We determine the group structure of the self-homotopy set of the double suspension of the real 7-dimensionnal projective space.</p

Miyauchi, Toshiyuki

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 48, Issue 1 2006 Article 9JANUARY 2006Self-homotopy of the Double Suspension ofthe Real 7-projective SpaceToshiyuki Miyauchi∗∗kyushu UniversityCopyright c©2006 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouSelf-homotopy of the Double Suspension ofthe Real 7-projective SpaceToshiyuki MiyauchiAbstractWe determine the group structure of the self-homotopy set of the double suspension of thereal 7-dimensionnal projective space.KEYWORDS: self-homotopy, real projective spaceMath. J. Okayama Univ. 48 (2006), 77{86SELF-HOMOTOPY OF THE DOUBLE SUSPENSION OFTHE REAL 7-PROJECTIVE SPACEToshiyuki MIYAUCHIAbstract. We determine the group structure of the self-homotopy setof the double suspension of the real 7-dimensionnal projective space.1. IntroductionIn this paper, all spaces, maps and homotopies are based. We use the samenotation as [10] and [5]. Let §nX be an n-fold suspension of a space X andPn be the n-dimensional real projective space. The purpose of the presentpaper is to determine the group structure of the homotopy set [§2P7;§2P7].We denote by °n : Sn ! Pn the covering map. According to [9], §2°6 = 0,§2P7 = §2P6 _ S9, and so[§2P7;§2P7] »= [§2P6;§2P6]© ¼9(§2P6)© ¼9(S9):Let Z be the group of integers and set Zn = Z=nZ. The notation (Zn)mmeans a direct sum of m-copies of Zn. Our result is stated as follows.Theorem 1.1. [§2P7;§2P7] »= Z© (Z8)2 © (Z2)7.In this paper we sometimes identify a map with its homotopy class. Form < n, let im;n : Pm ! Pn and pn;m : Pn ! Pn=Pm be the inclusionand collapsing maps, respectively. Especially, we write Mn = §n¡2P2,in = §n¡2i1;2 : Sn¡1 ! Mn and pn = §n¡2p2;1 : Mn ! Sn for n ¸ 2.We denote by [®; ¯] the Whitehead product of homotopy classes ® and ¯.To determine the group structure of ¼9(§2P6), we use the following.Theorem 1.2. [§2°5;§2i1;5] = 0 2 ¼9(§2P5).2. Some homotopy groupsWe denote by ¶X 2 [X;X] the identity class of a space X and let ¶n = ¶Sn .For the Hopf maps ´2 2 ¼3(S2) and º4 2 ¼7(S4), we set ´n = §n¡2´2,´2n = ´n´n+1, ´3n = ´n´n+1´n+2 for n ¸ 2 and ºn = §n¡4º4 for n ¸ 4. Werecall from [7] that there is an element e´2 2 ¼4(M3) such that p3 e´2 = ´3and §e´2 = e´3, where e´3 is a coextension of ´3. Let ´3 2 [M5; S3] be anextension of ´3 and set e´n = §n¡2 e´2 for n ¸ 2 and ´n = §n¡3´3 for n ¸ 3.Mathematics Subject Classi¯cation. Primary 55Q52; Secondary 55Q15, 55Q40.Key words and phrases. self-homotopy, real projective space.Thanks are due to Tomohisa Inoue, Professor Juno Mukai and Professor Norio Iwasefor the kind and useful advices.771Miyauchi: Self-homotopy of the Double Suspension of the Real 7-projectiveProduced by The Berkeley Electronic Press, 200678 T. MIYAUCHILet º 0 be a generator of the group ¼6(S3) »= Z12 and ¸2 be the attachingmap of the 7-cell of the Stiefel manifold V5;2 = M4 [¸2 e7. We recall that¼6(M4) = Z4f¸2g © Z2fe´3´5g [6]. We note ¼9(M4) = (Z2)3 [12, Theorem5.8] and, by use of these facts and the homotopy exact sequence of a pair(V5;2;M4), we determine the generators.Lemma 2.1. ¼9(M4) = Z2f¸2º6g © Z2f[¸2; i4]´8g © Z2fe´3º5´8g.Let s : S5 ! §2P3 = M4_S5 be the inclusion to the second factor. Then,we recall(2.1) §2°3 = 2s§ (§2i2;3)e´3:By the Hilton-Milnor theorem, we obtain(2.2) ¼i(§2P3) »= ¼i(M4)© ¼i(S5)© ¼i(M8)© ¼i(§(M7 ^M3));for i · 9. By Lemma 2.1 and the facts that ¼8(M4) = Z2f¸2´26g ©Z2f[i4; ¸2]g©Z2fe´3º5g [8, Lemma 2.4], ¼8(S5)=Z24fº5g, ¼8(M8)=Z2fi8´7g,¼9(S5)=Z2fº5´8g, ¼9(M8)=Z4fe´7g and ¼9(§(M7 ^M3))=Z2f§(i7 ^ i3)g,we have the following.Lemma 2.2.(1) ¼8(§2P3) = Z2f(§2i2;3)¸2´26g © Z2f(§2i2;3)[i4; ¸2]g©Z2f(§2i2;3)e´3º5g © Z24fsº5g © Z2f[§2i1;3; s]´7g;(2) ¼9(§2P3) = Z2f(§2i2;3)¸2º6g © Z2f(§2i2;3)[i4; ¸2]´8g©Z2f(§2i2;3)e´3º5´8g © Z2fsº5´8g©Z4f[§2i2;3; s]e´7g © Z2f[[§2i1;3; s];§2i1;3]g:Let X be a connected ¯nite CW-complex and X¤ = X [µ en for µ :Sn¡1 ! X a complex formed by attaching an n-cell. We denote by!(X¤;X)n 2 ¼n(X¤; X)the characteristic map of the n-cell en of X¤. Let CY be a cone of a spaceY . For an element ® 2 ¼m(Y ), we denote by b®0 2 ¼m+1(CY; Y ) an elementsatisfying @0(b®0) = ®, where @0 : ¼m+1(CY; Y ) ! ¼m(Y ) is the connectingbijection. For ® 2 ¼m(Sn¡1), we setb® = !(X¤;X)n ± b®0 2 ¼m+1(X¤; X):We note the following:@(b®) = µ ± ® and p¤b® = §®;where @ : ¼m+1(X¤; X) ! ¼m(X) is the boundary map and p : (X¤; X) !(Sn; ¤) is the collapsing map. Now we show the following.Lemma 2.3. 2((§2i3;4)¤¼9(§2P3)) = 0.2Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 9http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/9SELF-HOMOTOPY OF §2P7 79Proof. We consider the homotopy exact sequence of a pair (§2P4;§2P3):¼10(§2P4;§2P3)@10¡¡! ¼9(§2P3) (§2i3;4)¤¡¡¡¡¡! ¼9(§2P4):There exists an element [!6; s] 2 ¼10(§2P4;§2P3) for !6 = !(§2P4;§2P3)6 . Bythe relations [1, (3.5)] and (2.1), we have@10([!6; s]) = ¡[§2°3; s] = §[(§2i2;3)e´3; s] = §[§2i2;3; s]e´7:Hence, by Lemma 2.2 (2), we obtain 2((§2i3;4)¤¼9(§2P3)) = 0. This com-pletes the proof. ¤Since §2°3 ± ´35 = (§2i2;3)e´3 ± 4º5 = 0 by (2.1), there exists an elemente´35 2 f§2i3;4;§2°3; ´35g ½ ¼9(§2P4) such that (§2p4;3) e´35 = ´36. For thiselement, we show the following.Lemma 2.4. fe´3´25; ´7; 2¶8g = e´3º5´8 and the order of e´35 is two.Proof. By the properties of Toda brackets and by [3, Lemma 4.1], we havefe´3´25; ´7; 2¶8g ± p9 = ¡(e´3´25 ± f´7; 2¶8; p8g) = e´3´25´7 = e´3º5´8p9:Since p9¤ : ¼9(M4) ! [M9;M4] is a monomorphism by Lemma 2.1, weobtain the ¯rst. By (2.1), the relation (§2i2;4)e´3º5´8 = §2((§2i3;4)s) ±e´3º5´8 = 0 holds. So, by the ¯rst and Lemma 2.3, we have2 e´35 2 f§2i3;4;§2°3; ´35g ± 2¶9= ¡(§2i3;4 ± f§2°3; ´35; 2¶8g)¾ ¡(§2i2;4 ± fe´3´25; ´7; 2¶8g)= (§2i2;4)e´3º5´8 = 0 mod 2((§2i3;4)¤¼9(§2P3)) = 0:This leads to the second and completes the proof. ¤Next we compute the homotopy groups of the homotopy ¯bre of §2p4;3 :§2P4 ! S6 to determine ¼9(§2P4). Let K be the homotopy ¯bre of §2p4;3.By [2, Corollary 5.8], the 10-skeleton of K has a cellular decompositionK(10) = §2P3 [[¶§2P3 ;§2°3] C§6P3:For m < n, we denote by iKm;n : K(m) ! K(n) and iKm : K(m) ! K theinclusion maps and pKn;m : K(n) ! K(n)=K(m) the collapsing map.Lemma 2.5.(1) ¼8(K) = Z2fiK4 [i4; ¸2]g © Z2fiK4 e´3º5g © Z24fiK5 sº5g©Z2fiK5 [§2i1;3; s]´7g;(2) ¼9(K) = Z2fiK4 ¸2º6g © Z2fiK5 sº5´8g © Z2fiK5 [[§2i1;3; s];§2i1;3]g:3Miyauchi: Self-homotopy of the Double Suspension of the Real 7-projectiveProduced by The Berkeley Electronic Press, 200680 T. MIYAUCHIProof. We consider the homotopy exact sequence of a pair (K(8);§2P3):¼10(K(8);§2P3)@10¡¡! ¼9(§2P3)iK5;8¤¡¡¡! ¼9(K(8)) j¤¡! ¼9(K(8);§2P3)@9¡! ¼8(§2P3)iK5;8¤¡¡¡! ¼8(K(8)) j¤¡! ¼8(K(8);§2P3) @8¡! ¼7(§2P3):The group structures ¼8(K(8);§2P3) = Zf!8g and ¼9(K(8);§2P3) = Z2fb´7gare obtained by the Blakers-Massey theorem, where !8 = !(K(8);§2P3)8 . By(2.1) and the relation [i4; ¶M4 ] = ¸2p6 [8, Lemma 1.5], the attaching map ofthe 8-cell of K(8) is [¶§2P3 ;§2°3]±§6i1;3 = (§2i2;3)[i4; ¶M4 ]e´5 = (§2i2;3)¸2´6.So we have @8(!8) = (§2i2;3)¸2´6 and @9(b´7) = (§2i2;3)¸2´26. By (2.2),the order of these elements are two. Therefore, there exists an element' 2 ¼8(K(8)) such that pK8;5' = 2¶8. Here we note that ' is taken as arepresentative of the Toda bracket' 2 fiK5;8[§2i2;3;§2°3]; i8; 2¶7g:So, by Lemma 2.2 (1), we have¼8(K(8)) =Z2fiK4;8[i4; ¸2]g © Z2fiK4;8 e´3º5g © Z24fiK5;8sº5g© Z2fiK5;8[§2i1;3; s]´7g © Zf'g:(2.3)We have ¼10(K(8);§2P3) = Z2f b´27g © Z2f[!8;§2i1;3]g by the James exactsequence [4, Theorem 2.1]. Since @10( b´27) = (§2i2;3)¸2´36 = 0 and@10([!8;§2i1;3]) = [(§2i2;3)¸2´6;§2i1;3] = (§2i2;3)[¸2; i4]´8;we obtain¼9(K(8)) =Z2fiK4;8¸2º6g © Z2fiK4;8 e´3º5´8g © Z2fiK5;8sº5´8g© Z4fiK5;8[§2i2;3; s]e´7g © Z2fiK5;8[[§2i1;3; s];§2i1;3]g:(2.4)Note that ' is obtained in the follwoing diagram between the co¯bersequences:S7 §2P3 K(8) S8S7 M8 S8 S8:-[§2i1;3;§2°3] -iK5;8 -pK8;56=-i86[§2i2;3;§2°3]-p86'-2¶86=Write now the homotopy exact sequence of a pair (K(9);K(8)):¼10(K(9);K(8))@10¡¡! ¼9(K(8))iK8;9¤¡¡¡! ¼9(K(9)) j¤¡! ¼9(K(9);K(8))@9¡! ¼8(K(8))iK8;9¤¡¡¡! ¼8(K(9)) ¡! 0:4Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 9http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/9SELF-HOMOTOPY OF §2P7 81The group structures ¼9(K(9);K(8)) = Zf!9g and ¼10(K(9);K(8)) = Z2fb´8gare obtained by the Blakers-Massey theorem, where !9 = !(K(9);K(8))9 . Byuse of the exact sequence of a triple (K(9);K(8);§2P3),@0 : ¼9(K(9);K(8))! ¼8(K(8);§2P3)is the map of degree 2. So, by the commutative diagram¼9(K(9);K(8)) ¼8(K(8);§2P3)¼8(K(8));-@0?@9©©©©©©*j¤' is taken as the attaching map of 9-cell of K(9). Hence, by (2.3) and¼8(K(9))»= ¼8(K), we obtain (1) and j¤=0. We see that@10(b´8) = ' ± ´8 2 fiK5;8[§2i2;3;§2°3]; i8; 2¶7g ± ´8= iK5;8[§2i2;3;§2°3] ± fi8; 2¶7; ´7g3 iK5;8[§2i2;3;§2°3]e´7mod iK5;8[§2i2;3;§2°3] ± (¼8(M8) ± ´8 + i8 ± ¼9(S7)) = 0:Here we used [§2i2;3;§2°3]i8´27 = [§2i1;3;§2°3]´27 = (§2i2;3)¸2´36 = 0. Bythe fact that [§2i2;3;§2°3] = 2[§2i2;3; s] + (§2i2;3)[¶M4 ; e´3] and [¶M4 ; e´3] =e´3º5p8 § ¸2´6 [5, Lemma 1.2], we obtain@10(b´8) = 2iK5;8[§2i2;3; s]e´7 + iK4;8 e´3º5´8;and hence¼9(K(9)) =Z2fiK4;9¸2º6g © Z2fiK5;9sº5´8g © Z4fiK5;9[§2i2;3; s]e´7g© Z2fiK5;9[[§2i1;3; s];§2i1;3]g:(2.5)Let pM : §2P3 !M4 be the projection. Then,(2.6) ¶§2P3 = s§2p3;2 + (§2i2;3)pM ;(2.7) §2p3;2 ± s = ¶5; pM ± §2i2;3 = ¶M4 and pM ± s = 0:By (2.1) and (2.6), we have[¶§2P3 ;§2°3] = [s§2p3;2;§2°3] + [(§2i2;3)pM ;§2°3]= [s; (§2i2;3)~´3] ± §6p3;2 + (§2i2;3)[pM ; ~´3] + 2[(§2i2;3)pM ; s]:5Miyauchi: Self-homotopy of the Double Suspension of the Real 7-projectiveProduced by The Berkeley Electronic Press, 200682 T. MIYAUCHIBy (2.7), we have [pM ; ~´3] ± §6i2;3 = [¶M4 ; ~´3]. So, by use of the co¯bersequence M8§6i2;3¡¡¡¡! §6P3 §6p3;2¡¡¡¡! S9,[pM ; ~´3] ´ [¶M4 ; ~´3] ± §4pM mod ¼9(M4) ± §6p3;2:By the same reason,[(§2i2;3)pM ; s] ´ [§2i2;3; s] ± §4pM mod ¼9(§2P3) ± §6p3;2:Hence, by Lemma 2.2 (2) and (2.7), we conclude that[¶§2P3 ;§2°3] ± §4s ´ §[s; (§2i2;3)~´3] mod (§2i2;3) ± ¼9(M4) ± §6p3;2:The attaching map of the 10-cell of K(10) is iK5;9[¶§2P3 ;§2°3]§4s. By (2.1),we haveiK5;9[¶§2P3 ;§2°3]§4s ´ §iK5;9[s;§2i2;3]e´7 mod iK4;9¸2º6:So, by the homotopy exact sequence of a pair (K(10);K(9)) and (2.5), thegroup structure of ¼9(K) is obtained. This completes the proof. ¤Lemma 2.6.¼9(§2P4) =Z2f e´35g © Z2f(§2i2;4)¸2º6g © Z2f(§2i3;4)sº5´8g© Z2f(§2i3;4)[[s;§2i1;3];§2i1;3]g:Proof. We consder the exact sequence induced from the ¯bration §2p4;3 :§2P4 ! S6:¼10(S6) = 0! ¼9(K)! ¼9(§2P4)! ¼9(S6) ¢9¡¡! ¼8(K)! ¢ ¢ ¢ :By [8, Lemma 1.2], we obtain the relations ¢6(¶6) = §iK4 e´3 + 2iK5 s and¢9(º6) = ¢6(¶6) ± º5 = §iK4 e´3º5 + 2iK5 sº5:Using the second relation and Lemma 2.5 (1), we obtain Ker ¢9 = Z2f´36g.Therefore, by Lemma 2.4 and 2.5 (2) and by the fact that i ± iK5 = §2i3;4(i : K ! §2P4 is the inclusion), we obtain the result. This completes theproof. ¤Now we consider the homotopy exact sequence of a pair (§2P5;§2P4):¼10(§2P5;§2P4)@10¡¡! ¼9(§2P4) i¤¡! ¼9(§2P5)j¤¡! ¼9(§2P5;§2P4) @9¡! ¼8(§2P4);where i = §2i4;5 : §2P4 ! §2P5. By the James exact sequence, the groupstructures ¼9(§2P5;§2P4)=Z2f b´26g©Z2f[!7;§2i1;4]g and ¼10(§2P5;§2P4)=Z24fbº6g © Z2f[!7; (§2i1;4)´3]g are settled, where !7 = !(§2P5;§2P4)7 . Werecall, from [8, Lemma 1.3], the relation(2.8) §2°4 = (§2i3;4)s´5 + 2(§2i2;4)¸2:6Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 9http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/9SELF-HOMOTOPY OF §2P7 83By Lemma 2.6 and by the relation ´5º6 = 0, we obtain(2.9) @10(bº6) = (§2°4)º6 = ((§2i3;4)s´5 + 2(§2i1;4)¸2)º6 = 0:The equation (§2i3;4)[§2i2;3; s]e´7 = 0 is shown in the proof of Lemma 2.6.Then@10([!7; (§2i1;4)´3]) = [§2°4; (§2i1;4)´3] = [(§2i3;4)s´5; (§2i1;4)´3]= (§2i3;4)[s;§2i1;3]´27 = 2(§2i3;4)[s;§2i2;3]e´7 = 0:Therefore (§2i4;5)¤ : ¼9(§2P4)! ¼9(§2P5) is a monomorphism.By the fact that ¼8(§2P4) = Z4f(§2i3;4)sº5g©Z2f[(§2i3;4)s;§2i1;4]´7g©Z2f(§2i2;4)[i4; ¸2]g [8, Lemma 2.5] and by (2.8), we obtain@9( b´26) = (§2°4)´26 = (§2i3;4)s´35 = 4(§2i3;4)sº5 = 0and@9([!7;§2i1;4]) = [§2°4;§2i1;4] = [(§2i3;4)s´5;§2i1;4] = [(§2i3;4)s;§2i1;4]´7:Then there exists an element e´26 2 f§2i4;5;§2°4; ´26g ½ ¼9(§2P5) such that(§2p5;4) e´26 = ´27. We obtain2 e´26 2 f§2i4;5;§2°4; ´26g ± 2¶9 = ¡(§2i4;5 ± f§2°4; ´26; 2¶8g)andf§2°4; ´26; 2¶8g ½ f§2i3;4; (s´5 + 2(§2i2;3)¸2)´26; 2¶8g= f§2i3;4; s´35; 2¶8g= f§2i3;4;§2°3 ± 2º5; 2¶8g¾ f§2i3;4;§2°3; ´35g3 e´35mod 2¼9(§2P4) + (§2i3;4)¤¼9(§2P3) = (§2i3;4)¤¼9(§2P3);and hence we conclude that 2 e´26 ´ (§2i4;5) e´35 mod (§2i3;5)¤¼9(§2P3). Thus,by Lemma 2.6, we have the following.Lemma 2.7.¼9(§2P5) =Z4f e´26g © Z2f(§2i2;5)¸2º6g © Z2f(§2i3;5)sº5´8g© Z2f(§2i3;5)[[§2i1;3; s];§2i1;3]g;where 2 e´26 = (§2i4;5) e´35 for a suitable choice of e´35.7Miyauchi: Self-homotopy of the Double Suspension of the Real 7-projectiveProduced by The Berkeley Electronic Press, 200684 T. MIYAUCHI3. Proofs of main theoremsFirst, we show Theorem 1.2.From the fact that §2°5 2 f§2i4;5;§2°4; 2¶6g, §3°4 = (§3i3;4)(§s)´6, wesee that[§2°5;§2i1;5]p9 = [¶§2P5 ;§2i1;5] ± §4°5 ± p92 [¶§2P5 ;§2i1;5] ± f§4i4;5;§4°4; 2¶8g ± p9¾ [¶§2P5 ;§2i1;5]f(§4i3;5)§2s; ´7; 2¶8g ± p9= ¡([§2i3;5;§2i1;5]§2s ± f´7; 2¶8; p8g)3 [§2i3;5;§2i1;5](§2s)´7mod [§2i4;5;§2i1;5] ± ¼9(§4P4) ± p9:It is easily seen that ¼9(§4P4) = Z2f(§4i1;4)º5´8g © Z2f(§4i3;4)(§2s)´27g.Since [¶3; ¶3] = 0 and (§2i3;4)[s;§2i2;3]e´7 = 0, we obtain[§2i4;5;§2i1;5] ± (§4i1;4)º5´8 = (§2i1;5)[¶3; ¶3]º5´8 = 0and[§2i4;5;§2i1;5] ± (§4i3;4)(§2s)´27 = [§2i4;5;§2i1;5] ± (§4°4)´8 = 0:Then [§2i4;5;§2i1;5]±¼9(§4P4) = 0. By (2.6), the element [§2i3;5;§2i1;5]§2sis changed as follows.[§2i3;5;§2i1;5]§2s = (§2i3;5)[¶§2P3 ;§2i1;3]§2s= (§2i3;5)[s;§2i1;3] + (§2i2;5)[pM ; i4]§2s:By the fact that [pM ; i4] 2 [§4P3;M4]=(§4p3;2)¤¼7(M4)©(§2pM )¤[M6;M4],we obtain(§2i2;5)[pM ; i4]§2s 2 §2i2;5 ± (¼7(M4) ± §4p3;2 + [M6;M4] ± §2pM ) ± §2s= §2i2;5 ± ¼7(M4):We recall from [7, Lemma 2.2] that ¼7(M4) = Z2f¸2´6g © Z2fe´3´25g. Since(§2i2;4)¸2´6 = 0 [8, the proof of Lemma 2.2] and by (2.1), the group §2i2;5 ±¼7(M4) is 0. Then,[§2°5;§2i1;5]p9 = [§2i3;5;§2i1;5](§2s)´7 = (§2i3;5)[s;§2i1;3]´7:Here we consider an element [§2°4;§2i2;4] 2 [M9;§2P4]. Since 2¶M4 =i4´3p4 [11], (§2i2;4)¸2´6 = 0, (§2i3;4)[s;§2i2;3]e´7 = 0 and ´2 ^ ¶M2 = i4´3 +e´3p5, we obtain[§2°4;§2i2;4] = (§2i3;4)[s´5;§2i2;3] + (§2i2;4)[2¸2; ¶M4 ]= (§2i3;4)[s;§2i2;3] ± §(´4 ^ ¶M3) + (§2i2;4)[¸2; 2¶M4 ]8Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 9http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/9SELF-HOMOTOPY OF §2P7 85= (§2i3;4)[s;§2i2;3] ± §(i7´6 + e´6p8) + (§2i2;4)[¸2; i4´3p4]= (§2i3;4)[s;§2i1;3]´7 + (§2i2;4)[¸2´6; i4]p9= (§2i3;4)[s;§2i1;3]´7:Thus, we get that[§2°5;§2i1;5]p9 = (§2i3;5)[s;§2i1;3]´7 = (§2i4;5)[§2°4;§2i2;4] = 0:By use of the co¯bre sequence S8 i9¡!M9 p9¡! S9 2¶9¡¡! S9, we have[§2°5;§2i1;5] 2 2¼9(§2P5) = Z2f2 e´26g:Let l1 : P4=P3 = S4 ! P5=P3 = S4_S5 be the canonical inclusion map. ByLemma 2.7 and by the relations p5;3 ± i4;5 = l1 ± p4;3 and p5;3 ± i1;5 = 0, weobtain§2p5;3 ± 2 e´26 = §2p5;3 ± §2i4;5 ± e´35= §2l1 ± §2p4;3 ± e´35= (§2l1)´36 6= 0 2 ¼9(S6 _ S7) »= Z24 © Z2and§2p5;3 ± [§2°5;§2i1;5] = 0:Therefore we have [§2°5;§2i1;5] = 0 and the proof of Theorem 1.2 is com-plete.By [8, the proof of Lemma 2.5], we have a relation (§2°5)´7 = 0 and wecan de¯ne a coextension e´07 2 ¼9(§2P6) of ´7 as follows:e´07 2 f§2i5;6;§2°5; ´7g:Since 2e´07 2 f§2i5;6;§2°5; ´7g ± 2¶9 = ¡(§2i5;6 ± f§2°5; ´7; 2¶8g) and§2p5;4 ± f§2°5; ´7; 2¶8g ½ f2¶7; ´7; 2¶8g = ´27;we obtain 2e´07 ´ (§2i5;6) e´26 mod §2i5;6 ± 2¼9(§2P5) + §2i4;6 ± ¼9(§2P4) =§2i4;6 ± ¼9(§2P4). From the exact sequence of a pair (§2P6;§2P5) and byTheorem 1.2, we see that (§2i5;6)¤ : ¼9(§2P5) ! ¼9(§2P6) is a monomor-phism. Thus, e´07 is of order 8 and the group structure of ¼9(§2P6) is givenas follows.Lemma 3.1.¼9(§2P6) =Z8fe´07g © Z2f(§2i2;6)¸2º6g © Z2f(§2i3;6)sº5´8g© Z2f(§2i3;6)[[§2i1;3; s];§2i1;3]g;where 2e´07 = (§2i5;6) e´26 for a suitable choice of e´26.9Miyauchi: Self-homotopy of the Double Suspension of the Real 7-projectiveProduced by The Berkeley Electronic Press, 200686 T. MIYAUCHIWe denote by s1 : S9 ! §2P7 = §2P6 _ S9 the inclusion map to thesecond factor and by q1 : §2P7 = §2P6 _ S9 ! §2P6 the map collapsing S9to one point. Finally we obtain the following.Theorem 3.2.[§2P7;§2P7] =Zfs1§2p7;6g © Z8f(§2i6;7)q1g © Z8f(§2i6;7)e´07§2p7;6g© Z2f(§2i3;7)[s;§2i2;3](§2p6;4)q1g© Z2f(§2i2;7)¸2(§2¹p4;3)q1g © Z2f(§2i2;7)[¸2; i4](§2p6;5)q1g© Z2f(§2i3;7)sº5(§2p6;5)q1g © Z2f(§2i2;7)¸2º6§2p7;6g© Z2f(§2i3;7)sº5´8§2p7;6g© Z2f(§2i3;7)[[§2i1;3; s];§2i1;3]§2p7;6g:References[1] A. L. Blakers and W. S. Massey. Products in homotopy theory. Ann. of Math. 58(1953), 295-324.[2] B. Gray. On the homotopy groups of mapping cones. Proc. London Math. Soc. (3)26 (1973), 497-520.[3] T. Inoue and J. Mukai. A note on the Hopf homomorphism of a Toda bracket andits application. Hiroshima Math. J. 33-3 (2003), 379-389.[4] I. M. James. On the homotopy groups of certain pairs and triads. Quart. J. Math.Oxford (2), 5 (1954), 260-270.[5] Y. Miyauchi and J. Mukai. Self-homotopy of the double suspension of the real 6-projective space. Kyushu J. Math. 59 (2005) 101-116.[6] J. Mukai. On the attaching map in the Stiefel manifold of 2-frames. Math. J. OkayamaUniv. 33 (1991), 177-188.[7] J. Mukai. Note on existence of the unstable Adams map. Kyushu J. Math. 49 (1995),271-279.[8] J. Mukai. Some homotopy groups of the double suspension of the real projective spaceRP6. Matem¶atica Contempora^nea 13 (1997), 235-249.[9] E. Rees. Embeddings of real projective spaces. Topology 10 (1971), 309-312.[10] H. Toda. Composition methods in homotopy groups of spheres. Ann. of Math. Studies49 (1962), Princeton University Press.[11] H. Toda. Order of the identity class of a suspension space. Ann. of Math. 78 (1963),300-325.[12] J. Wu. Homotopy theory of the suspensions of the projective plane. Mem. Amer.Math. Soc. 769 (2003).Toshiyuki MiyauchiGraduate School of MathematicsKyushu UniversityFukuoka, 812-8581 Japane-mail address: miyauchi@math.kyushu-u.ac.jp(Received May 20, 2005 )10Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 9http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/9

Self-homotopy of the Double Suspension of the Real 7-projective Space

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