Extendibility and stable extendibility of vector bundles over real projective spaces

Extendibility and stable extendibility of vector bundles over real projective spaces

Teiichi Kobayashi and Kazushi Komatsu

(Received April 3, 2000) (Revised June 5, 2000)

Abstract. The purpose ofthis paper is to study the extendibility and the stable extendibility ofvector bundles ofreal projective spaces and those oftheir complex- i®cations. We determine the dimensionmfor which the complexi®cation of the tangent bundle ofthen-dimensional real projective spaceRPⁿis extendible toRP^m fornˆ6 or n>7, and determine the dimensionn for which the square of the tangent bundle of RPⁿ or its complexi®cation is extendible toRP^m for every m>n.

1. Introduction

Let Xbe a space and Abe its subspace. At-dimensionalF-vector bundle z over A is called extendible (respectively stably extendible) to X, ifthere is a t-dimensional F-vector bundle over X whose restriction to A is equivalent (respectively stably equivalent) to z as F-vector bundles, where F is the real number ®eld R, the complex number ®eld Cor the quaternion number ®eld H (cf. [8] and [3]). Let Rⁿ be the n-dimensional Euclidean space, RPⁿ be the n-dimensional real projective space and t…RPⁿ† be the tangent bundle of RPⁿ.

First, we study the question: Determine the dimension m withm>n for which a vector bundle over RPⁿ is extendible to RP^m. We have obtained the complete answer for the tangent bundle t…RPⁿ† in [4, Theorem 6.6].

For an R-vector bundle and a C-vector bundle over RPⁿ we have Theorem1. Let zbe a t-dimensional R-vector bundle over RPⁿ. If n<t, z is extendible to RP^m for every m with n<mUt.

Theorem 2. Let z be a t-dimensional C-vector bundle over RPⁿ. If n<2t‡1, z is extendible to RP^m for every m with n<mU2t‡1.

For the complexi®cation ofthe tangent bundle t…RPⁿ†, we have

2000 Mathematics Subject Classi®cation. Primary 55R50; secondary 55N15.

Key words and phrases. vector bundle, extendible, stably extendible, tangent bundle, tensor product, K-theory, KO-theory, real projective space.

Theorem 3. Let ctbe the complexi®cation ct…RPⁿ† of the tangent bundle t…RPⁿ†. Then ctis extendible to RP^2n‡1, but is not stably extendible to RP^2n‡2 if nˆ6 or n>7. If nˆ1;3 or 7, ct is extendible to RP^m for every m with m>n.

Second, we study the question: Determine the dimension n for which a vector bundle over RPⁿ is extendible to RP^m for every m with m>n. We have obtained the complete answer for the normal bundle n associated to an immersion ofRPⁿ inR^2n‡1 in [7, Theorem A], and its complexi®cationcnin [7, Theorem 4.4]. For the square t²ˆt…RPⁿ†nt…RPⁿ† ofthe tangent bundle t…RPⁿ† and its complexi®cation ct², we have

Theorem 4. Let t² be the square t…RPⁿ†nt…RPⁿ† of the tangent bundle of RPⁿ. Then the following three conditions are equivalent:

(1) t² is extendible to RP^m for every m with m>n, (2) t² is stably extendible to RP^m for every m with m>n, (3) 1UnU16.

Theorem 5. Let ct² be the complexi®cation c…t…RPⁿ†nt…RPⁿ†† of the square of the tangent bundle of RPⁿ. Then the following three conditions are equivalent:

(1) ct² is extendible to RP^m for every m with m>n, (2) ct² is stably extendible to RP^m for every m with m>n, (3) 1UnU17.

This note is arranged as follows. We prove Theorem 1 in § 2. In § 3 we study the Whitney sum decomposition ofthe square ofthe tangent bundle of the real projective space and prove Theorem 4. In § 4 we prove Theorem 2 and the part ofthe extendibility ofTheorem 3. In § 5 we prove the part ofthe non-extendibility ofTheorem 3, study the Whitney sum decomposition ofthe complexi®cation ofthe square ofthe tangent bundle, and prove Theorem 5.

2. Extendibility of an R-vector bundle over the real projective space

Let x_n be the canonical R-line bundle over RPⁿ. The ring structure of KO…RPⁿ† is determined in [1]. We recall the results which are necessary for our proofs. We use the same letter for a vector bundle and its isomorphism class.

(2.1) [1, Theorem 7.4]. (1) The reduced KO-groupKO…RPg ⁿ† is isomorphic to the cyclic group Z=2^f…n†, generated by x_n 1, where f…n† is the number of integers s such that 0<sUn and s10;1;2 or 4 mod 8. (2) …x_n†²ˆ1.

Let ddenote dim_RF, where FˆR;C orH. For a real numberx, let dxe denote the smallest integer n with xUn. The following facts are known (cf.

[2, Theorems 1.2 and 1.5, p. 99±p. 100]).

(2.2). Let mˆ d…n‡1†=d 1e. Then every t-dimensional F-vector bundle over an n-dimensional CW-complex X is isomorphic to al…t m† for some m- dimensional F-vector bundle a over X if mUt, where l denotes the Whitney sum and t m denotes the …t m†-dimensional trivial F-vector bundle over X.

(2.3). Let lˆ d…n‡2†=d 1e. If a and b are two t-dimensional F-vector bundles over an n-dimensional CW-complex X such thatlUt and alkˆblk for some k-dimensional trivial F-bundle k over X, then aˆb.

Using (2.1), (2.2) and (2.3), we prove Theorem 1.

Proof of Theorem 1. It follows from (2.1)(1) that z is stably equivalent to kx_n for some non-negative integer k. So there are trivialR-bundles u andv over RPⁿ ofdimensions u and v respectively such that zluˆkx_nlv. Let n<m. Then

zluˆi…kx_mlv†;

where i:RPⁿ!RP^m is the inclusion. Note that k‡v mVk‡v m u

ˆt‡u m uˆt mV0 if mUt. By (2.2) there is an R-vector bundle a over RP^m ofdimension m …ˆ d…m‡1†=1 1e† such that

kx_mlvˆal…k‡v m†;

since mUt. Hence we have

zluˆi…al…k‡v m†† ˆi…al…k‡v m u††lu:

Therefore, by (2.3),

zˆi…al…k‡v m u††;

since d…n‡2†=1 1e ˆn‡1Ut. q.e.d.

Theorem 2.4. Let w be an integer >1 and let t^w be the w-fold tensor product t…RPⁿ†^w of the tangent bundle tˆt…RPⁿ†. Then t^w is extendible to RP^m for every m with n<mUn^w.

Proof. Since dimt^wˆn^w, the result follows from Theorem 1. q.e.d.

3. Extendibility and stable extendibility of the square of the tangent bundle of the real projective space

Lemma 3.1. Lett² ˆt…RPⁿ†nt…RPⁿ†be the square of the tangent bundle tˆt…RPⁿ† of RPⁿ. Then the equality

t²ˆ …a2^f…n† 2n 2†x_n‡ …n‡1†²‡1 a2^f…n†

holds in KO…RPⁿ†, where a is any integer.

Proof. Since tl1ˆ …n‡1†x_n, we have, by (2.1), t²ˆ …n‡1†²…x_n†² 2…n‡1†x_n‡1ˆ …a2^f…n† 2n 2†x_n‡ …n‡1†²‡1 a2^f…n† inKO…RPⁿ†for any

integer a. q.e.d.

Theorem 3.2. Let t…RPⁿ†²ˆt…RPⁿ†nt…RPⁿ† be the square of the tangent bundle of RPⁿ. Then we have the Whitney sum decompositions:

t…RP¹†² ˆ1, t…RP²†²ˆ2x₂l2, t…RP³†²ˆ9, t…RP⁴†² ˆ6x₄l10, t…RP⁵†²ˆ4x₅l21, t…RP⁶†²ˆ2x₆l34, t…RP⁷†² ˆ49, t…RP⁸†²ˆ14x₈l50, t…RP⁹†²ˆ12x₉l69, t…RP¹⁰†²ˆ42x₁₀l58, t…RP¹¹†² ˆ40x₁₁l81, t…RP¹²†²ˆ102x₁₂l42, t…RP¹³†²ˆ100x₁₃l69, t…RP¹⁴†² ˆ98x₁₄l98, t…RP¹⁵†²ˆ96x₁₅l129 and t…RP¹⁶†²ˆ222x₁₆l34.

Proof. For 2UnU8, put aˆ2 and for 9UnU16, put aˆ1.

Then the equalities above follow from Lemma 3.1 and (2.3). The casenˆ1 is

clear. q.e.d.

Remark 3.3. t…RP¹⁷†²ˆ476x₁₇ 187.

The following result is Theorem 4.1 in [6] which is the stably extendible version ofTheorem 6.2 in [4].

(3.4). Let z be a t-dimensional R-vector bundle over RPⁿ. Assume that there is a positive integer l such that z is stably equivalent to …t‡l†x_n and t‡l<2^f…n†. Then n<t‡l and zis not stably extendible to RP^m for every m with mVt‡l.

Lemma 3.5. De®ne l…n† ˆ2^f…n† n² 2n 2. Then l…n†>0 if nV17.

Proof. As is seen in the table below, the inequality l…n†>0 holds for integers n with 17UnU24 clearly.

n 17 18 19 20 21 22 23 24

8s‡i iˆ1 iˆ2 iˆ3 iˆ4 iˆ5 iˆ6 iˆ7 iˆ0 sˆ2 sˆ2 sˆ2 sˆ2 sˆ2 sˆ2 sˆ2 sˆ3 l…n† 187 662 623 1606 1563 1518 1471 3470

For larger values of n, we have l…n†>0 by induction on s. q.e.d.

Theorem 3.6. t²ˆt…RPⁿ†nt…RPⁿ† is not stably extendible to RP^m for every m with mV2^f…n† 2n 2 if nV17.

Proof. According to Lemma 3.1, t² is stably equivalent to …2^f…n† 2n 2†x_n. Setting zˆt², tˆn² andlˆl…n† ˆ2^f…n† n² 2n 2 in (3.4), we see that t² is not stably extendible to RP^m for every m with mV2^f…n† 2n 2, since 2^f…n† n² 2n 2>0 f or nV17 by Lemma 3.5 and 2^f…n† 2n 2<

2^f…n†. q.e.d.

Example 3.7. t²ˆt…RP¹⁷†nt…RP¹⁷† is not stably extendible to RP^m for every m with mV476.

Proof of Theorem 4. It is clear that (1) implies (2). The fact that (2) implies (3) is a consequence ofTheorem 3.6. Since x_n and the trivial bundles over RPⁿ are extendible to RP^m for every m with m>n, Theorem 3.2 shows

that (3) implies (1). q.e.d.

4. Extendibility of a C-vector bundle over the real projective space

The ring structure of K…RPⁿ† is determined in [1]. We recall the result which is necessary for our proofs.

(4.1) [1, Theorem 7.3]. The reduced K-groupK…RP~ ⁿ† is isomorphic to the cyclic group Z=2^bn=2c, generated by c…x_n 1†, where bn=2c denotes the integral part of n=2.

Using (4.1), (2.2) and (2.3), we prove Theorem 2.

Proof of Theorem 2. It follows from (4.1) that z is stably equivalent to kcx_n for some non-negative integer k. So there are trivial C-bundles u and v over RPⁿ ofdimensions u and v respectively such that zluˆkcx_nlv. Let n<m. Then

zluˆi…kcx_mlv†;

where i:RPⁿ !RP^m is the inclusion. Note that k‡v lVk‡v l u

ˆt‡u l uˆt lV0 iflUt. By (2.2) there is aC-vector bundlea over RP^m ofdimension l …ˆ d…m‡1†=2 1e† such that

kcx_mlvˆal…k‡v l†;

since lUt. Hence we have

zluˆi…al…k‡v l†† ˆi…al…k‡v l u††lu:

The inequalities n<mU2t‡1 show thatd…n‡2†=2 1eUd…m‡1†=2 1e ˆ lUt. Therefore, by (2.3), we obtain

zˆi…al…k‡v l u††: q.e.d.

Theorem 4.2. Let w be a positive integer and let ct^w be the complex- i®cation of the w-fold tensor product t^wˆt…RPⁿ†^w. Then ct^w is extendible to RP^m for every m with n<mU2n^w‡1

Proof. Since dimct^wˆn^w, the result follows from Theorem 2. q.e.d.

Corollary 4.3. ctˆct…RPⁿ† is extendible to RP^2n‡1.

5. Extendibility and stable extendibility of the complexi®cation of the square of the tangent bundle of the real projective space

Lemma 5.1. Let ct²ˆc…t…RPⁿ†nt…RPⁿ†† be the complexi®cation of the square of the tangent bundle of RPⁿ. Then the equality

ct² ˆ …b2^bn=2c 2n 2†cx_n‡ …n‡1†²‡1 b2^bn=2c holds in K…RPⁿ†, where b is any integer.

Proof. Complexifying the equality t²ˆ 2…n‡1†x_n‡ …n‡1†²‡1 (cf.

Lemma 3.1) and using (4.1), we have the equality above. q.e.d.

Theorem 5.2. Let ct…RPⁿ†²ˆc…t…RPⁿ†nt…RPⁿ†† be the complex- i®cation of the square t…RPⁿ†nt…RPⁿ†. Then we have the Whitney sum decompositions:

ct…RP¹†²ˆ1; ct…RP²†² ˆ4, ct…RP³†²ˆ9, ct…RP⁴†²ˆ6cx₄l10, ct…RP⁵†²ˆ25, ct…RP⁶†²ˆ2cx₆l34, ct…RP⁷†²ˆ49,

ct…RP⁸†²ˆ14cx₈l50, ct…RP⁹†²ˆ12cx₉l69, ct…RP¹⁰†²ˆ42cx₁₀l58, ct…RP¹¹†²ˆ40cx₁₁l81, ct…RP¹²†²ˆ102cx₁₂l42, ct…RP¹³†²ˆ100cx₁₃l69, ct…RP¹⁴†²ˆ98cx₁₄l98, ct…RP¹⁵†²ˆ96cx₁₅l129, ct…RP¹⁶†²ˆ222cx₁₆l34 and ct…RP¹⁷†²ˆ220cx₁₇l69.

Proof. Since bn=2cUf…n†, complexifying the equalities in Theorem 3.2, we have the equalities above except for nˆ2;5 and 17. Considering the relations 2cx₂ 2ˆ0 f or nˆ2, 4cx₅ 4ˆ0 f or nˆ5 and 256cx₁₇ 256ˆ0 fornˆ17 (cf. (4.1)) and using (2.3), we have the equalities above from those in

Theorem 3.2 and Remark 3.3. q.e.d.

The following result is Theorem 2.1 in [6] which is the stably extendible version ofTheorem 4.2 for d ˆ1 in [5].

(5.3). Let z be a t-dimensional C-vector bundle over RPⁿ. Assume that there is a positive integerlsuch thatzis stably equivalent to…t‡l†cx_n and t‡l

<2^bn=2c. Thenbn=2c<t‡l and z is not stably extendible to RP^m for every m with mV2t‡2l.

Theorem 5.4. Let ctˆct…RPⁿ† be the complexi®cation of the tangent bundle t…RPⁿ†. If nˆ6 or n>7,ctis not stably extendible to RP^m for every m with mV2n‡2.

Proof. Putting zˆct…RPⁿ†, tˆn and lˆ1, we have the result from (5.3), since ct…RPⁿ†l1ˆ …n‡1†cx_n and n‡1<2^bn=2c if nˆ6 or

n>7. q.e.d.

Proof of Theorem 3. The ®rst part follows from Corollary 4.3 and Theorem 5.4. The second part is a consequence ofthe fact that t…RPⁿ† is

trivial for nˆ1;3 and 7. q.e.d.

Lemma 5.5. De®ne l…n† ˆ2^bn=2c n² 2n 2. Then n²‡l…n†<2^bn=2c, and if nV18, l…n†>0.

Proof. Since 2^bn=2c n² l…n† ˆ2n‡2>0, we have n²‡l…n†<2^bn=2c. The inequality l…n†>0 holds for nˆ18 and 19 clearly. For larger values of

n, we have l…n†>0 by induction. q.e.d.

Theorem 5.6. ct²ˆc…t…RPⁿ†nt…RPⁿ†† is not stably extendible to RP^m for every m with mV2^bn=2c‡1 4n 4 if nV18.

Proof. According to Lemma 5.1,ct² is stably equivalent to …2^bn=2c 2n 2†cx_n. Setting zˆct², tˆn² and lˆl…n† ˆ2^bn=2c n² 2n 2 in (5.3), we see that ct² is not stably extendible to RP^m for every m with mV2^bn=2c‡1 4n 4, since l…n†>0 and t‡l…n†<2^bn=2c then by Lemma 5.5. q.e.d.

Example 5.7. ct²ˆc…t…RP¹⁸†nt…RP¹⁸††is not stably extendible to RP^m for every m with mV948.

Proof of Theorem 5. It is clear that (1) implies (2). The fact that (2) implies (3) is a consequence ofTheorem 5.6. Since cx_n and the trivial bundles over RPⁿ are extendible to RP^m for every m with m>n, Theorem 5.2 shows

that (3) implies (1). q.e.d.

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T. Kobayashi Asakura-ki 292-21 Kochi, 780-8066 Japan

kteiichi@lime.ocn.ne.jp K. Komatsu Kochi University Department of Mathematics

Faculty of Science Kochi, 780-8520 Japan komatsu@math.kochi-u.ac.jp