendible, quaternionic projective space, Pontrjagin class, Stiefel-Whitney class.

an odd prime p) is not stably extendible toRP^nþ1 (resp.L^2nþ2ðpÞ) ifn01;3 or 7 (resp. nb2p2) as R-vector bundle.

The tangent bundle TðHPⁿÞ of the quaternionic projective space HPⁿ is anR-vector bundle of dimension 4n, and we show the following result applying the Stiefel-Whitney classes of R-vector bundles over HPⁿ.

Theorem A. For nb2, TðHPⁿÞ is not stably extendible to HP^nþ1. Since TðHP¹Þ ¼TðS⁴Þ is stably trivial, it is stably extendible to HP^k for any kb2. We remark that TðHP¹Þ is extendible to HP² (see Lemma 2.4).

Hence, the unextendibility of TðHPⁿÞ to HP^nþ1 agrees with the stable unex- tendibility of it.

Stable extendibility of the normal bundles of RPⁿ is studied in [8], and in [9] it is remarked that the stable extendibility of the normal bundle associated to an immersion f :MJR^l of a manifoldM does not depend on the map f but only on the existence of the immersion of M in R^l. Let n^k be the normal bundle associated to an immersion HPⁿJR^4nþk if it exists. Then, we obtain the following by a similar method used for the proof of Theorem A.

Theorem B. Assume that ka4nþ3 and n¼2^m1 for some mb2.

Then, if an immersion HPⁿJR^4nþk exists, its normal bundle n^k is not stably extendible to HP^nþ1.

Thomas [15] has studied the so called Chern vectors whose components are given from the Chern classes, and apply it to obtain some condition on the extendibility of C-vector bundles over CPⁿ. Analogously, we have the Pontr- jagin classes ofH-vector bundles (cf. [13, Chapter V]), and we can consider the notion of the Pontrjagin vectors and apply it to study the stable unextendibility of H-vector bundles overHPⁿ. We need some notations to express the result.

First, let x be the canonical H-line bundle over HPⁿ, and xAH⁴ðHPⁿ;ZÞ the Euler class of x. Then, the cohomology ring HðHPⁿ;ZÞ is isomorphic to the truncated polynomial ring Z½x=ðx^nþ1Þ, and the i-th Pontrjagin class PiðVÞ ¼ ð1ÞⁱC2iðc⁰ðVÞÞ of an H-vector bundle V can be represented as an integer p_iðVÞ multiple of xⁱ, namely P_iðVÞ ¼ p_iðVÞxⁱ. Here, c⁰ðVÞ denotes the underlying C-vector bundle of V, and Cjðc⁰ðVÞÞ is the j-th Chern class of it. Then, we define the Pontrjagin vector of V as the integral vector ðp1ðVÞ;. . .;pnðVÞÞAZⁿ.

Next, let s_k :Z^k !Z for kb1 be the map defined recursively using the Newton’s relations as follows: s1ðm1Þ ¼m1; for kb2,

skðm1;. . .;mkÞ ¼X^k1

i¼1

ð1Þ^iþ1miskiðm1;. . .;mkiÞ þ ð1Þ^kþ1kmk: ð1Þ

Thirdly, let gk :Z^k !Z for kb1 be the map defined recursively by g₁ðm1Þ ¼m₁ and, for kb2,

g_kðm1;. . .;m_kÞ ¼gk1ðm2;. . .;m_kÞ ðk1Þ²gk1ðm1;. . .;mk1Þ:

ð2Þ

Then, we show the following, and throughout the paper aðiÞ ¼1 or 2 according as i is an even or odd integer.

Theorem C. A necessary and su‰cient condition for an integral vector

q¼ ðq1;. . .;qnÞto be a Pontrjagin vector of anH-vector bundle overHPⁿ is that

the congruences

g_iðs1ðq1Þ;. . .;s_iðq1;. . .;q_iÞÞ10 modð2iÞ! aðiÞ

hold for all i with 1aian.

Now, for anH-vector bundleaof dimensionroverHPⁿ withran, we set

sk ¼skðp1ðaÞ;. . .;pkðaÞÞ for 1akar; and

ð3Þ

trþi¼srþiðp1ðaÞ;. . .;prðaÞ;0;. . .;0

|fflfflfflffl{zfflfflfflffl}

i

Þ for ib1:

ð4Þ

Then, we have the following result from Theorem C.

Theorem D. An H-vector bundle a of dimension r overHPⁿ with ran is not stably extendible to HP^m for m>n if

grþiðs1;. . .;s_r;trþ1;. . .trþiÞD0 mod ð2ðrþiÞÞ! aðrþiÞ

for some i with 1aiamr.

When an H-vector bundle a over HPⁿ is of dimension n, a is stably extendible to HP^m if and only if it is extendible to HP^m, and we have the following.

Corollary E. An H-vector bundle a of dimension n over HPⁿ is not extendible to HP^nþ1 if and only if the following holds:

gnþ1ðs1;. . .;sn;tnþ1ÞD0 mod ð2ðnþ1ÞÞ!

aðnþ1Þ

:

The paper is organized as follows. In § 2 we prove Theorems A and B, and in § 3 we study the Pontrjagin vectors and prove Theorem C. In § 4, we prove Theorem D and Corollary E, and, as an example, we show in Prop- osition 4.1 some condition under which an H-vector bundle of dimension n stably equivalent to ðnþkÞx for some k>0 is not extendible.

2. Proof of Theorems A and B

LetKOⁱðBÞ;KⁱðBÞandKSpⁱðBÞforiAZbe the reduced real, complex and symplecticK-group of a compact spaceB, respectively, which are often denoted by KOKOfⁱðBÞ and so on. Then KSp⁰ðBÞ is an abelian group formed by stable classes ½V ðdimVÞ_H of virtual dimensions 0 for H-vector bundles V over B, and KSp⁰ðBÞ ¼KO⁴ðBÞby definition. We refer to the classical works (cf. [1], [14, Chapter 11, 13]) for the general properties on these K-theories and use them without comments.

By the multiplications induced from the tensor products of vector bundles, KO⁴ðBÞ ¼0

iAZKO⁴ⁱðBÞ is a graded KO⁴ðS⁰Þ-algebra. To assign an R- vector bundle (resp.H-vector bundle)V the C-vector bundleVn_RC(resp. the underlying C-vector bundle c⁰ðVÞ) induces a homomorphism c:KO⁰ðBÞ ! K⁰ðBÞ (resp. c:KO⁴ðBÞ ¼KSp⁰ðBÞ !K⁰ðBÞ), and c is extended to a ring homomorphism c:KO⁴ðBÞ !K⁰ðBÞ using the Bott periodicity, which also satisfies cðayÞ ¼cðaÞcðyÞ for aAKO⁴ⁱðS⁰Þ and yAKO^4jðBÞ. Here, ayA KO^4ðiþjÞðBÞ denotes the element defined by the KO⁴ðS⁰Þ-algebra structure of KO⁴ðBÞ, and we regard cðaÞAK⁰ðS⁰Þ as an integer by K⁰ðS⁰ÞGZ. If KO⁴ⁱðBÞ is a free abelian group for someiAZ, then c:KO⁴ⁱðBÞ !K⁰ðBÞ is a monomorphism.

As for the ring KO⁴ðS⁰Þ, we have an isomorphism KO⁴ðS⁰ÞG Z½u;v;w=ðu²4v;vw1Þ, where uAKO⁴ðS⁰Þ, vAKO⁸ðS⁰Þ and w¼v¹A KO⁸ðS⁰Þ. Then, as is known, we can take u and v to satisfy cðuÞ ¼2 and cðvÞ ¼1 regardingK⁰ðS⁰ÞasZ. We setg_2i¼vⁱAKO⁸ⁱðS⁰Þandg2iþ1¼uvⁱA KO⁸ⁱ⁴ðS⁰Þ for iAZ, and thus we have cðg2iÞ ¼1 and cðg2iþ1Þ ¼2.

Now, let X ¼ ½x1HAKSp⁰ðHPⁿÞ ¼KO⁴ðHPⁿÞ be the stable class of the canonical H-line bundle x over HPⁿ. Then, cðXÞ ¼ ½c⁰ðxÞ 2CA K⁰ðHPⁿÞ by the definition of c. Since HðHPⁿ;ZÞGZ½x=ðx^nþ1Þ and KO^4iþ1ðS⁰Þ ¼0 for any iAZ, the Atiyah-Hirzebruch spectral sequence

E₂^p;q¼HH~^pðHPⁿ;ZÞnKO^qðS⁰Þ )KO^pþqðHPⁿÞ

collapses. Remark that E₂^4p;4p¼Zfx^png_pg for any p with 1apan.

Then, g1X AKO⁰ðHPⁿÞ is represented by xng1 modulo Zfxⁱngig for all i with 2aian in the E2-term of the spectral sequence, because iðg1XÞA KO⁰ðHP¹ÞGZ is a generator for the inclusion i:HP¹¼S⁴!HPⁿ and the E2-term E₂^p;qð1Þ ¼HH~^pðHP¹;ZÞnKO^qðS⁰Þ of the Atiyah-Hirzebruch spectral sequence on HP¹ satisfies that E₂^4;4ð1Þ ¼Zfxng₁g and E₂^p;pð1Þ ¼0 if p04. Since ðg1XÞ^2mþe¼4^mg2mþeX^2mþe in KO⁰ðHPⁿÞ and ðxng1Þ^2mþe¼ 4^mx^2mþeng2mþe in E4ð2mþeÞ;4ð2mþeÞ

2 for e¼0 or 1, gpX^p is represented by x^pngp modulo Zfxⁱngig for all i with pþ1aian in the E2-term of the

spectral sequence. In this way, we see that KO⁰ðHPⁿÞis a free abelian group with basis g₁X;g₂X²;. . .;g_nXⁿ. That is, we have

KO⁰ðHPⁿÞGZfg1X;g2X²;. . .;gnXⁿg:

ð5Þ

The tangent bundle TðHPⁿÞ of HPⁿ satisfies TðHPⁿÞ þxn_HxGðnþ1ÞxR; ð6Þ

where x andx_R denote the quaternionic conjugate bundle and the underlying real vector bundle of x, respectively. Then, we have the following.

Lemma 2.1. In KO⁰ðHPⁿÞ, ½TðHPⁿÞ 4nR ¼ ðn1Þg1Xg2X². Proof. We remark that the underlying C-vector bundle c⁰ðxÞ of x is self conjugate and cðxRÞG2c⁰ðxÞ. Also, there is a relation cðxn_HxÞ ¼c⁰ðxÞn_C c⁰ðxÞ ¼c⁰ðxÞ². Thus, we have the following equalities:

cðg1XÞ ¼cðg1ÞcðXÞ ¼2½c⁰ðxÞ 2C ¼cð½xR4RÞ;

cðg2X²Þ ¼cðg2ÞcðXÞ²¼ ½c⁰ðxÞ 2C²¼ ½c⁰ðxÞ²4C 4½c⁰ðxÞ 2C

¼cð½xn_Hx4R 2g1XÞ:

Since the homomorphism c:KO⁰ðHPⁿÞ !K⁰ðHPⁿÞ is a monomorphism, we have ½xR4R ¼g₁X and ½xn_Hx4R ¼2g₁Xþg₂X². Hence, by (6),

½TðHPⁿÞ 4nR ¼ ðnþ1Þ½xR4R ½xn_Hx4R ¼ ðn1Þg1Xg2X²

as is required. r

Let wðVÞ ¼1þw₁ðVÞ þ AHðB;Z=2Þ be the total Stiefel-Whitney class of an R-vector bundle V over a compact space B. Then, by the stable and multiplicative properties of the Stiefel-Whitney classes, we can also have the Stiefel-Whitney class wðaÞAHðB;Z=2Þ of aAKO⁰ðBÞ. As for the ele- ments giXⁱ of KO⁰ðHPⁿÞ in (5), we have the following lemma, where we denote the mod 2 reduction of xAH⁴ðHPⁿ;ZÞ by the same letter x.

Lemma 2.2. wðg1XÞ ¼1þx, wðg2X²Þ ¼ ð1þxÞ² and wðgmX^mÞ11 ðmodx^mþ1Þ for mb3.

Proof. First, we shall prove the following congruence, where k is a positive integer and e¼0 or 1:

Cðc⁰ðxÞ^2kþeÞ11þex^4k ðmod 2Þ:

ð7Þ

Here, Cðc⁰ðxÞ^2kþeÞ denotes the total Chern class of the ð2kþeÞ-fold tensor product of c⁰ðxÞ.

Let h be the canonicalC-line bundle over CP²ⁿ. Then, for the canonical projection p:CP²ⁿ!HPⁿ, we have pðc⁰ðxÞÞ ¼hþh, where h is the complex conjugate bundle of h, and pðxÞ ¼t² for the Euler class tAH²ðCP²ⁿ;ZÞ of h. Since hh¼hn_Ch¼1C, we have

pðc⁰ðxÞ^2kþ1Þ ¼X^k

i¼0

2kþ1 i

ðh^2ðkiÞþ1þh^2ðkiÞþ1Þ:

Since Cðh^jÞ ¼1þjt, Cðh^jÞ ¼1jt and p:HðHPⁿ;ZÞ !HðCP²ⁿ;ZÞ is a monomorphism, we have the congruence

Cðc⁰ðxÞ^2kþ1Þ ¼Y^k

i¼0

ð1 ð2ðkiÞ þ1Þ²xÞ^ai1ð1þxÞ^a ðmod 2Þ

fora_i¼ 2kþ1 i

anda¼Pk

i¼0a_i¼4^k, and thus we have (7) fore¼1. The congruence (7) for e¼0 is similarly shown.

Recall that cðXÞ ¼ ½c⁰ðxÞ 2C. Then, for e¼0 or 1, we have CðcðXÞ^2kþeÞ11 ðmodð2;x^4kÞÞ:

ð8Þ

In fact, for e¼0, cðXÞ^2k ¼P2k

i¼0bið2Þⁱ½c⁰ðxÞ^2ki2_C^2ki, where bi¼ 2k i

. Then, using (7), we have the following congruences:

CðcðXÞ^2kÞ1 Y^k1

j¼0

ð1þx^4kj1Þ^22jþ1b2jþ11ð1þx^4kÞ^d ðmod 2Þ;

where d ¼ ð1=2ÞPk1

j¼0 b2jþ1 is an integer. Thus, we have CðcðXÞ^2kÞ11 ðmodð2;x^4kÞÞ as is required in (8) for e¼0. The congruence (8) for e¼1 is similarly shown.

Now, we conclude the proof of the congruence wðgmX^mÞ11 ðmodx^mþ1Þ for mb3. Generally, for an element a¼ ½V ðdimVÞ_RAKO⁰ðBÞ, we have CðcðaÞÞ ¼CðcðVÞÞ1wðVÞ²¼wðaÞ² ðmod 2Þ. Thus,

wðgmX^mÞ²1CðcðgmX^mÞÞ ¼CðcðgmÞcðXÞ^mÞ ¼CðaðmÞcðXÞ^mÞ

¼CðcðXÞ^mÞ^aðmÞ ðmod 2Þ;

where aðmÞ ¼1 or 2 according as mis an even or odd integer. When mb3, CðcðXÞ^mÞ^aðmÞ11 ðmodð2;x^2mþ2ÞÞ by (8) since 2mþ2a2^m. Therefore, wðgmX^mÞ²11 ðmodx^2mþ2Þ. This congruence is valid onHP^N for any Nb1 by the same reason, in particular on HP²ⁿ. Hence, we have wðgmX^mÞ11 ðmodx^mþ1Þ on HPⁿ for mb3 as is required.

In the case of m¼1 and 2, we have

wðg1XÞ²1Cðcðg1ÞcðXÞÞ ¼Cð2½c⁰ðxÞ 2CÞ ¼Cðc⁰ðxÞÞ² 1ð1þxÞ² ðmod 2Þ;

wðg2X²Þ²1Cðcðg2ÞcðXÞ²Þ ¼Cð½c⁰ðxÞ 2C²Þ ¼Cðc⁰ðxÞ²ÞCðc⁰ðxÞÞ⁴ 1ð1þxÞ⁴ ðmod 2Þ;

which is still valid on HP²ⁿ. Thus, wðg1XÞ ¼1þx and wðg2X²Þ ¼ ð1þxÞ²

on HPⁿ, and we have completed the proof. r

Corollary 2.3. Let g be an R-vector bundle over HP^nþ1 for nb2. If the restriction ig over HPⁿ satisfies igþsRGTðHPⁿÞ þtR for some s;tb0, then it follows wðgÞ ¼ ð1þxÞ^nþ1 in HðHP^nþ1;Z=2Þ.

Proof. The kernel of the homomorphism i:KO⁰ðHP^nþ1Þ !KO⁰ðHPⁿÞ is a free abelian group of rank 1 with generator g_nþ1X^nþ1 by (5). Thus, by Lemma 2.1, the stable class of g satifies

½grR ¼ ðn1Þg1Xg₂X²þagnþ1X^nþ1 in KO⁰ðHP^nþ1Þ

for some integer a, whereris the dimension of g. Then, using Lemma 2.2, we have

wðgÞ ¼wðg1XÞⁿ¹wðg2X²Þ¹wðgnþ1X^nþ1Þ^a¼ ð1þxÞ^nþ1

as is required. r

Now we complete the proofs of Theorem A and Theorem B.

Proof of Theorem A. Assume that TðHPⁿÞ is stably extendible to HP^nþ1 for somenb2. Then, there exists a 4n-dimensional R-vector bundle b over HP^nþ1 whose restriction to HPⁿ is stably equivalent to TðHPⁿÞ. Then, by Corollary 2.3, we have wðbÞ ¼ ð1þxÞ^nþ1, which contradicts that b is of dimension 4n. Thus, we have the required result. r Proof ofTheoremB. Assume thatn¼2^m1 formb2 andka4nþ3.

Since n_kþTðHPⁿÞ is equivalent to the trivial bundle ð4nþkÞ_R, we have

½n^kkR ¼ ½TðHPⁿÞ 4nR ¼g2X² ðn1Þg1X

in KO⁰ðHPⁿÞby Lemma 2.1. By the same reason as in Corollary 2.3, if there exists a k-dimensional R-vector bundle g over HP^nþ1 satisfying that ig is stably equivalent to n^k, then wðgÞ ¼ ð1þxÞ^ðnþ1Þ, and thus

w_4ðnþ1ÞðgÞ ¼ 2nþ1 nþ1

x^nþ1:

Hence, when n¼2^m1, w4ðnþ1ÞðgÞ00 which contradicts that g is of dimen- sion k with ka4nþ3. Thus, we have completed the proof. r

As the last of this section, we show the following mentioned in § 1.

Lemma 2.4. The tangent bundle TðHP¹Þ of HP¹ is extendible to HP². Proof. Let ½X;Y denote the homotopy set of maps from a space X to a space Y, and BG the classifying space of a group G. Then, for the inclu- sion map i:HP¹!HP², we shall show that i:½HP²;BOð4Þ ! ½HP¹;BOð4Þ is surjective, which establishes that TðHP¹Þ is extendible to HP². As is known, we have an isomorphism Spinð4ÞGSpð1Þ Spð1Þ of the Lie groups, and thus BSpinð4Þ ¼BSpð1Þ BSpð1Þ ¼HP^yHP^y. Then, for i¼1 or 2, we have the canonical bijection½HPⁱ;BOð4ÞA½HPⁱ;BSpinð4ÞA½HPⁱ;HP^y

½HPⁱ;HP^yA½HPⁱ;HP² ½HPⁱ;HP², where the last bijection is induced by the cellular approximation. Thus, it is su‰cient to show that i:½HP²;HP²

! ½HP¹;HP² is surjective.

Letj:S⁷!HP¹ be the attaching map of the top cell ofHP². Then, the cofiber sequsence S⁷ !^j HP¹!ⁱ HP² induces the exact sequence of the homo- topy sets ½HP²;HP² !ⁱ

½HP¹;HP² !^j

½S⁷;HP². However, ½HP¹;HP² ¼

½S⁴;HP² is a free abelian groupZ with a base of the homotopy class ofi, and thus j ¼0. Hence, i is surjective, and we have completed the proof. r 3. Pontrjagin vectors

First, we define an H-vector bundles xðkÞ recursively as follows:

xð1Þ ¼x;

xð2iÞ ¼ ðxð2i1Þn_HxÞn_R1H for ib1;

xð2iþ1Þ ¼ ðxð2i1Þn_HxÞn_Rx for ib1:

Here, the H-vector bundle structures of xð2iÞ and xð2iþ1Þ are given by 1H

and x, respectively. Thus, xðkÞ is an H-vector bundle of dimension 2^k=aðkÞ, where aðkÞ ¼1 or 2 according as k is an even or odd integer. We have c⁰ðxð2iÞÞ ¼cðxð2i1Þn_HxÞn_C2C¼2c⁰ðxð2i1ÞÞn_Cc⁰ðxÞandc⁰ðxð2iþ1ÞÞ

¼cðxð2i1Þn_HxÞn_Cc⁰ðxÞ ¼c⁰ðxð2i1ÞÞn_Cc⁰ðxÞ². For instance, c⁰ðxð2ÞÞ

¼2c⁰ðxÞ² and c⁰ðxð3ÞÞ ¼c⁰ðxÞ³. Thus, proceeding recursively, we have the relation

c⁰ðxðkÞÞ ¼ 2 aðkÞc⁰ðxÞ^k ð9Þ

for any kb1. Furtheremore, we have

Lemma 3.1. f½xðiÞ ð2ⁱ=aðiÞÞ_H j1aiang forms a basis of the free abelian group KSp⁰ðHPⁿÞ.

Proof. By the similar reason as for (5), KSp⁰ðHPⁿÞ ¼KO⁴ðHPⁿÞ is a free abelian group with basis fX;g1X²;. . .;gn1Xⁿg. Let c:KO⁴ðHPⁿÞ ! K⁰ðHPⁿÞ be the homomorphism mentioned in the first part of § 2. Then, we have

cðgi1XⁱÞ ¼cðgi1ÞcðXÞⁱ¼aði1Þ½c⁰ðxÞ 2Cⁱ¼ 2

aðiÞ½c⁰ðxÞ 2Cⁱ

¼Xⁱ

j¼1

i

j ð2Þ^ijaðjÞ aðiÞ

2

aðjÞ½c⁰ðxÞ^j2_C^j

for any i with 1aian. Here, we have ð2=aðjÞÞ½c⁰ðxÞ^j2_C^j ¼cð½xðjÞ ð2^j=aðjÞÞ_HÞ for jb1 by (9), and the homomorphism c:KSp⁰ðHPⁿÞ ! K⁰ðHPⁿÞ is a monomorphism. Thus, each gi1Xⁱ is written using ½xðjÞ ð2^j=aðjÞÞ_H for 1ajai as follows:

gi1Xⁱ¼ ½xðiÞ ð2ⁱ=aðiÞÞ_H þXⁱ¹

j¼1

bj½xðjÞ ð2^j=aðjÞÞ_H

for some integers b_j. Since fgi1Xⁱj1aiang is a basis of KSp⁰ðHPⁿÞ, f½xðiÞ ð2ⁱ=aðiÞÞ_H j1aiangis also a basis of it from these equalities. Thus,

we obtain the required result. r

Let p:CP²ⁿ!HPⁿ be the canonical projection, h the canonical C-line bundle over CP²ⁿ and h the complex conjugate bundle of h as in the proof of Lemma 2.2. Then, p induces a monomorphism p:K⁰ðHPⁿÞ !K⁰ðCP²ⁿÞ, and we have the following.

Lemma 3.2. There exists a basis fYij1aiang of KSp⁰ðHPⁿÞ which satisfies pðcðYiÞÞ ¼ ð2=aðiÞÞ½hⁱþhⁱ2C.

Proof. We put Z_i¼ ½xðiÞ ð2ⁱ=aðiÞÞ_H for 1aian to the basis of KSp⁰ðHPⁿÞin Lemma 3.1. Then, using the relation pðc⁰ðxÞÞ ¼hþh and (9), we have

pðcðZiÞÞ ¼ 2

aðiÞpð½c⁰ðxÞⁱ2_CⁱÞ ¼ 2

aðiÞð½ðhþhÞⁱ2_CⁱÞ

¼ 2

aðiÞ½hⁱþhⁱ2C þ X

0<j<i=2

i j

2

aði2jÞ½h^i2jþh^i2j2C:

Then, since p and c are both monomorphisms, fð2=aðiÞÞ½hⁱþhⁱ2C j 1aiang forms a basis of pcðKSp⁰ðHPⁿÞÞ, and thus we have a basis fYij1aiang ofKSp⁰ðHPⁿÞwhich satisfies pðcðYiÞÞ ¼ ð2=aðiÞÞ½hⁱþhⁱ2C,

as is required. r

As mentioned in § 1, the Pontrjagin vector of an H-vector bundle V over HPⁿ is defined to be an integral vector

pðVÞ ¼ ðp₁ðVÞ;p₂ðVÞ;. . .;p_nðVÞÞAZⁿ;

where the integers p_iðVÞ satisfy P_iðVÞ ¼ p_iðVÞxⁱ for the Pontrjagin classes PiðVÞAH⁴ⁱðHPⁿ;ZÞ of V, respectively.

Since the total Pontrjagin class PðVÞ ¼1þP₁ðVÞ þP₂ðVÞ þ satisfies the multiplicative property PðVþWÞ ¼PðVÞPðWÞ and PðkHÞ ¼1, we can also define the Pontrjagin vector pðaÞAZⁿ of a¼ ½VdimVHAKSp⁰ðHPⁿÞ by setting pðaÞ ¼ pðVÞ.

In (1), we have introduced a map sk :Z^k !Z defined by the Newton’s relation as follows:

s_kðm1;. . .;m_kÞ ¼X^k1

i¼1

ð1Þ^iþ1m_iskiðm1;. . .;mkiÞ þ ð1Þ^kþ1km_k for m_iAZ. Then, for an integral vector q¼ ðq1;. . .;q_nÞAZⁿ, we set

sðqÞ ¼ ðs1ðq1Þ;s2ðq1;q2Þ;. . .;snðq1;. . .;qnÞÞAZⁿ:

Then, it defines a monomorphism s:Zⁿ!Zⁿ between the free abelian groups Zⁿ, and we set sðaÞ ¼sðpðaÞÞ for aAKSp⁰ðHPⁿÞ. As for the basis fYij 1aiang of KSp⁰ðHPⁿÞ in Lemma 3.2, we have the following.

Lemma 3.3. sðYkÞ ¼ ð2=aðkÞÞðk²;k⁴;. . .;k²ⁿÞ for 1akan.

Proof. By the definition of Yk, we have PðYkÞ ¼ ð1þk²xÞ^2=aðkÞ for kb0. Thus, using the Newton’s relation, we have s_iðYkÞ ¼ ð2=aðkÞÞk²ⁱ as is

required. r

Let A be the nn matrix whose j-th column is sðYjÞ for 1ajan.

Hence, by Lemma 3.3, A is represented as

A¼ 0 BB BB BB B@

1 22² 3² ð2=aðnÞÞ n² 1 22⁴ 3⁴ ð2=aðnÞÞ n⁴ 1 22⁶ 3⁶ ð2=aðnÞÞ n⁶

... ...

... .. .

... 1 22²ⁿ 3²ⁿ ð2=aðnÞÞ n²ⁿ

1 CC CC CC CA ð10Þ :

We also denote byAðqÞthe n ðnþ1Þmatrix whose firstnn submatrix is A and the last column is sðqÞ. That is,

AðqÞ ¼ ðA sðqÞÞ:

Since fYij1aiang is a basis of KSp⁰ðHPⁿÞ and s:Zⁿ!Zⁿ is a monomorphism, a necessary and su‰cient condition for an integral vector qAZⁿ to be a Pontrjagin vector of some H-vector bundle over HPⁿ is that sðqÞis a linear combination offsðYiÞ j1aiangwith integer coe‰cients. It is equivalent to say that there exists an integral vector b¼ ðb1;. . .;bnÞ satisfying

sðqÞ ¼Xⁿ

i¼1

bisðYiÞ ¼Ab;

and thus we have the following.

Proposition 3.4. When AðqÞ is transformed into an integral matrix ðB uÞ by row operations within integral matrices, a necessary and su‰cient condition for an integral vectorq¼ ðq1;. . .;qnÞAZⁿ to be a Pontrjagin vector is that there exists an integral vector bAZⁿ satisfying

Bb¼u:

ð11Þ

Let gk :Z^k!Zbe the map given in (2). Then, the following is easy by the induction on k.

Lemma 3.5. gkðm²;m⁴;. . .;m^2kÞ ¼Qk1

i¼0ðm²i²Þ.

Now, we transform AðqÞ step by step through elementary row operations within integral matrices to make A an upper triangular matrix. As the first step, subtracting the i-th row from the ðiþ1Þ-th row proceeding upward con- secutively from i¼n1 to i¼1, AðqÞ is transformed into the following matrix, since g1ðk1Þ ¼k1 and g2ðk1;k2Þ ¼k2k1.

0 BB BB BB BB B@

g₁ð1Þ 2g₁ð2²Þ g₁ð3²Þ _aðnÞ² g₁ðn²Þ g₁ðs1Þ 0 2g2ð2²;2⁴Þ g2ð3²;3⁴Þ _aðnÞ² g2ðn²;n⁴Þ g2ðs1;s2Þ 0 2g2ð2⁴;2⁶Þ g2ð3⁴;3⁶Þ _aðnÞ² g2ðn⁴;n⁶Þ g2ðs2;s3Þ

...

...

...

.. .

...

... 0 2g2ð2²ⁿ²;2²ⁿÞ g2ð3²ⁿ²;3²ⁿÞ _aðnÞ² g2ðn²ⁿ²;n²ⁿÞ g2ðsn1;snÞ

1 CC CC CC CC CA

;

where we abbreviate siðq1;. . .;qiÞ simply by si for 1aian.

As the next step, subtract the 2² times of i-th row from the ðiþ1Þ-th row proceeding upward consecutively from i¼n1 to i¼2. Then, the matrix

is transformed into the following matrix, since g3ðk1;k2;k3Þ ¼g2ðk2;k3Þ 2²g₂ðk1;k₂Þ.

g1ð1Þ 2g1ð2²Þ g1ð3²Þ _aðnÞ² g1ðn²Þ g1ðs1Þ 0 2g2ð2²;2⁴Þ g2ð3²;3⁴Þ _aðnÞ² g2ðn²;n⁴Þ g2ðs1;s2Þ 0 0 g3ð3²;3⁴;3⁶Þ _aðnÞ² g3ðn²;n⁴;n⁶Þ g3ðs1;s2;s3Þ

...

...

...

.. .

...

... 0 0 g3ð3²ⁿ⁴;3²ⁿ²;3²ⁿÞ _aðnÞ² g3ðn²ⁿ⁴;n²ⁿ²;n²ⁿÞ g3ðsn2;s_n1;snÞ 0

BB BB BB BB B@

1 CC CC CC CC CA :

Proceeding such way, aftern1 steps we reach a matrix MðqÞwhose first n columns form an upper triangular matrix, as follows.

Lemma 3.6. By elementary row operations within integral matrices, the matrix AðqÞ is transformed into a matrix MðqÞwhose ði;jÞ element m_i;j satisfies

mi;j¼

2

aðjÞg_iðj²;j⁴;. . .;j²ⁱÞ for 1aiajan;

0 for i> j;

giðs1ðq1Þ;. . .;siðq1;. . .;qiÞÞ for 1aian and j¼nþ1:

8>

<

>:

Using Lemma 3.5, the elements mi;i for 1aian and mi;j for 1ai<

jan can be written, respectively, as follows:

m_i;i¼ 2

aðiÞi²ði²1Þ. . .ði² ði1Þ²Þ ¼ð2iÞ! aðiÞ; ð12Þ

mi;j¼ 2

aðjÞj²ðj²1Þ. . .ðj² ði1Þ²Þ ¼ð2jÞðjþi1Þ!

aðjÞðjiÞ! : ð13Þ

Then, we have the following.

Lemma 3.7. For 1aiajan, mi;i is a factor of mi;j.

Proof. We show that mi;j=mi;i is an integer. Using (12) and (13), we have

m_i;j mi;i

¼aðiÞj aðjÞi

jþi1 2i1

:

By Feder and Gitler [3], it is shown that j

i

jþi1 2i1

AZ:

Therefore, if j is even or if both i and j are odd, then aðiÞ=aðjÞ ¼1 or 2 and thus m_i;j=m_i;i is an integer. We also have

j i

jþi1 2i1

¼ 2j iþj

jþi 2i

:

Hence, if j is odd and i is even, then the odd integer iþj divides j jþi 2i

. Thus, m_i;j

mi;i

is still an integer, which concludes the proof. r

Now, we complete the proof of Theorem C.

Proof of Theorem C. By Proposition 3.4, a necessary and su‰cient condition for an integral vectorq¼ ðq1;. . .;q_nÞto be a Pontrjagin vector of an H-vector bundle over HPⁿ is that there exists an integral vector b satisfying (11) when we takeðB uÞ ¼MðqÞ. Then, by Lemmas 3.6 and 3.7, suchb exists if and only if each mi;i is a factor of giðs1ðq1Þ;. . .;siðq1;. . .;qiÞÞ for 1aian.

Thus, using (12), we obtain the required result. r

4. Proof of Theorem D and Corollary E

Now, we prove Theorem D and Corollary E using Theorem C.

Proof of Theorem D. Let a be an H-vector bundle of dimension r over HPⁿ with ran, and assume that a is stably extendible to HP^m for some m with m>n. Then, there exists an H-vector bundle b of dimension r over HP^m whose restriction toHPⁿ is stably equivalent to a. The Pontrjagin vector of b is represented as pðbÞ ¼ ðp₁ðaÞ;. . .;p_rðaÞ;0;. . .;0Þ. Hence, by Theorem C, we have grþiðs1;. . .;sr;trþ1;. . .;trþiÞ10 ðmodð2ðrþiÞÞ!=aðrþiÞÞ for any i with 1aiamr, where s₁;. . .;s_r;trþ1;. . .;t_m are the integers given in (3) and (4) for a under consideration. Thus, taking the contraposition, we have the

required result. r

Proof of Corollary E. Let a be an H-vector bundle of dimension n over HPⁿ. Then, the extendibility of a is equivalent to the stable extendibility of it by a stability property of vector bundle (cf. [5, § 8, Theorem 1.5]). Thus, it is su‰cient to prove the converse of Theorem D for a when r¼n andm¼ nþ1. We assume that gnþ1ðs1;. . .;sn;tnþ1Þ10 ðmodð2ðnþ1ÞÞ!=aðnþ1ÞÞ.

Then, by Theorem C, there exists an H-vector bundle b over HP^nþ1 with P_iðbÞ ¼P_iðaÞ for 1aian and P_nþ1ðbÞ ¼0. By a stability property (cf. [5,

§ 8, Theorem 1.2]), we can assume that b is of dimension nþ1 as an H- vector bundle. Then, the restriction ib over HPⁿ is stably equivalent to aþ1H, since they have the same Pontrjagin classes (cf. [6, Lemma 4]). On the other hand, regarding b as an oriented vector bundle, the Euler class of b is Pnþ1ðbÞ up to sign, and thus the primary obstruction class to construct a cross section of the associated sphere bundle ofb isPnþ1ðbÞ(cf. [10, Theorem 12.5]).

Hence, the equality Pnþ1ðbÞ ¼0 shows that b admits an everywhere nonzero section, and thus bGgþ1H for some H-vector bundle g of dimension n over HP^nþ1. Then, by the stability property again, it follows thatigGa. Thus,a is extendible to HP^nþ1, and we have completed the proof. r As a special case of Corollary E, we have the following, where x denotes the canonical H-line bundle as before.

Proposition 4.1. Let a be an H-vector bundle of dimension n over HPⁿ. If aþkHGðnþkÞx for some kb1, then the necessary and su‰cient condition for a not to be extindible to HP^nþ1 is that the following holds:

nþk nþ1

D0 ðmodaðnÞð2nþ1Þ!Þ:

Proof. Put b¼ ðnþkÞx over HP^nþ1. Then, piðaÞ ¼ piðbÞ and s_iðaÞ ¼s_iðbÞ for 1aian. Since PðbÞ ¼PððnþkÞxÞ ¼ ð1þxÞ^nþk, p_iðbÞ for 1aianþ1 is equal to the value of thei-th elementary symmetric polynomial with nþk variables substituted 1 for all variables. Thus, using an algebraic property concerning the symmetric polynomial and the Newton polynomial, we haves_iðbÞ ¼1ⁱþ þ1ⁱ

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

nþk

¼nþkfor 1aianþ1. Then, by the definition of tnþ1, we have

tnþ1¼Xⁿ

i¼1

ð1Þ^iþ1piðaÞsnþ1iðaÞ ¼Xⁿ

i¼1

ð1Þ^iþ1piðbÞsnþ1iðbÞ

¼snþ1ðbÞ þ ð1Þ^nþ1ðnþ1Þpnþ1ðbÞ ¼ ðnþkÞ þ ð1Þ^nþ1ðnþ1Þ nþk nþ1

: Since the map gk;Z^k!Zis a homomorphism between the abelian groups and since g_kð1;1;. . .;1Þ ¼0 and g_kð0;0;. . .;0;1Þ ¼1 for any k, we have

g_nþ1ðs1;s₂;. . .;s_n;t_nþ1Þ

¼ ðnþkÞgnþ1ð1;1;. . .;1Þ þ ð1Þ^nþ1ðnþ1Þ nþk

nþ1

gnþ1ð0;0;. . .;0;1Þ

¼ ð1Þ^nþ1ðnþ1Þ nþk nþ1

:

Thus, by Corollary E, a necessary and su‰cient condition for a not to be extendible to HP^nþ1 is that

nþk nþ1

D0 ðmodaðnÞð2nþ1Þ!Þ

as is required. r

For example, when 1akan, a vector bundlea overHPⁿ of dimensionn which is stably equivalent to ðnþkÞx is not extendible to HP^nþ1.

References

[ 1 ] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Di¤erential Geometry, Proc. of Symp. in Pure Math. 3 (1961), 7–38.

[ 2 ] J. F. Adams and Z. Mahmud, Maps between classifying spaces, Invent. Math.35(1976), 1–41.

[ 3 ] S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat.

Mexicana 18 (1973), 33–37.

[ 4 ] F. Hirzebruch, Topological methods in algebraic geometry, Grundlehren der mathemati- schen Wissenschaften, Vol. 131, Springer-Verlag, New York Inc., 1978.

[ 5 ] D. Husemoller, Fiber bundles, Third edition, Graduate Texts in Mathematics 20, Springer- Verlag, New York Inc., 1994.

[ 6 ] M. Imaoka and K. Kuwana, Stably extendible vector bundle over the quaternionic pro- jective space, Hiroshima Math. J.29 (1999), 273–279.

[ 7 ] T. Kobayashi, H. Maki and T. Yoshida, Stably extendible vector bundles over the real projective spaces and the lens spaces, Hiroshima Math. J. 29 (1999), 631–638.

[ 8 ] T. Kobayashi, H. Maki and T. Yoshida, Extendibility and stable extendibility of normal bundles associated to immersions of real projective spaces, Osaka J. Math. 39 (2002), 315–324.

[ 9 ] T. Kobayashi and T. Yoshida, Extendibility, stable extendibility and span of some vector bundles over real projective spaces, Mem. Fac. Sci. Kochi Univ. (Math.)23(2002), 45–56.

[10] J. W. Milnor and J. D. Stashe¤, Characteristic classes, Annals of Mathematics Studies 76, Princeton Press, 1974.

[11] E. Rees, On submanifolds of projective space, J. London Math. Soc.19(1979), 159–162.

[12] R. L. E. Schwarzenberger, Extendible vector bundles over real projective space, Quart. J.

Math. Oxford (2) 17 (1966), 19–21.

[13] R. E. Stong, Notes on cobordism theory, Math. Notes, Princeton University Press, Princeton, 1968.

[14] R. M. Switzer, Algebraic Topology-Homotopy and Homology, Grundlehren der mathe- matischen Wissenschaften Vol. 212, Springer-Verlag, Berlin, 1975.

[15] A. Thomas, Almost complex structures on complex projective spaces, Trans. Amer. Math.

Soc. 193 (1974), 123–132.

Mitsunori Imaoka

Department of Mathematics Education Graduate Course of Education

Hiroshima University Higashi-Hiroshima 739-8524, Japan

E-mail: imaoka@hiroshima-u.ac.jp