Extendibility and stable extendibility of vector bundles over lens spaces mod 3

35

Extendibility and stable extendibility of vector bundles over lens spaces mod 3

Dedicated to the Memory of Professor Masahiro Sugawara Teiichi Kobayashi and Kazushi Komatsu

(Received August 27, 2004)

Abstract. In this paper, we prove that the tangent bundletðLⁿð3ÞÞ of theð2nþ1Þ- dimensional mod 3 standard lens space Lⁿð3Þis stably extendible to L^mð3Þfor every mbn if and only if 0ana3. Combining this fact with the results obtained in [6], we see that tðL²ð3ÞÞ is stably extendible to L³ð3Þ, but is not extendible to L³ð3Þ.

Furthermore, we prove that thet-fold power oftðLⁿð3ÞÞ and its complexification are extendible to L^mð3Þ for every mbn if tb2, and have a necessary and su‰cient condition that the squaren²of the normal bundlenassociated to an immersion ofLⁿð3Þ in the Euclidean ð4nþ3Þ-space is extendible to L^mð3Þ for everymbn.

1. Definitions and results

The extension problem is one of the fundamental problems in topology.

We study the problem for F-vector bundles over standard lens spaces mod 3, where F is either the real number field R or the complex number field C.

First, we recall the definitions of extendibility and stable extendibility according to [12] and [2]. Let X be a space and A be 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) to z as F-vector bundles, that is, if z is equivalent (respectively stably equivalent) to the induced bundle ia of ak-dimensional F-vector bundle a overX under the inclusion map i:A!X. For simplicity, we use the same letter for an F-vector bundle and its equivalence class, and use a non-negative integer k for the k-dimensional trivial F-vector bundle.

For a non-negative integer n and an integer q>1, let LⁿðqÞ denote the ð2nþ1Þ-dimensional standard lens space modq and L₀ⁿðqÞ its 2n-skeleton (cf.

[3], [4] and [11]). For a positive integern, leth_n stand for the canonicalC-line

2000 Mathematics Subject Classification. Primary 55R50; Secondary 55S25.

Key words and phrases.extendible, stably extendible, tangent bundle, tensor product, immersion, normal bundle, KO-theory,K-theory, lens space.

n

striction of h_n to L₀ⁿðqÞ. For a di¤erentiable manifoldM, let tðMÞdenote the tangent bundle of M.

On extendibility and stable extendibility of tangent bundles over standard lens spaces mod 3, we have

Theorem 1. As for the tangent bundle tðLⁿð3ÞÞ of Lⁿð3Þ. The following three conditions are equivalent:

(1) tðLⁿð3ÞÞ is stably extendible to L^mð3Þ for every mbn.

(2) tðLⁿð3ÞÞ is stably extendible to L^2nþ2ð3Þ.

(3) 0ana3:

Combining Theorem 1 with Theorem 5.1 of [6], we obtain

Corollary 2. tðL²ð3ÞÞis stably extendible to L³ð3Þ, but is not extendible to L³ð3Þ.

Another example of an R-vector bundle that is stably extendible but is not extendible is given by the tangent bundle tðSⁿÞ of the n-sphere Sⁿ in the ðnþ1Þ-sphere S^nþ1 for n01;3;7 (cf. [10, Proof of Theorem 2.2]).

Let c:KRðXÞ !KCðXÞ be the complexification. Then we have

Theorem 3. The complexification ctðLⁿð3ÞÞ of tðLⁿð3ÞÞ is stably extendible to L^mð3Þfor every m with nama2nþ1, but ctðLⁿð3ÞÞis not stably extendible to L^2nþ2ð3Þ if nb6.

For a positive integer t, let tðLⁿðqÞÞ^t¼tðLⁿðqÞÞn ntðLⁿðqÞÞ (t-fold) be the t-fold power oftðLⁿðqÞÞ, wherendenotes the tensor product. Then we prove

Theorem 4. tðLⁿð3ÞÞ^t is extendible to L^mð3Þ for every mbn if tb2.

Theorem 5. The complexification ctðLⁿð3ÞÞ^t of tðLⁿð3ÞÞ^t is extendible to L^mð3Þ for every mbn if tb2.

As for the normal bundle n associated to an immersion of Lⁿð3Þ in the Euclidean ð4nþ3Þ-space R^4nþ3, it was proved in [9, Theorem B] that the following three conditions are equivalent:

(1) n is extendible to L^mð3Þ for every mbn.

(2) n is stably extendible to L^mð3Þ for every mbn.

(3) 0ana5:

For the square n² ¼nnn of n, we have

Theorem 6. Let n be the normal bundle associated to an immersion of Lⁿð3Þ in R^4nþ3 and n² its square. Then the following three conditions are equivalent:

(1) n² is extendible to L^mð3Þ for every mbn.

(2) n² is stably extendible to L^mð3Þ for every mbn.

(3) 0ana13 or n¼15.

Theorem 2 of [5] is the result which corresponds to Theorem 6 for extendibility of the square of the normal bundle associated to an immersion of the real projective n-space RPⁿ in R^2nþ1.

This paper is arranged as follows. In Section 2 we recall results that are necessary for our proofs. In Section 3 we prove Theorem 1, Corollary 2 and Theorem 3. In Section 4 we prepare lemmas and prove Theorems 4 and 5. In Section 5 we give Whitney sum decompositions of the squaresn² of the normal bundles n associated to immersions of Lⁿð3Þ in R^4nþ3 for 0ana13 and n¼15 and prove Theorem 6.

2. Preliminaries

For a positive integer n, let h_n denote the canonical C-line bundle over Lⁿð3Þands_n¼h_n1 its stable class. Letr:K_CðXÞ !K_RðXÞbe the forgetful map and Z=q denote the cyclic group of order q, where q is an integer>1.

For a real number x, let bxc denote the largest integer s with sax.

The ring structure of the reduced Grothendieck ring KK~_RðLⁿð3ÞÞ is de- termined in [3] as follows (cf. [4] and [11]).

Theorem 2.1 (cf. [3, Theorem 2] and [4, Proposition 2.11]).

K~

K_RðLⁿð3ÞÞG KK~RðLⁿ₀ð3ÞÞ þZ=2 for n10 mod 4;

K~

K_RðLⁿ₀ð3ÞÞ otherwise;

(

where þ denotes the direct sum. The group KK~RðLⁿ₀ð3ÞÞ is isomorphic to the cyclic group Z=3^bn=2cof order3^bn=2cand is generated by rs_n. Moreover, the ring structure is given by

ðrsnÞ²¼ 3rsn; namely ðrh_nÞ²¼rh_nþ2; and ðrsnÞ^bn=2cþ1¼0:

The ring structure of the reduced Grothendieck ring KK~CðLⁿð3ÞÞ is de- termined in [3] as follows (cf. [4] and [11]).

Theorem 2.2 (cf. [3, Theorem 1] and [4, Lemma 2.4]).

K~

K_CðLⁿð3ÞÞGKK~_CðLⁿ₀ð3ÞÞG Z=3^bn=2cþZ=3^bn=2c for even n;

Z=3^bn=2cþ1þZ=3^bn=2c for odd n:

The first summand is generated by s_n and the second summand is generated by s²_n. Moreover, the ring structure is given by

s_n³¼ 3s_n²3sn; namely h_n³¼1; and s_n^nþ1 ¼0:

for our proofs.

Theorem 2.3 (cf. [6, Theorem 1.1] and [8, Theorem 3.1]). Let p be an odd prime and z be a k-dimensional R-vector bundle over LⁿðpÞ. Assume that there is a positive integer l such thatzis stably equivalent to a sum ofbk=2c þl non-trivial2-dimensional R-vector bundles andbk=2c þl<p^bn=ðp1Þc. Then n<

2bk=2c þ2l and z is not stably extendible to L^mðpÞ for every m with mb 2bk=2c þ2l.

Theorem 2.4 (cf. [7, Theorem 1.1] and [9, Theorem 4.5]). Let p be a prime and z be a k-dimensional C-vector bundle over LⁿðpÞ. Assume that there is a positive integer l such that z is stably equivalent to a sum of kþl non- trivial C-line bundles and kþl<p^bn=ðp1Þc. Then n<kþl andzis not stably extendible to L^mðpÞ for every m with mbkþl.

Let d ¼1 or 2 according as F ¼R or C. For a real number x, let dxe denote the smallest integer s with xas. The following results are known.

Theorem 2.5 (cf. [1, Theorem 1.2, p. 99]). Let m¼ dðnþ1Þ=d1e.

Then each k-dimensional F-vector bundle over an n-dimensional CW-complex X is equivalent to alðkmÞ for some m dimensional F-vector bundle a over X if mak.

Theorem 2.6 (cf. [1, Theorem 1.5, p. 100]). Let m¼ dðnþ2Þ=d1e.

Then two k-dimensional F-vector bundles over an n-dimensional CW-complex which are stably equivalent are equivalent if mak.

3. Proofs of Theorem 1, Corollary 2 and Theorem 3

Let q be any integer>1. As for extendibility of tðLⁿðqÞÞ, the following result is obtained.

Theorem 3.1 ([6, Theorems 5.1 and 5.3]). For any integer q>1, the following three conditions are equivalent:

(1) tðLⁿðqÞÞ is extendible to L^mðqÞ for every mbn.

(2) tðLⁿðqÞÞ is extendible to L₀^nþ1ðqÞ.

(3) n¼0;1 or 3.

As for stable extendibility of tðLⁿðqÞÞ, the following result is obtained.

Theorem 3.2 ([8, Theorem 4.3]). Let p be an odd prime. Then tðLⁿðpÞÞ is not stably extendible to L^2nþ2ðpÞ, if nb2p2.

Proof of Theorem 1. Obviously, (1) implies (2). It follows from Theorem 3.2 that (2) implies (3), since n<2p2 for p¼3 if and only if

0ana3. Hence it remains to prove that (3) implies (1). If n¼0;1 or 3, (1) holds by Theorem 3.1. Let n¼2. Then rh₂2 is of order 3 by Theorem 2.1, and so 3rh₂¼6 in KRðL²ð3ÞÞ. As is well-known,

tðL²ð3ÞÞl1¼3rh₂

So we have tðL²ð3ÞÞ ¼3rh₂1¼5 in K_RðL²ð3ÞÞ. Hence tðL²ð3ÞÞ is stably trivial, and so tðL²ð3ÞÞ is stably extendible to L^mð3Þ for every mb2, as

desired. r

Proof of Corollary 2. The former part follows from Theorem 1. By Theorem 3.1, tðL²ð3ÞÞis not extendible toL₀³ð3Þ, and hence is not extendible to

L³ð3Þ. So we obtain the latter part. r

Proof of Theorem 3. As is well-known, tðLⁿð3ÞÞl1¼ ðnþ1Þrh_n:

Applying the complexification c:KRðLⁿð3ÞÞ !KCðLⁿð3ÞÞ to the both sides of the equality, we have

ctðLⁿð3ÞÞl1¼ ðnþ1Þcrh_n¼ ðnþ1Þðh_nþh²_nÞ;

sincecrh_n¼h_nþh¹_n (cf. [1, Proposition 11.3, p. 191]) andh_n³¼1 (cf. Theorem 2.2).

Supposema2nþ1. Then dimfðnþ1Þðh_mþh_m²Þg dð2mþ1þ1Þ=21e

¼2nþ2mb1. Hence, by Theorem 2.5, there is a ð2nþ1Þ-dimensional C-vector bundle a over L^mð3Þ such that

ðnþ1Þðh_mþh_m²Þ ¼al1:

Let nam and i:Lⁿð3Þ !L^mð3Þ be the standard inclusion. Then, applying i to the both sides of the equality above, we have

ðnþ1Þðh_nþh_n²Þ ¼ial1;

since ih_m¼h_n. Hence ctðLⁿð3ÞÞ is stably equivalent to ia. Now, both ctðLⁿð3ÞÞ and ia are ð2nþ1Þ-dimensional. So ctðLⁿð3ÞÞ is stably extendible to L^mð3Þ. Thus the former part of the theorem is proved.

Put p¼3,z¼ctðLⁿð3ÞÞ, k¼2nþ1 andl¼1 in Theorem 2.4. Then the latter part of the theorem follows from Theorem 2.4, since 2nþ2<3^bn=2c if

and only if nb6. r

4. Proofs of Theorems 4 and 5

In Sections 4 and 5, h denotes the canonical C-line bundle h_n over Lⁿð3Þ andNthe set of all positive integers. We prepare some lemmas for our proofs.

g:N!N such that

tðLⁿð3ÞÞ^t¼gðtÞrhþ ð2nþ1Þ^t2gðtÞ in KRðLⁿð3ÞÞ;

namely tðLⁿð3ÞÞ^t ð2nþ1Þ^t¼gðtÞðrh2Þ in KK~_RðLⁿð3ÞÞ. Furthermore, the function gðtÞ is uniquely determined modulo 3^bn=2c.

Proof. We prove the first part of the lemma by induction on t. Since tðLⁿð3ÞÞ ¼ ðnþ1Þrh1 in KRðLⁿð3ÞÞ, we may define gð1Þ ¼nþ1. Assume that there exists gðtÞ for every tb1 such that tðLⁿð3ÞÞ^t¼gðtÞrhþ ð2nþ1Þ^t 2gðtÞ and gð1Þ ¼nþ1. Then, by Theorem 2.1,

tðLⁿð3ÞÞ^tþ1¼ fgðtÞrhþ ð2nþ1Þ^t2gðtÞgfðnþ1Þrh1g

¼gðtÞðnþ1ÞðrhÞ²þ fð2nþ1Þ^tðnþ1Þ 2gðtÞðnþ1Þ gðtÞgrh ð2nþ1Þ^tþ2gðtÞ

¼ fð2nþ1Þ^tðnþ1Þ gðtÞðnþ2Þgrh ð2nþ1Þ^tþ2gðtÞðnþ2Þ:

Now set

gðtþ1Þ ¼ ð2nþ1Þ^tðnþ1Þ gðtÞðnþ2Þ.

Then we have ð2nþ1Þ^tþ2gðtÞðnþ2Þ ¼ ð2nþ1Þ^tþ12gðtþ1Þ, as desired.

Suppose there are two functions f;g:N!N such that fðtÞrhþ ð2nþ1Þ^t2fðtÞ ¼gðtÞrhþ ð2nþ1Þ^t2gðtÞ:

Then ðfðtÞ gðtÞÞðrh2Þ ¼0, and so fðtÞ gðtÞ10 mod 3^bn=2c by Theorem

2.1. So we have the latter part. r

Lemma 4.2. There is a function g:N!N defined in Lemma 4.1 which satisfies the inequalities:

ð2nþ1Þ^t1<gðtÞ<2¹ð2nþ1Þ^t for nb3 and tb2:

Proof. We prove the lemma by induction on t. Define gð1Þ ¼nþ1.

Next, by Theorem 2.1,

tðLⁿð3ÞÞ²¼ fðnþ1Þrh1g² ¼ ðn²1Þrhþ2n²þ4nþ3:

Define gð2Þ ¼n²1. Then clearly 2nþ1<gð2Þ<2¹ð2nþ1Þ² for nb3.

Assume that there exists gðtÞ for tb2 which satisfies the inequalities:

ð2nþ1Þ^t1<gðtÞ<2¹ð2nþ1Þ^t fornb3. As in the proof of Lemma 4.1, set gðtþ1Þ ¼ ð2nþ1Þ^tðnþ1Þ gðtÞðnþ2Þ. Then, by the inductive assumption,

gðtþ1Þ>ð2nþ1Þ^tðnþ1Þ 2¹ð2nþ1Þ^tðnþ2Þ

¼2¹ð2nþ1Þ^tn>ð2nþ1Þ^t

and

gðtþ1Þ<ð2nþ1Þ^tðnþ1Þ ð2nþ1Þ^t1ðnþ2Þ

¼ ð2nþ1Þ^t1ð2n²þ2n1Þ

<ð2nþ1Þ^t1ð2n²þ2nþ1=2Þ ¼2¹ð2nþ1Þ^tþ1:

Thus the inequalities: ð2nþ1Þ^t <gðtþ1Þ<2¹ð2nþ1Þ^tþ1 hold. r Proof of Theorem4. Ifn¼0;1 or 3,tðLⁿð3ÞÞis extendible toL^mð3Þfor every mbn by Theorem 3.1. Hence tðLⁿð3ÞÞ^t is extendible to L^mð3Þ for every mbn, where tb1. If n¼2, we see in the proof of Theorem 1 that tðL²ð3ÞÞ is stably trivial. So tðL²ð3ÞÞ^t is stably trivial, where tb1. If, in addition, tb2,tðL²ð3ÞÞ^t is trivial by Theorem 2.6, sincedðdimL²ð3Þ þ2Þ 1e

¼6a5^t¼dimtðL²ð3ÞÞ^t. Hence tðL²ð3ÞÞ^t is extendible to L^mð3Þ for every mbn, if tb2. We may therefore devote our attention to the case where nb4.

According to Lemmas 4.1 and 4.2, there is a positive integer gðtÞ such that tðLⁿð3ÞÞ^t¼gðtÞrhþ ð2nþ1Þ^t2gðtÞ in K_RðLⁿð3ÞÞ and ð2nþ1Þ^t2gðtÞ

>0 for nb3 and tb2. Since dðdimLⁿð3Þ þ2Þ 1e ¼2nþ2að2nþ1Þ^t¼ dimtðLⁿð3ÞÞ^t for nb1 and tb2, we have the equality

tðLⁿð3ÞÞ^t¼gðtÞrhlfð2nþ1Þ^t2gðtÞg

of R-vector bundles by Theorem 2.6. Since rh and the trivial R-vector bundle over Lⁿð3Þ are extendible to L^mð3Þ for every mbn, tðLⁿð3ÞÞ^t is extendible to

L^mð3Þ for every mbn, as desired. r

Complexifying the equality in Lemma 4.1, we have

Lemma 4.3. For the function g:N!N in Lemmas 4.1 and 4.2, ctðLⁿð3ÞÞ^t¼gðtÞðhþh²Þ þ ð2nþ1Þ^t2gðtÞ in KCðLⁿð3ÞÞ:

Proof. Sincecrh¼hþh², the result follows from the equality in Lemma

4.1. r

Proof of Theorem5. As is well-known, tðL¹ð3ÞÞis trivial. In the proof of Theorem 1, we see that tðL²ð3ÞÞ is stably trivial. So ctðL¹ð3ÞÞ^t and ctðL²ð3ÞÞ^t are stably trivial for any tb1. Furthermore, ctðL¹ð3ÞÞ^t and ctðL²ð3ÞÞ^t are trivial for any tb1 by Theorem 2.6, since dðdimL¹ð3Þ þ2Þ= 21e ¼2a3^t¼dimctðL¹ð3ÞÞ^t and dðdimL²ð3Þ þ2Þ=21e ¼3a5^t¼ dimctðL²ð3ÞÞ^t hold for any tb1. Hence we have the results for n¼1 and n¼2, since the trivial C-bundle over Lⁿð3Þ is extendible to L^mð3Þ for every mbn.

Since dðdimLⁿð3Þ þ2Þ=21e ¼nþ1að2nþ1Þ^t¼dimctðLⁿð3ÞÞ^t holds for any tb1, it follows from Lemma 4.3 that the equality

ctðLⁿð3ÞÞ^t ¼gðtÞðhlh²Þlfð2nþ1Þ^t2gðtÞg

of C-vector bundles holds by Theorem 2.6. Since h, h² and the trivial C- vector bundle over Lⁿð3Þare extendible to L^mð3Þfor every mbn,ctðLⁿð3ÞÞ^t is extendible to L^mð3Þ for every mbn, as desired. r

5. Proof of Theorem 6

First, we study the square of the normal bundle associated to an im- mersion of Lⁿð3Þ in R^4nþ3.

Theorem 5.1. Let n¼nðf_nÞ be the normal bundle associated to an im- mersion fn:Lⁿð3Þ !R^4nþ3 and n²¼nðfnÞ² its square. Then we have the Whitney sum decompositions:

nðf₀Þ²¼4; nðf₁Þ²¼16; nðf₂Þ²¼36;

nðf₃Þ²¼2rh₃l60; nðf₄Þ²¼5rh₄l90; nðf₅Þ²¼144;

nðf6Þ²¼8rh₆l180; nðf7Þ²¼11rh₇l234; nðf8Þ²¼324;

nðf9Þ²¼29rh₉l342; nðf10Þ²¼125rh₁₀l234;

nðf11Þ²¼207rh₁₁l162; nðf12Þ²¼275rh₁₂l126;

nðf13Þ²¼86rh₁₃l612; nðf15Þ²¼395rh₁₅l234:

Proof. If n¼0, the result is clear. Hence we assume that n>0. Let t¼tðLⁿð3ÞÞ denote the tangent bundle of Lⁿð3Þ. Then tl1¼ ðnþ1Þrh and tln¼4nþ3. Hence n¼ ðnþ1Þrhþ4nþ4. By Theorem 2.1, we have

n²¼ ðnþ1Þ²ðrhÞ²2ðnþ1Þð4nþ4Þrhþ ð4nþ4Þ²

¼ ðnþ1Þ²ðrhþ2Þ 8ðnþ1Þ²rhþ16ðnþ1Þ²

¼ fa3^bn=2c7ðnþ1Þ²grhþ18ðnþ1Þ²2a3^bn=2c

in K_RðLⁿð3ÞÞ, where a is any integer. If a3^bn=2c7ðnþ1Þ²b0 and 18ðnþ1Þ²2a3^bn=2cb0, then we have the equality

n²¼ fa3^bn=2c7ðnþ1Þ²grhlf18ðnþ1Þ²2a3^bn=2cg

of R-vector bundles, since dðdimLⁿð3Þ þ2Þ 1e ¼2nþ2að2nþ2Þ²¼dimn² by Theorem 2.6.

Puta¼28 forn¼1,a¼21 forn¼2,a¼38 forn¼3,a¼20 forn¼4, a¼28 for n¼5,a¼13 forn¼6,a¼17 forn¼7,a¼7 forn¼8,a¼9 for n¼9, a¼4 for n¼10, a¼5 for n¼11, a¼2 for n¼12, a¼2 for n¼13 and a¼1 for n¼15. Then we can check easily that the two inequalities a3^bn=2c7ðnþ1Þ²b0 and 18ðnþ1Þ²2a3^bn=2cb0 hold, if 1ana13 or

n¼15. r

Theorem 3.1 of [5] is the result corresponding to Theorem 5.1 for the square of the normal bundle associated to an immersion of RPⁿ in R^2nþ1.

Theorem 5.2. Under the assumption of Theorem 5.1, the following two equalities hold in K_RðL¹⁴ð3ÞÞ and K_RðL¹⁶ð3ÞÞ, respectively.

nðf₁₄Þ²¼612rh₁₄324; nðf₁₆Þ²¼4538rh₁₆7920:

Proof. Putting a¼1 for n¼14 and 16 in the proof of Theorem 5.1, we

have the desired equalities. r

Using Theorem 2.3, we prove

Theorem 5.3. Let n be the normal bundle associated to an immersion of Lⁿð3Þ in R^4nþ3. Then the square n² of n is not stably extendible to L^mð3Þ for m¼2f3^bn=2c7ðnþ1Þ²g, if n¼14 or nb16.

Proof. We see in the proof of Theorem 5.1 that n² is stably equivalent to f3^bn=2c7ðnþ1Þ²grh. Note that 3^bn=2c9ðnþ1Þ²>0 if n¼14 or nb16.

Then, putting p¼3, z¼n², k¼4ðnþ1Þ² and l¼3^bn=2c9ðnþ1Þ² in The- orem 2.3, we see that n² is not stably extendible to L^mð3Þ for m¼2f3^bn=2c 7ðnþ1Þ²g, if n¼14 or nb16, by Theorem 2.3. r Corollary 5.4. Under the assumption of Theorem5.1,nðf14Þ² and nðf16Þ² are not stably extendible to L¹²²⁴ð3Þ and L⁹⁰⁷⁶ð3Þ, respectively.

Proof. The results follow from Theorems 5.2 and 5.3. r Proof of Theorem 6. Clearly (1) implies (2). It follows from Theorem 5.3 that (2) implies (3). It follows from Theorem 5.1 that (3) implies (1), since rh and trivial R-vector bundles over Lⁿð3Þ are extendible to L^mð3Þ for every

mbn. r

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T. Kobayashi Asakura-ki 292-21 Kochi 780-8066 Japan E-mail: kteiichi@lime.ocn.ne.jp

K. Komatsu

Department of Mathematics Faculty of Science

Kochi University Kochi 780-8520 Japan E-mail: komatsu@math.kochi-u.ac.jp