Stable extendibility of vector bundles over real projective spaces and bounds for the Schwarzenberger numbers b(k)
Teiichi Kobayashi and Toshio Yoshida
(Received April 8, 2003) (Revised July 7, 2003)
Abstract. For a non-negative integerk, R. L. E. Schwarzenberger defined in [7] an integer bðkÞb0 which we call the Schwarzenberger number of k. Let z be a k- dimensionalF-vector bundle over the real projectiven-spaceRPn, whereF is either the real number fieldRor the complex number fieldC. ThenbðkÞis closely related to the problem to find the dimension m withmbnwhich has the property that zis stably equivalent to a sum ofk F-line bundles ifzis stably extendible toRPm. The problem forF¼Rhas been studied in [7], [5] and [4], and that forF¼Chas been studied in [6]
and [4]. In this note we obtain further results on the problem and determine bounds for the Schwarzenberger numbers bðkÞ.
1. Introduction
Throughout this note, F denotes either the real number field R or the complex number field C, and N is the set of all non-negative integers. Let X be a space and A its subspace. A k-dimensional F-vector bundle z over A is said to be extendible (respectively stably extendible) to X, if there is a k-dimensional F-vector bundle over X whose restriction to A is equivalent (respectively stably equivalent) toz, that is, ifzis equivalent (respectively stably equivalent) to the induced bundle ih of a k-dimensional F-vector bundle h over X under the inclusion map i:A!X (cf. [7, p. 20], [8, p. 191] and [3, p. 273]).
For a positive integeri, writei¼ ð2aþ1Þ2nðiÞ, whereaAN, and for kAN define an integer bðkÞAN by
bðkÞ ¼minfinðiÞ 1jk<ig which we call the Schwarzenberger number of k.
Let zbe a k-dimensional F-vector bundle over the real projective n-space RPn where k>0. We study the problem to find the dimensionmwith mbn which has the property that zis stably equivalent to a sum ofk F-line bundles if z is stably extendible to RPm.
2000 Mathematics Subject Classification. Primary 55R50; Secondary 55N15.
Key words and phrases. vector bundle, extendible, stably extendible, KO-theory,K-theory, real projective space, Schwarzenberger number.
Let fðnÞ denote the number of integers s with 0<san and s10;1;2 or 4 mod 8. For F ¼R, we have
Theorem 1. Let z be a k-dimensional R-vector bundle over RPn, where k>0, and consider the following four conditions.
(1) z is stably extendible to RPm for every mbn.
(2) z is stably extendible to RPm, where mbn, mb2k1 and fðmÞb fðnÞ þbðkÞ.
(3) z is stably extendible to RPm, where m¼2fðnÞ1.
(4) z is stably equivalent to a sum of k R-line bundles.
Then all the four conditions are equivalent. Moreover, when k¼1 or n¼1;3 or 7, the conditions always hold.
Let ½x denote the largest integer q with qax. For F ¼C, we have Theorem 2. Let z be a k-dimensional C-vector bundle over RPn, where k>0, and consider the following four conditions.
(1) z is stably extendible to RPm for every mbn.
(2) z is stably extendible to RPm, where mbn, mb4k1 and fðmÞb
½n=2 þbð2kÞ þ1.
(3) z is stably extendible to RPm, where m¼2½n=2þ11.
(4) z is stably equivalent to a sum of k C-line bundles.
Then all the four conditions are equivalent. Moreover, when k¼1 or n¼1;2 or 3, the conditions always hold.
Concerning bounds for the Schwarzenberger numbers bðkÞ, we obtain Theorem 3. Let k be a positive integer, let aðkÞ denote the number of the non-zero terms of the 2-adic expansion of k, and let bðkÞ denote the Schwar- zenberger number of k. Then the inequalities kaðkÞabðkÞak hold.
This note is arranged as follows. We study some properties of bðkÞ in Section 2. We prove Theorems 1 and 2 in Section 3, and prove Theorem 3 in Section 4.
2. Some properties of b(k)
Lemma 2.1. Let k be a positive integer and t be any integer with k<2t. Then bðkÞ ¼minfinðiÞ 1jk<ia2tg.
Proof. Clearly it su‰ces to prove that
minfinðiÞ 1j2t<igbminfinðiÞ 1jk<ia2tg:
Comparing 2tnð2tÞ 1 with inðiÞ 1 for i¼a2tþb, where ab1 and 0<b<2t, and with inðiÞ 1 for i¼a2t, where a>1, we have
a2tþbnða2tþbÞ 1 f2tnð2tÞ 1g
¼ ða1Þ2tþbnðbÞ 1þtþ1>0;
and
a2tnða2tÞ 1 f2tnð2tÞ 1g ¼ ða1Þ2tnðaÞ>a1nðaÞb0;
since
jnðjÞ 1¼ ð2xþ1Þ2ynðð2xþ1Þ2yÞ 1b2yy1b0;
where j¼ ð2xþ1Þ2y ðx;yANÞ. We therefore obtain the desired inequality.
r Remark. It seems to us that in line 11 of [7, p. 21], the last inequality i<2t should be replaced by the inequality ia2t. In fact, minfinðiÞ 1j k<i<2tg is not necessarily equal to bðkÞ (for example, if ðk;tÞ ¼ ð6;3Þ, minfinðiÞ 1jk<i<2tg ¼6 and bðkÞ ¼4), the first inequality in line 11 of [7, p. 21] holds also fori¼2t and minfinðiÞ 1jk<ia2tg is equal tobðkÞ by Lemma 2.1.
Lemma 2.2. bð2rÞ ¼2r for rb2, bð2Þ ¼1 and bð1Þ ¼0.
Proof. We prove the first equality. Suppose rb2. Since bð2rÞa2r (cf. [7, Examples]), it su‰ces to prove that minfinðiÞ 1j2r<ia2rþ1gb2r by Lemma 2.1. If i¼2rþb, where 0<b<2r, we have inðiÞ 1¼2rþ bnðbÞ 1b2r, and ifi¼2rþ1, we haveinðiÞ 1¼2rþ1 ðrþ1Þ 1b2r for rb2. Hence we obtain the desired inequality. The other equalities are
easily verified. r
Corollary 2.3. bð0Þ ¼0, and bðkÞ>0 for k>1.
Proof. By definition, jbk implies bðjÞbbðkÞ. Hence the results fol-
low from Lemma 2.2. r
Lemma 2.4. Let j¼1;2;3 or 4 and let k¼2rj, where rb0 for j¼1, rb1 for j¼2, and rb3 for j¼3 or 4. Then
bðkÞ ¼2rr1:
Moreover,
bð2r5Þ ¼2rr1 for rb6; bð2r5Þ ¼2r7 for 3ara5:
Proof. By definition and by Lemma 2.2, we have
bð2r1Þ ¼minf2rnð2rÞ 1;bð2rÞg ¼minf2rr1;2rdg
¼2rr1;
where d¼0 ifrb2 and d¼1 ifr¼0 or 1. By the result above, we have, for rb1,
bð2r2Þ ¼minf2r1nð2r1Þ 1;bð2r1Þg
¼minf2r2;2rr1g ¼2rr1:
Similarly, we have, for rb3,
bð2r3Þ ¼minf2r2nð2r2Þ 1;bð2r2Þg
¼minf2r4;2rr1g ¼2rr1;
bð2r4Þ ¼minf2r3nð2r3Þ 1;bð2r3Þg
¼minf2r4;2rr1g ¼2rr1:
Moreover,
bð2r5Þ ¼minf2r4nð2r4Þ 1;bð2r4Þg
¼minf2r7;2rr1g
¼2rr1 for rb6; ¼2r7 for 3ara5: r
3. Proofs of Theorems 1 and 2
Proof of Theorem1. Clearly (1) implies (2) and (3). In [7, Theorem 3]
R. L. E. Schwarzenberger proved that (2) implies (4) (cf. Remark in Section 2).
In the original result of Schwarzenberger, the R-vector bundle z is assumed to be extendible, but his result is also valid if we only assume that z is stably extendible instead of extendible (cf. [3, Section 1]). We proved in [4, Theorem 3.1(i)] that (3) implies (4) for n01;3;7, and in [4, Theorem 3.2] that (4) is equivalent to (1). We therefore proved the theorem for the case n01;3;7.
When n¼1;3 or 7, it su‰ces to prove (4). In fact, (4) for n¼1;3 or 7 follows from [4, Theorem 3.1 (ii)]. The latter part for k¼1 is clear (cf. [4,
Theorem 3.2]). r
Remark. If k>1, then bðkÞ>0 by Corollary 2.3 and so the inequality fðmÞbfðnÞ þbðkÞ implies the inequality m>n.
Proof of Theorem 2. Clearly (1) implies (2) and (3). We proved in [6, Theorem 2.2 and Remark] that (3) implies (4) for n>3, and in [4, Theorem 3.2] that (4) is equivalent to (1). Hence for the proof for the case n>3, it su‰ces to prove that (2) implies (4). Though the proof is parallel to that of [7, Theorem 3], for completeness we prove that (2) implies (4) below.
Assume that (2) holds for z. Then there is a k-dimensional C-vector bundle h over RPm such that ih is stably equivalent to z, where i:RPn! RPm is the standard inclusion. According to [1, Theorem 7.3], we have, for some integer q with 0aq<2½m=2, hk¼qcðxm1Þ in the reduced K-group K~
KðRPmÞ, and so
zk¼ihk¼qcðixm1Þ ¼qcðxn1Þ
in KKðRP~ nÞ, where xN is the canonical R-line bundle over RPN and c denotes the complexification. Let r denote the forgetful map. Then
rh2k¼rcqðxm1Þ ¼2qðxm1Þ;
since rc¼2. In the terminology of [2, Section 2], the element 2qðxm1Þ has geometrical dimension not exceeding 2k. If qak, then z is stably equivalent to a sum of k C-line bundles. If q>k, then, by [1, Theorem 7.4] and [2, Proposition 2.3], for all i>2k,
gið2qðxm1ÞÞ ¼C2q;iðxm1Þi¼ ð1Þi12i1C2q;iðxm1Þ ¼0;
where gi is the Grothendiek operator and Cj;i is the binomial coe‰cient j!=ððjiÞ!i!Þ, and so, by [1, Theorem 7.4],
i1þnðC2q;iÞbfðmÞ for all i with 2k<ia2q:
Since rh is stably equivalent to 2qxm, the total Stiefel-Whitney class wðrhÞ of rh is ð1þxmÞ2q, where xm is the generator of H1ðRPm;Z2Þ. On the other hand, by [7, Theorem 2],
wðrhÞ ¼wðsxmlð2ksÞÞ ¼ ð1þxmÞs for some s with 0asa2k;
since mb4k1, where l denotes the Whitney sum. Hence 2q¼ ð2aþ1Þ2tþs for some aAN and tAN with m<2t: It follows from the inequalities 3am<2t that tb2. So s is even and 0as=2ak. Now,
nðCð2aþ1Þ2tþs;iÞ ¼nðC2tþs;iÞanðC2t;iÞ for all i with s<ia2t; and nðC2t;iÞ ¼tnðiÞ. Therefore
i1þtnðiÞbfðmÞ for all i with 2k<ia2t;
and sot1bfðmÞ bð2kÞ 1b½n=2 by Lemma 2.1 and by the assumption.
Therefore zk¼ ðs=2Þcðxn1Þ, since cðxn1Þ is of order 2½n=2 by [1, The- orem 7.3]. Hence z is stably equivalent to a sum of k C-line bundles and so (4) holds. Thus we have completed the proof of the former part.
To prove the theorem for the case n¼1;2 or 3, it su‰ces to prove (4) (cf. [4, Theorem 3.2]). By [1, Theorem 7.3] there is an integer l such that zk¼ ðkþlÞcðxn1Þ, where 0akþl<2½n=2. If l>0,½n=2<kþl by [6, Theorem 2.1]. This contradicts the inequality kþl<2½n=2 if n¼1;2 or 3.
Hence la0, and so (4) holds. The latter part for k¼1 is clear (cf. [4, The-
orem 3.2]). r
Remark. For k>0, bð2kÞ>0 by Corollary 2.3 and so the inequality fðmÞb½n=2 þbð2kÞ þ1 implies the inequality m>n.
4. Proof of Theorem 3
Lemma 4.1. For 0<i<2r, að2ri1Þ ¼að2riÞ þnðiÞ 1.
Proof. Let nðiÞ ¼s. Then nð2riÞ ¼s and 2ri¼ ð2aþ1Þ2s, for some aAN. Hence 2ri1¼a2sþ1þ2s1þ2s2þ þ2þ1 and so að2ri1Þ ¼aða2sþ1þ2s1Þ þs1¼aða2sþ1þ2sÞ þs1¼að2riÞ þnðiÞ
1. r
Theorem 3 is a consequence of the following result.
Theorem 4.2. Let k be a positive integer. Then we have the following.
( i ) If k02r1 and k02r2, kaðkÞ<bðkÞake, wheree¼0 for k even and e¼1 for k odd.
(ii) If k¼2r1 or k¼2r2, bðkÞ ¼kaðkÞ.
Proof. (i) The inequality bðkÞake follows from [7, Examples].
By Lemma 2.2, clearly the inequality kaðkÞ<bðkÞ holds for k¼2r for rb2. We prove the inequality kaðkÞ<bðkÞ fork02r1 and k02r2 by a downward induction on k¼2ri, where rb3 and 3ai<2r1. If i¼3, by Lemma 2.4, bð2r3Þ ¼2rr1. On the other hand, að2r3Þ ¼ r1, and so kaðkÞ<bðkÞ holds for k¼2r3.
Suppose that the inequality kaðkÞ<bðkÞ holds for k¼2ri, where 4aiþ1<2r1. Since bð2ri1Þ ¼minf2rinð2riÞ 1;bð2riÞg, bð2ri1Þ ¼2rinð2riÞ 1 or bð2ri1Þ ¼bð2riÞ.
If bð2ri1Þ ¼2rinð2riÞ 1, we have, by Lemma 4.1, bð2ri1Þ f2ri1að2ri1Þg
¼2rinð2riÞ 1 f2ri1 ðað2riÞ þnðiÞ 1Þg
¼að2riÞ 1>0;
since að2riÞ>1 for 3ai<2r1. If bð2ri1Þ ¼bð2riÞ, we have, by Lemma 4.1 and by the inductive assumption,
bð2ri1Þ f2ri1að2ri1Þg
¼bð2riÞ f2ri1 ðað2riÞ þnðiÞ 1Þg
¼bð2riÞ f2riað2riÞg þnðiÞ>0:
So the inequality kaðkÞ<bðkÞ holds for k¼2ri1.
(ii) Note that if k¼2r1 or k¼2r2, r¼aðkÞ or r¼aðkÞ þ1, respectively. By Lemma 2.4, if k¼2r1, bðkÞ ¼2rr1¼kaðkÞ, and
if k¼2r2, bðkÞ ¼2rr1¼kaðkÞ. r
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T. Kobayashi Asakura-ki 292-21 Kochi 780-8066 Japan E-mail: [email protected]
T. Yoshida
Department of Mathematics Faculty of Integrated Arts and Sciences
Hiroshima University Higashi-Hiroshima 739-8521 Japan E-mail: [email protected]
