35 (2005), 471–483
Stable extendibility of mt
nover real projective spaces
Hironori Yamasaki
(Received December 21, 2004) (Revised April 1, 2005)
Abstract. The purpose of this paper is to study the stable extendibility of them-times Whitney sum mtn of the tangent bundle tn¼tðRPnÞ of the n-dimensional real pro- jective spaceRPn. We determine the dimensionNfor whichmtnis stably extendible to RPN but is not stably extendible to RPNþ1 for ma10.
1. Introduction
Let X be a space and Aits subspace. A t-dimensional real vector bundle zoverA is said to beextendible (respectivelystably extendible) to X, if there is a t-dimensional real vector bundle overX whose restriction to A is equivalent (respectively stably equivalent) to z, that is, z is equivalent (respectively stably equivalent) to the induced bundle ih of a t-dimensional real vector bundle h overX under the inclusion map i:A!X (cf. [10, p. 20] and [3, p. 273]). Let RPn denote then-dimensional real projective space. For a real vector bundlez over RPn, define an integer sðzÞ by
sðzÞ ¼maxfmjmbn and z is stably extendible to RPmg;
where we put sðzÞ ¼y if z is stably extendible to RPm for every mbn.
The following theorem is known.
Theorem 1 ([7, Theorem 4.2]). For the tangent bundle tn¼tðRPnÞ of RPn,
sðtnÞ ¼y if n¼1;3 or 7; and sðtnÞ ¼n if n01;3;7:
The purpose of this paper is to study sðmtnÞ formb2. Our main results are as follows.
We write simply sðm;nÞ instead of sðmtnÞ.
Theorem 2. (1) If 1ana8, then sð2;nÞ ¼y. (2) If nb9, then sð2;nÞ ¼2nþ1.
2000 Mathematics Subject Classification. Primary 55R50; Secondary 55N15.
Keywords and Phrases. vector bundle, tangent bundle, Whitney sum, real projective space, stably extendible, KO-theory.
Theorem 3. (1) If 1ana8, then sð3;nÞ ¼y. (2) If nb9, then
(a) sð3;nÞ ¼3n for n10;1 mod 4, (b) sð3;nÞ ¼3nþ1 for n12 mod 4, (c) sð3;nÞ ¼3nþ2 for n13 mod 4.
Theorem 4. (1) If 1ana9, then sð4;nÞ ¼y. (2) If nb10, then sð4;nÞ ¼4nþ3.
Theorem 5. (1) If 1ana9, then sð5;nÞ ¼y. (2) If nb10, then
(a) sð5;nÞ ¼5n for n10;2;3;5 mod 8, (b) sð5;nÞ ¼5nþ1 for n16 mod 8, (c) sð5;nÞ ¼5nþ2 for n11 mod 8, (d) sð5;nÞ ¼5nþ3 for n14 mod 8, (e) sð5;nÞ ¼5nþ4 for n17 mod 8.
Theorem 6. (1) If 1ana11, then sð6;nÞ ¼y. (2) If nb12, then
(a) sð6;nÞ ¼6nþ1 for n10;1 mod 4, (b) sð6;nÞ ¼6nþ3 for n12 mod 4, (c) sð6;nÞ ¼6nþ5 for n13 mod 4.
Theorem 7. (1) If 1ana11, then sð7;nÞ ¼y. (2) If nb12, then
(a) sð7;nÞ ¼7n for n10;1 mod 8,
(b) sð7;nÞ ¼7nþi1 for n1i mod 8 with 2aia7.
Theorem 8. (1) If 1ana11 or n¼15, then sð8;nÞ ¼y. (2) If n¼12;13;14 or nb16, then sð8;nÞ ¼8nþ7.
Theorem 9. (1) If 1ana11, n¼14 or 15, then sð9;nÞ ¼y. (2) If n¼12;13 or nb16, then
(a) sð9;nÞ ¼9n for n10;2;4;6;7;9;11;13 mod 16, (b) sð9;nÞ ¼9nþ1 for n114 mod 16,
(c) sð9;nÞ ¼9nþ2 for n15 mod 16, (d) sð9;nÞ ¼9nþ3 for n112 mod 16, (e) sð9;nÞ ¼9nþ4 for n13 mod 16, (f ) sð9;nÞ ¼9nþ5 for n110 mod 16, (g) sð9;nÞ ¼9nþ6 for n11 mod 16, (h) sð9;nÞ ¼9nþ7 for n18 mod 16, (i) sð9;nÞ ¼9nþ8 for n115 mod 16.
Theorem 10. (1) If 1ana15, then sð10;nÞ ¼y. (2) If nb16, then
(a) sð10;nÞ ¼10nþ1 for n10;2;3;5 mod 8, (b) sð10;nÞ ¼10nþ3 for n14;6;7;14 mod 16, (c) sð10;nÞ ¼10nþ4 for n11 mod 16, (d) sð10;nÞ ¼10nþ5 for n19 mod 16, (e) sð10;nÞ ¼10nþ7 for n112 mod 16, (f ) sð10;nÞ ¼10nþ9 for n115 mod 16.
This paper is arranged as follows. In § 2 we state some known theorems that are used to prove Theorems 2–10. In § 3 we state some applications. In
§ 4 we study on mtn. In § 5 we prove Theorem 10. In § 6 and § 7 we prove Theorems 2–9.
2. Some known theorems
Let xn denote the canonical real line bundle over RPn.
Theorem 2.1 ([1, Theorem 7.4]). (1) The reduced KO-group KOKOðRPf nÞ is isomorphic to the cyclic group Z=2fðnÞ, generated by xn1, where fðnÞ is the number of integers s such that 0<san and s10;1;2 or 4 mod 8.
(2) ðxnÞ2ð¼xnnxnÞ ¼1, where n denotes the tensor product.
For a real vector bundle z, we denote by Spanzthe maximum number of linearly independent cross-sections of z.
Theorem 2.2 ([5, Theorem 1]). Let l, n and t be integers with tb0 and 0atþl<2fðnÞ, and letzbe a t-dimensional real vector bundle over RPn which is stably equivalent to ðtþlÞxn. Then the following hold.
(1) sðzÞ ¼y if and only if la0.
(2) Let lb1 and mbn. Then, sðzÞbm if and only if Spanða2fðnÞþtþlÞxmba2fðnÞþl for some integer ab0.
For a non-negative integer t and a positive integer l, define an integer eðt;lÞ as follows.
eðt;lÞ ¼min j
t<j and tþl j
11 mod 2
;
where nr denotes the binomial coe‰cient n!=ðr!ðnrÞ!Þ. Clearly t<eðt;lÞa tþl.
Theorem2.3 ([6, Theorem 2]). Let zbe a t-dimensional real vector bundle over RPn and assume that there is a positive integer l satisfying the following properties:
(1) z is stably equivalent to ðtþlÞxn, (2) tþl<2fðnÞ.
Then n<eðt;lÞ and sðzÞ<eðt;lÞ.
The following theorems are useful.
n
Theorem 2.4 ([12, Theorem 2.4], [4, (1.1) and Section 4]). Let nþ1¼ ð2bþ1Þ2cþ4d, where b, c, d are non-negative integers and 0aca3. Then
Spanðnþ1Þxn¼2cþ8d:
Theorem 2.5 ([9, Theorem 1.1]). Let k¼8lþp, n¼8mþq, where 0amal and 0ap;qa7. If the binomial coe‰cient ml is odd, then Spankxn¼ ðknÞ þ j, with j given by Table I below:
Theorem 2.6 ([9, Theorem 3.1]). (A) Let k¼8lþp, n¼8mþq, where 1<m<l and ml is even. Then for 0apa6 and 1aqa7, Spankxnb ðknÞ þj, with j given by Table II below:
Table I
p 0 1 2 3 4 5 6 7
q
0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0
2 2 1 0 0 2 1 0 0
3 3 2 1 0 3 2 1 0
4 4 3 2 1 0 0 0 0
5 5 4 3 2 1 0 1 0
6 6 5 4 3 2 1 0 0
7 7 6 5 4 3 2 1 0
Table II
p 0 1 2 3 4 5 6
q
1 3 4 3 3 4 3 3
2 3 5 4 4 5 4 3
3 4 6 5 5 6 5 4
4 5 4 3 5 5 4 3
5 6 5 4 3 5 4 3
6 7 6 5 4 6 5 4
7 8 7 6 5 4 3 3
(B) Furthermore, if ml 12 mod 4, then Spankxn¼ ðknÞ þj except when ðp;qÞ ¼ ð3;4Þ.
Theorem 2.7 ([11, Lemma 2.6, p. 5]). Let a¼a020þa121þ þal2l and b¼b020þb121þ þbl2l ð0aai;bi<2Þ. Then
a
b 10 mod 2 if and only if ai¼0 and bi¼1 for some i:
The following theorem may be well-known. For completeness, we give a proof.
Theorem 2.8. Let a¼Pl
i¼0
ai2i and b¼Pl
i¼0
bi2i ð0aai;bi<2Þ. Then
a
b 12 mod 4 if ab can be described as follows:
a
b ¼ mþ2pþ1þ0þr nþ0þ2pþs
;
where m;n10 mod 2pþ2; 2p>r;sb0; mn ; rs 11 mod 2.
Proof. For a positive integer N, let nðNÞ denote the non-negative in- teger such that N¼ ð2qþ1Þ2nðNÞ, where q is a non-negative integer. Clearly, N11 mod 2 if and only if nðNÞ ¼0, and N12 mod 4 if and only if nðNÞ ¼1. It is known that (cf. [2, Lemma 4.8]), for a positive integer M, nðM!Þ ¼MaðMÞ, where aðMÞ is the number of the non-zero terms in the 2-adic expansion of M. Hence we have
n a
b ¼aðabÞ þaðbÞ aðaÞ:
Thus ab 11 mod 2 if and only if aðabÞ þaðbÞ aðaÞ ¼0, and ab 12 mod 4 if and only if aðabÞ þaðbÞ aðaÞ ¼1. Therefore, if ab is described
as in the theorem, we see that n ab ¼1. r
3. Some applications
Proposition 3.1. Let z be a t-dimensional real vector bundle over RPn. Then
sðzÞ ¼sðznxnÞ:
Proof. If z is stably extendible to RPm ðmbnÞ, then there is a t- dimensional real vector bundle a over RPm such that z is stably equivalent to ia, where i:RPn!RPm is the standard inclusion. Since znxn is stably equivalent to ianxn¼iðanxmÞ, znxn is stably extendible to RPm
n
ðmbnÞ. On the other hand, by using Theorem 2.1(2) and the above result, we see that z¼ ðznxnÞnxn is stably extendible to RPm ðmbnÞ if znxn is
stably extendible to RPm. r
In the same way as the above proof, we have the following.
Remark. Let zbe at-dimensional real vector bundle over RPn. Then,z is extendible to RPm ðmbnÞ if and only if znxn is extendible to RPm.
For a non-negative integer t and a positive integer l, we define an integer mðt;l;nÞ as the maximum of integers m satisfying
tþlbmþ1¼ ð2bþ1Þ2cþ4d >n and 2cþ8dbl;
where b, c and d are non-negative integers with 0aca3. We remark that the above mðt;l;nÞ does not necessarily exist, even if tþl>n.
Proposition3.2. Let zbe a t-dimensional real vector bundle over RPn and assume that there is a positive integer l satisfying the following properties:
(1) z is stably equivalent to ðtþlÞxn, (2) tþl<2fðnÞ.
If moreover mðt;l;nÞ exists, then mðt;l;nÞasðzÞ.
Proof. Put mðt;l;nÞ ¼m. Then, SpanðtþlÞxmbSpanðmþ1Þxm¼2cþ 8dbl by the definition and Theorem 2.4. Therefore, by using Theorem 2.2(2),
we have sðzÞbm. r
Proposition 3.3. Let tb0, lb1 and tþl<2fðnÞ. Then eðt;lÞ ¼eðt;2fðnÞtlÞ:
Proof. We put 2fðnÞl¼2q1þ2q2þ þ2qm ðq1>q2> >qmb0Þ, e¼eðt;lÞ and ~ee¼eðt;2fðnÞtlÞ.
The proof is given by induction ont. Whent¼0, by Theorem 2.7 we see that e¼ee~¼2qm and hence the proposition holds. Now we may assume that e¼~eefor all t0 with 0at0<t, and consider the case fortby putting t¼2pþs, where pb0 and 0as<2p.
Now in order to apply Theorem 2.7 more easily, we put e0ðh;kÞ ¼ eðk;hkÞ, that is
e0ðh;kÞ ¼min j
h
j 11 mod 2 and j>k
:
(1) Let qm>p. Then e0ð2q1þ þ2qm;2pþsÞ ¼2qm by Theorem 2.7.
And e0ð2fðnÞ2q1 2qmþ2pþs;2pþsÞ ¼2qm by Theorem 2.7. Hence e¼~ee.
(2) Let qm¼p. Then mb2 and e0ð2q1þ þ2qm1þ2qm;2qmþsÞ ¼ 2qm1. And e0ð2fðnÞ2q1 2qm1þs;2qmþsÞ ¼2qm1. Hence ~ee¼e.
(3) Let qi>p>qiþ1 ð1aiam1Þ. Thene0ð2q1þ þ2qm;2pþsÞ ¼ 2qi. And e0ð2fðnÞ2q1 2qi þ2p2qiþ1 2qmþs;2pþsÞ ¼2qi. Hence e¼~ee.
(4) Let p¼qi ð1aiam1Þ. If 2qiþ1þ þ2qmas then ib2 and e0ð2q1þ þ2qm;2qiþsÞ ¼2qi1. And e0ð2fðnÞ2q1 2qi12qiþ1 2qmþs;2qiþsÞ ¼2qi1. Hence e¼~ee.
If 2qiþ1þ þ2qm >s then e0ð2q1þ þ2qm;2qiþsÞ ¼2qiþe0ð2q1þ þ2qm;sÞ. Ande0ð2fðnÞ2q1 2qi12qiþ1 2qmþs;2qiþsÞ ¼2qiþ e0ð2fðnÞ2q1 2qi12qiþ1 2qmþs;sÞ ¼2qi þe0ð2fðnÞ2q1 2qmþs;sÞ. Hence we see that e¼~ee by using the assumption of induction.
r Theorem 3.4. Let z be a t-dimensional real vector bundle over RPn and assume that there is a positive integer l satisfying the following properties:
(1) z is stably equivalent to ðtþlÞxn, (2) tþl<2fðnÞ.
If moreover mðt;l;nÞ exists, then mðt;l;nÞasðzÞ<eðt;lÞ ¼eðt;2fðnÞtlÞ.
Proof. The proof follows from Theorem 2.3 and Propositions 3.2 and
3.3. r
4. Study on mtn
Lemma 4.1. Let mtn be the m-times Whitney sum of the tangent bundle tn¼tðRPnÞ of RPn. Then the equality
mtn¼mðnþ1Þxnm holds in KOðRPnÞ.
Proof. Since tnl1 is equivalent to ðnþ1Þxn, the equality holds. r Proposition 4.2. If mb2fðnÞ, sðmtnÞ ¼y.
Proof. Letm¼b2fðnÞþg, wherebb1 and 0<g<2fðnÞ. Then Lemma 4.1 implies mtn¼ ðb2fðnÞþgÞðnþ1Þxn ðb2fðnÞþgÞ in KOðRPnÞ. So, we see that mtn is stably equivalent to gðnþ1Þxn by using Theorem 2.1. Then, since gðnþ1Þ<mn, we see that sðmtnÞ ¼y by using Theorem 2.2(1). r Proposition 4.3. Let 0<m<2fðnÞ. Then sðmtnÞ ¼y if and only if 2fðnÞamðnþ1Þ.
Proof. Since mtn¼mðnþ1Þxnm holds in KOðRPnÞ, we see that mtnnxn is stably equivalent to ð2fðnÞmÞxn by using Theorem 2.1. Now
n
0<2fðnÞm<2fðnÞ by the assumption. Hence, by Theorem 2.2(1), 2fðnÞ mmna0 if and only if sðmtnÞ ¼sðmtnnxnÞ ¼y. r
Proposition 4.4.
sðmtnÞbmn
Proof. Propositions 4.2 and 4.3 imply that sðmtnÞ ¼y if mðnþ1Þb 2fðnÞ.
Let mðnþ1Þ<2fðnÞ. Then, since SpanðmnþmÞxmnbðmnþmÞ mn¼ m, we see that sðmtnÞbmn by using Theorem 2.2(2). r
5. Proof of Theorem 10
In this section we give a proof of Theorem 10. Before proving the theorem, we prepare two lemmas.
Lemma 5.1. eð10n;10Þ ¼10nþ2 for n10;2;3;5 mod 8, ¼10nþ4 for n16 mod 8, ¼10nþ6 for n11 mod 8, ¼10nþ8 for n14 mod 8,
¼10nþ10 for n17 mod 8.
Proof. The results follow from the definition of eð10n;10Þ and Theorem
2.7. r
Lemma 5.2. mð10n;10;nÞb10nþ5 for n19 mod 16, b10nþ7 for n112 mod 16,b10nþ9 for n115 mod 16.
Proof. Let n¼16kþ9 ðkb1Þ and put 10nþ6ð¼25ð5kþ3ÞÞ ¼ ð2bþ1Þ2cþ4d ð0aca3Þ. Then we see 2cþ8db10. So we get mð10n;10;
nÞb10nþ5 by the definition of mð10n;10;nÞ.
For n112;15 mod 16, by putting 10nþ8, 10nþ10¼ ð2bþ1Þ2cþ4d re-
spectively, we obtain the results similarly. r
Proof of Theorem 10. We recall the following 10tn¼ ð10nþ10Þxn10AKOðRPnÞ:
(1) Since 10b2fðnÞ for 1ana7, sð10tnÞ ¼y for 1ana7 by Prop- osition 4.2. And since 10<2fðnÞ and 2fðnÞ1010na0 for 8ana15, sð10tnÞ ¼y for 8ana15 by Proposition 4.3.
(2) Let nb16. Then 0a10nþ10<2fðnÞ and 10tn is stably equivalent to ð10nþ10Þxn.
(a) Let n10;2;3;5 mod 8. Since 10nþ10 is even and 10nþ1 is odd, we have Spanð10nþ10Þx10nþ1bð10nþ10Þ ð10nþ1Þ þ1¼10 by Theorems 2.5 and 2.6(A). Hence, by Theorem 2.2(2), sð10tnÞb10nþ1. On the other hand, we see eð10n;10Þ ¼10nþ2 by Lemma 5.1. Hence sð10tnÞ<10nþ2 by Theorem 2.3. Therefore sð10tnÞ ¼10nþ1.
(b) Let n¼16kþ4 ðkb1Þ. We see 10nþ10¼8ð20kþ6Þ þ2 and 10nþ3¼8ð20kþ5Þ þ3. Here 20kþ620kþ5
10 mod 2. So, by Theorem 2.6(A), we see Spanð10nþ10Þx10nþ3b10nþ1010n3þ5¼12b10. Hence, by using Theorem 2.2(2), we obtainsð10tnÞb10nþ3. On the other hand, we see a2fðnÞþ10nþ10¼8ða2fð16kþ4Þ3þ20kþ6Þ þ2 and 10nþ4¼8ð20kþ5Þ þ 4. Here, by Theorem 2.8,a2fð16kþ4Þ320kþ5þ20kþ6
12 mod 4 for any ab0. Hence, by Theorem 2.6(B), Spanða2fðnÞþ10nþ10Þx10nþ4¼a2fðnÞþ10nþ1010n 4þ3<a2fðnÞþ10 for any ab0. So we obtain sð10tnÞ<10nþ4 by using Theorem 2.2(2). Therefore we have sð10tnÞ ¼10nþ3.
Let n¼8kþ6 ðkb2Þ. We see 10nþ10¼8ð10kþ8Þ þ6 and 10nþ3¼8ð10kþ7Þ þ7. Here 10kþ810kþ7
10 mod 2. So, by Theorem 2.6(A), we see Spanð10nþ10Þx10nþ3b10nþ1010n3þ3¼10. Hence we obtain sð10tnÞb10nþ3. On the other hand, we see eð10n;10Þ ¼ 10nþ4 by Lemma 5.1. Hence sð10tnÞ<10nþ4. Therefore we have sð10tnÞ ¼10nþ3.
Let n¼16kþ7 ðkb1Þ. We see 10nþ10¼8ð20kþ10Þ and 10nþ3¼ 8ð20kþ9Þ þ1. Here 20kþ1020kþ9
10 mod 2. So, by Theorem 2.6(A), we see Spanð10nþ10Þx10nþ3b10nþ1010n3þ3¼10. Hence we obtainsð10tnÞ b10nþ3. On the other hand, we see a2fðnÞþ10nþ10¼8ða2fð16kþ7Þ3 þ20kþ10Þ and 10nþ4¼8ð20kþ9Þ þ2. Here, by Theorem 2.8,
a2fð16kþ7Þ3þ20kþ10 20kþ9
12 mod 4 for any ab0. Hence, by using Theorem 2.6(B), Spanða2fðnÞþ10nþ10Þx10nþ4 ¼a2fðnÞþ10nþ1010n4þ3<a2fðnÞ þ10 for any ab0. So we obtain sð10tnÞ<10nþ4. Therefore we havesð10tnÞ ¼ 10nþ3.
(c) Let n¼16kþ1 ðkb1Þ. We see 10nþ10¼8ð20kþ2Þ þ4 and 10nþ4¼8ð20kþ1Þ þ6. Here 20kþ220kþ1
10 mod 2. So, by Theorem 2.6(A), we see Spanð10nþ10Þx10nþ4b10nþ1010n4þ6¼12b10. Hence we obtain sð10tnÞb10nþ4. On the other hand, we see a2fðnÞþ10nþ10¼ 8ða2fð16kþ1Þ3þ20kþ2Þ þ4 and 10nþ5¼8ð20kþ1Þ þ7. Here, by Theo- rem 2.8, a2fð16kþ1Þ320kþ1þ20kþ2
12 mod 4 for any ab0. Hence, by Theorem 2.6(B), Spanða2fðnÞþ10nþ10Þx10nþ5¼a2fðnÞþ10nþ1010n5þ4<a2fðnÞ þ10 for any ab0. So we obtain sð10tnÞ<10nþ5. Therefore we have sð10tnÞ ¼10nþ4.
(d) Let n¼16kþ9 ðkb1Þ. Then eð10n;10Þ ¼10nþ6 by Lemma 5.1.
Hence sð10tnÞ<10nþ6 by Theorem 2.3. And we get mð10n;10;nÞb10nþ5 by Lemma 5.2. Therefore, by Proposition 3.2, sð10tnÞb10nþ5.
(e) Let n¼16kþ12 ðkb1Þ. Then eð10n;10Þ ¼10nþ8 by Lemma 5.1. Hence sð10tnÞ<10nþ8. And we getmð10n;10;nÞb10nþ7 by Lemma 5.2. Therefore sð10tnÞb10nþ7.
n
(f ) Let n¼16kþ15 ðkb1Þ. Then eð10n;10Þ ¼10nþ10 by Lemma 5.1. Hence sð10tnÞ<10nþ10. And we get mð10n;10;nÞb10nþ9 by
Lemma 5.2. Therefore sð10tnÞb10nþ9. r
6. Proof of Theorem 6
In this section we prove Theorem 6, since the method of the proofs of Theorems 2–5, 7–9 is simpler than and similar to that of Theorem 6.
Proof of Theorem 6. We recall the following 6tn¼ ð6nþ6Þxn6AKOðRPnÞ:
(1) Since 6b2fðnÞ for 1ana3, sð6tnÞ ¼y for 1ana3 by Propo- sition 4.2. And since 6<2fðnÞ and 2fðnÞ66na0 for 4ana11, sð6tnÞ ¼ y for 4ana11 by Proposition 4.3.
(2) Let nb12. Then 0a6nþ6<2fðnÞ and 6tn is stably equivalent to ð6nþ6Þxn.
(a) Let n10;1 mod 4. We use the method similar to the proof of Theorem 10(a). Since 6nþ6 is even and 6nþ1 is odd, we have Spanð6nþ6Þx6nþ1bð6nþ6Þ ð6nþ1Þ þ1¼6 by Theorems 2.5 and 2.6(A).
Hence, by Theorem 2.2(2), sð6tnÞb6nþ1. On the other hand, eð6n;6Þ ¼ 6nþ2 by Theorem 2.7. Hence sð6tnÞ<6nþ2 by Theorem 2.3. Therefore sð6tnÞ ¼6nþ1.
(b) We use the method similar to the proof of Theorem 10(2)(d). Let n¼4kþ2 ðkb3Þ. Then eð6n;6Þ ¼6nþ4. Hence sð6tnÞ<6nþ4 by The- orem 2.3. Also putting 6nþ4¼23ð3kþ2Þ ¼ ð2bþ1Þ2cþ4d ð0aca3Þ, we see 2cþ8db6. So we get mð6n;6;nÞb6nþ3. Therefore, by Proposition 3.2, sð6tnÞb6nþ3.
(c) Let n¼4kþ3 ðkb3Þ. Then eð6n;6Þ ¼6nþ6. Hence sð6tnÞ<
6nþ6. Putting 6nþ6¼23ð3kþ3Þ ¼ ð2bþ1Þ2cþ4d ð0aca3Þ, we see 2cþ 8db6. So we get mð6n;6;nÞb6nþ5. Therefore sð6tnÞb6nþ5. r
7. Proofs of Theorems 2–5, 7–9
In this section we give an outline of the proofs of Theorems 2–5, 7–9 in the way similar to the proofs of Theorem 6(1) and Theorem 6(2)(b) by using Theorem 2.3, Propositions 3.2, 4.2–4.4.
Proof of Theorem 2. We recall the following 2tn¼ ð2nþ2Þxn2AKOðRPnÞ:
(1) Since 2b2fðnÞ for n¼1, sð2tnÞ ¼y for n¼1 by Proposition 4.2. And since 2<2fðnÞ and 2fðnÞ22na0 for 2ana8, sð2tnÞ ¼y for 2ana8 by Proposition 4.3.
(2) Let nb9. Then 0a2nþ2<2fðnÞ and 2tn is stably equivalent to ð2nþ2Þxn.
We use the method similar to the proof of Theorem 6(2)(b). Now eð2n;2Þ ¼2nþ2. Hence sð2tnÞ<2nþ2 by Theorem 2.3. Also putting 2nþ 2ð¼2ðnþ1ÞÞ ¼ ð2bþ1Þ2cþ4d ð0aca3Þ, we see 2cþ8db2. So we get mð2n;2;nÞb2nþ1. Therefore, by Proposition 3.2, sð2tnÞb2nþ1. r
Proof of Theorem 3. We recall the following 3tn¼ ð3nþ3Þxn3AKOðRPnÞ:
(1) Since 3b2fðnÞ for n¼1, sð3tnÞ ¼y forn¼1. And since 3<2fðnÞ and 2fðnÞ33na0 for 2ana8, sð3tnÞ ¼y for 2ana8.
(2) Let nb9. Then 0a3nþ3<2fðnÞ and 3tn is stably equivalent to ð3nþ3Þxn.
(a) Let n10;1 mod 4. Then eð3n;3Þ ¼3nþ1. Hence sð3tnÞ<3nþ1 by Theorem 2.3. And we get sð3tnÞb3n by Proposition 4.4.
(b), (c) Now eð3n;3Þ ¼3nþ2 for n12 mod 4, ¼3nþ3 for n13 mod 4. Hence sð3tnÞ<3nþ2 for n12 mod 4, <3nþ3 for n13 mod 4.
Also let n¼4kþ2 ðkb2Þ and put 3nþ2ð¼22ð3kþ2ÞÞ ¼ ð2bþ1Þ2cþ4d ð0aca3Þ. Then we see 2cþ8db3. So mð3n;3;nÞb3nþ1. For n13 mod 4, we see similarly mð3n;3;nÞb3nþ2. Therefore sð3tnÞb3nþ1 for
n12 mod 4, b3nþ2 for n13 mod 4. r
Proof of Theorem 4. We recall the following 4tn¼ ð4nþ4Þxn4AKOðRPnÞ:
(1) Since 4b2fðnÞ for 1ana3, sð4tnÞ ¼y for 1ana3. And since 4<2fðnÞ and 2fðnÞ44na0 for 4ana9, sð4tnÞ ¼y for 4ana9.
(2) Let nb10. Then 0a4nþ4<2fðnÞ and 4tn is stably equivalent to ð4nþ4Þxn.
Now eð4n;4Þ ¼4nþ4. Also we get mð4n;4;nÞb4nþ3. r Proof of Theorem 5. We recall the following
5tn¼ ð5nþ5Þxn5AKOðRPnÞ:
(1) Since 5b2fðnÞ for 1ana3, sð5tnÞ ¼y for 1ana3. And since 5<2fðnÞ and 2fðnÞ55na0 for 4ana9, sð5tnÞ ¼y for 4ana9.
(2) Let nb10. Then 0a5nþ5<2fðnÞ and 5tn is stably equivalent to ð5nþ5Þxn.
n
(a) Let n10;2;3;5 mod 8. Then eð5n;5Þ ¼5nþ1. And we get sð5tnÞ b5n by Proposition 4.4.
(b), (c), (d), (e) Now eð5n;5Þ ¼5nþ2 for n16 mod 8, ¼5nþ3 for n11 mod 8, ¼5nþ4 for n14 mod 8, ¼5nþ5 for n17 mod 8. Also we get mð5n;5;nÞb5nþ1 for n16 mod 8,b5nþ2 for n11 mod 8, b5nþ3
for n14 mod 8, b5nþ4 for n17 mod 8. r
Proof of Theorem 7. We recall the following 7tn¼ ð7nþ7Þxn7AKOðRPnÞ:
(1) Since 7b2fðnÞ for 1ana3, sð7tnÞ ¼y for 1ana3. And since 7<2fðnÞ and 2fðnÞ77na0 for 4ana11, sð7tnÞ ¼y for 4ana11.
(2) Let nb12. Then 0a7nþ7<2fðnÞ and 7tn is stably equivalent to ð7nþ7Þxn.
(a) Let n10;1 mod 8. Then eð7n;7Þ ¼7nþ1. Also we get sð7tnÞb 7n by Proposition 4.4.
(b) Now eð7n;7Þ ¼7nþi for n1i mod 8 with 2aia7. Also we get mð7n;7;nÞb7nþi1 for n1i mod 8 with 2aia7. r
Proof of Theorem 8. We recall the following 8tn¼ ð8nþ8Þxn8AKOðRPnÞ:
(1) Since 8b2fðnÞ for 1ana7, sð8tnÞ ¼y for 1ana7. And since 8<2fðnÞ and 2fðnÞ88na0 for 8ana11 or n¼15, sð8tnÞ ¼y for 8ana11 or n¼15.
(2) Let n¼12;13;14 or nb16. Then 0a8nþ8<2fðnÞ and 8tn is stably equivalent to ð8nþ8Þxn.
Now eð8n;8Þ ¼8nþ8. Also we get mð8n;8;nÞb8nþ7. r Proof of Theorem 9. We recall the following
9tn¼ ð9nþ9Þxn9AKOðRPnÞ:
(1) Since 9b2fðnÞ for 1ana7, sð9tnÞ ¼y for 1ana7. And since 9<2fðnÞ and 2fðnÞ99na0 for 8ana11, n¼14 or 15, sð9tnÞ ¼y for 8ana11, n¼14 or 15.
(2) Let n¼12;13 or nb16. Then 0a9nþ9<2fðnÞ and 9tn is stably equivalent to ð9nþ9Þxn.
(a) Let n10;2;4;6;7;9;11;13 mod 16. Then eð9n;9Þ ¼9nþ1. Also we get sð9tnÞb9n by Proposition 4.4.
(b), (c), (d), (e), (f ), (g), (h), (i) Noweð9n;9Þ ¼9nþ2 forn114 mod 16,
¼9nþ3 for n15 mod 16, ¼9nþ4 for n112 mod 16, ¼9nþ5 for n13 mod 16,¼9nþ6 for n110 mod 16,¼9nþ7 for n11 mod 16,¼9nþ8 for
n18 mod 16, ¼9nþ9 for n115 mod 16. Also we get mð9n;9;nÞb9nþ1 for n114 mod 16, b9nþ2 for n15 mod 16,b9nþ3 for n112 mod 16, b9nþ4 for n13 mod 16,b9nþ5 for n110 mod 16, b9nþ6 for n11 mod 16, b9nþ7 for n18 mod 16, b9nþ8 for n115 mod 16. r Remark. According to Theorem 2.2 of [8], mtn is extendible to RPN if and only if mtn is stably extendible to RPN, provided nb1 and m>1.
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Hironori Yamasaki Department of Mathematics Graduate School of Science
Hiroshima University Higashi-Hiroshima 739-8526 Japan [email protected]
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