Homotopy types of m-twisted CP

35 (2005), 143–157

Homotopy types of m-twisted CP

⁴

’s

Dedicated to the memory of Prof. Masahiro Sugawara Kohhei Yamaguchi

(Received April 8, 2004) (Revised September 21, 2004)

Abstract. We study the homotopy type classification problem of n dimensional m- twisted complex projective spaces for the casen¼4. In particular, we determine the number of homotopy types of m-twisted CP⁴’s when mb1 is an odd integer.

1. Introduction

Let nb2 be an integer and let M be a simply-connected 2n dimensional finite Poincare´ complex. For an integer mb0, M is called an m-twisted CPⁿ if there is an isomorphism HðM;ZÞGHðCPⁿ;ZÞ with the condition x₂x₂¼mx₄, where x_2k AH^2kðM;ZÞGZdenotes the corresponding generator (k¼1;2). Any m-twisted CPⁿ is homotoy equivalent to a CW complex of the form

MFS²U_mh

2e⁴Ue⁶U Ue²ⁿ²Ue²ⁿ ðhomotopy equivalenceÞ;

and it has the homotopy type of 2n dimensional closed topological manifolds ([6]). Let M_mⁿ be the set consisting of all the homotopy equivalence classes ofm-twistedCPⁿ’s. For example, whenn¼2,M₁²¼ f½CP²gandM_m² ¼q if m01. When n¼3, the following result is known:

Theorem 1.1 ([11] (cf. [4])). If mb0 is an integer, cardðM_m³Þ ¼ 1 if m11 ðmod 2Þ;

3 if m10 ðmod 2Þ;

where cardðVÞ denotes the cardinal number of a set V.

In general, it is known that M_m^2kþ10 q for any m;kb2 (cf. [2]), and we have infinitely many non-trivial examples ofm-twistedCP^2kþ1’s. On the other

2000 Mathematics Subject Classification. Primary 55P10, 55P15; Secondly 57P10.

Key Words and Phrases. Poincare´ complex, homotopy type, Whitehead product.

Partially supported by Grant-in-Aid for Scientific Research (No. 13640067 (C)(2)), The Ministry of Education, Culture, Sports, Science and Technology, Japan.

hand, it is even not known whether M^2k_m is an empty set or not if m01 and kb2. As the first step of this question, we would like to study the setM_mⁿ for the case n¼4. Then if ða;bÞ denotes the greatest common divisor of positive integers a and b, the following result has been known.

Theorem 1.2 ([6]).

( i ) If m¼0, 3acardðM_m⁴Þa2⁷3².

( ii ) If m11 ðmod 2Þ, 1acardðM_m⁴Þam ðm;3Þ.

(iii) If m10 ðmod 8Þ and m00, 3acardðM_m⁴Þa2⁵3m ðm;3Þ.

(iv) If m10 ðmod 2Þ and mD0 ðmod 8Þ, M_m⁴ ¼q.

In this paper we shall investigate the set M_m⁴ when m11 ðmod 2Þ, and our main results are stated as follows:

Theorem 1.3 (The main Theorem). Let mb1 be an odd integer, and let M₀, M₁, M₁ denote the m-twisted CP⁴’s defined in Definition 4.

( i ) If mD0 ðmod 3Þ, M_m⁴ ¼ f½M0g.

(ii) If m10 ðmod 3Þ, M⁴_m¼ f½M0;½M1;½M1g, such that the first mod 3 reduced power operation P¹:H⁴ðMe;Z=3Þ !H⁸ðMe;Z=3Þ is an iso- morphism if e¼G1 and is trivial if e¼0.

Corollary 1.4. If mb1 is an odd integer,

cardðM_m⁴Þ ¼ ðm;3Þ ¼ 1 if mD0 ðmod 3Þ;

3 if m10 ðmod 3Þ:

Remark. Let mb3 be an odd integer with m10 ðmod 3Þ, and let Ap denote the modp Steenrod algebra. Although M1 and M1 are not homotopy equivalent, there are isomorphisms

HðM1;ZÞGHðM1;ZÞ (as graded rings)

HðM1;Z=pÞGHðM1;Z=pÞ (asAp-modules for any prime pb2).

This paper is organized as follows. In § 2, we compute some Whitehead products and in § 3, we consider the group of self-homotopy equivalences EðXmÞ of the 6-skeletonXmofm-twistedCP⁴’s. In § 4, we study the leftEðXmÞ-action on p₇ðXmÞgiven by composite of maps, which is the key point for determining the homotopy types of m-twisted CP⁴’s. In particular, we determine the set M⁴_m explicitly when ðm;6Þ ¼1. Finally, in § 5, we compute the EðXmÞ-action on p₇ðXmÞ explicitly and determine the set M_m⁴ when ðm;6Þ ¼3.

2. Whitehead products

For an integer mb1, let Lm and P⁴ðmÞdenote the CW complexes defined by L_m ¼S²U_mh

2e⁴ and P⁴ðmÞ ¼S³U_mi

3e⁴, respectively. If q: ~LL_m!L_m

denotes the 2-connective covering of Lm, it is known that there is a homotopy equivalence LL~_mFP⁴ðmÞ4S⁵ ([13]). If we identify LL~_m¼P⁵ðmÞ4S⁵, then the map q is also identified with the map

q¼ ðf_m;b_mÞ:P⁴ðmÞ4S⁵!L_m ðup to homotopyÞ:

ð1Þ

It follows from [[6], Lemma 3.3] that there is a homotopy commutative di- agram

S^{3 mi}!³ S^{3 i0}! P⁴ðmÞ ^q!

0 m S⁴

^h2

??

?y ^fm

??

?y

S^{3 mh}!² S^{2 i}! Lm ^qm!S⁴; ð2Þ

where two horizontal sequences are cofiber sequences.

Let oAp₆ðS³ÞGZ=12 and a₁ð3Þ ¼4oAp₆ðS³Þ_ð3ÞGZ=3 denote the generators. If m10 ðmod 3Þ, we denote by aae11ð3ÞAp7ðP⁴ðmÞÞ the coexten- sion of a1ð3Þ which satisfies the condition q_m⁰ aae11ð3Þ ¼Ea1ð3Þ.

Lemma 2.1 ([6], [11]). If mb1 is an odd integer, there are isomorphisms p₅ðLmÞ ¼Zb_m; p₅ðP⁴ðmÞÞ ¼0;

p₆ðLmÞ ¼Z=ðm;3Þ iðh₂oÞlZ=mf_mslZ=2b_mh₅; p₆ðP⁴ðmÞÞ ¼Z=ðm;3Þ i⁰olZ=ms;

p7ðLmÞ ¼Z=ðm;3Þ fmomlZ=2bmh²₅lZ=m ½bm;iðh₂Þ; p7ðP⁴ðmÞÞ ¼Z=ðm;3Þ om;

8>

>>

>>

<

>>

>>

>:

where we can take o_m¼aae₁₁ð3ÞAfi⁰;mi₃;a₁ð3Þg if m10 ðmod 3Þ.

Proof. The assertions follow from [6] except the last equality. If m10 ðmod 3Þ, by the proof of [[6], Proposition 2.9], the induced homo- morphism

Z=3om¼p7ðP⁴ðmÞÞⁱ!

0

1 p7ðP⁴ðmÞ;S³Þ ^a_G^mp7ðD⁴;S³ÞGZ=12

is injective, wherea_mAp₄ðP⁴ðmÞ;S³ÞGZdenotes the characteristic map of the top celle⁴ in P⁴ðmÞ. BecauseD⁰ðEoÞ ¼mði3oÞin the sequence (5) of [6], we have q_m⁰ o_m¼Ea₁ð3Þ and the condition o_m¼aae₁₁ð3ÞAfi⁰;mi₃;a₁ð3Þg is also

satisfied. r

Definition 1. For an integer mb1, let Xm be the space defined by Xm¼LmU_mb

me⁶. There is a cofiber sequence, S^{5 mb}!^m Lm!^j Xm !S⁶. Lemma 2.2 ([6]). If mb1 is an odd integer, there are isomorphisms

p₆ðXmÞ ¼Z=ðm;3Þ jðiðh₂oÞÞlZ=m jðf_msÞ;

p₇ðXmÞ ¼Zj_mlZ=ðm;3Þ jðf_mo_mÞlZ=m jð½bm;iðh₂ÞÞ;

and the following equality holds:

j1ðj_mÞ ¼ ½b_m;i_rþb_mh₅⁰: ð3Þ

Here b_mAp6ðXm;LmÞGZ denotes the characteristic map of the top cell e⁶ in X_m, ½;_r a relative Whitehead product, h_k⁰ Apkþ2ðD^kþ1;S^kÞGZ=2 the gen- erator ðkb3Þ, j1:ðXm;Þ ! ðXm;LmÞ is the inclusion and j1:p7ðXmÞ ! p7ðXm;LmÞ ¼Z ½b_m;i_rlZ=2b_mh₅⁰ the induced homomorphism.

Lemma 2.3. If m11 ðmod 2Þ, ½bm;iðh₂Þ ¼ ½½bm;i;iAp₇ðLmÞ.

Proof. It follows from the Jacobi identity ([[10], Corollary 7.14]) that

½½bm;i;i þ ½½i;i;bm þ ½½i;bm;i ¼0:

Because½i;bm ¼ ½bm;i, we have 2½½bm;i;i þ ½½i;i;bm ¼0. Then using ½i;i ¼ i ½i2;i2 ¼ið2h₂Þ ¼2iðh₂Þ, we have

2½½bm;i;i þ2½iðh₂Þ;bm ¼2½½bm;i;i 2½bm;iðh₂Þ ¼0:

Since the order of ½bm;iðh₂Þ is just m (by [[6], Corollary 3.5]) and m11 ðmod 2Þ, ½½bm;i;i ½bm;iðh₂Þ ¼0. r

Lemma 2.4. If m11 ðmod 2Þ, fms¼ ½bm;i þbmh₅Ap6ðLmÞ.

Proof. It follows from [[6], Proposition 5.1] that there is a unit x_mAðZ=mÞ such that ½bm;i ¼x_m f_msþb_mh₅. Since the order of sAp6ðP⁴ðmÞÞ is m([8]), by changing the generator s7!x¹_m s, we may assume

that x_m¼1 and the assertion follows. r

Corollary 2.5. If m11 ðmod 2Þ, ½fms;i ¼ ½bm;iðh₂Þ.

Proof. It follows from Lemmas 2.3 and 2.4 that

½fms;i ¼ ½bm;iðh₂Þ þ ½bmh₅;i:

ð4Þ

Since the orders of ½bm;iðh₂Þ and s are m, we see ½bmh₅;i ¼0 and the

assertion follows. r

Now we remark the following general fact concerning m-twisted CP⁴’s.

Lemma 2.6. Let mb0 be an integer and M an m-twisted CP⁴. Then if mD0 ðmod 3Þ, P¹ :H⁴ðM;Z=3Þ !^G H⁸ðM;Z=3Þ is an isomorphism.

Proof. If y_2l AH^2lðM;Z=3ÞGZ=3 denotes the mod 3 generator (1ala4), ðy2Þ²¼my4, y2y4¼my6 and ðy4Þ²¼y2y6¼ y8 by [[6], (0.2)].

Hence, P¹ðy2Þ ¼ ðy2Þ³ ¼ ðmy4Þ y₂ ¼m²y₆¼Gy₆ and

mP¹ðy4Þ ¼P¹ðmy4Þ ¼P¹ððy2Þ²Þ ¼2y2P¹ðy2Þ ¼G2y2y6 ¼Hy8: Because mD0 ðmod 3Þ, this implies that P¹ðy₄Þ ¼Gy₈. r 3. Groups of self-homotopy equivalences

For a connected space X, we denote by EðXÞ the set consisting of all based homotopy classes of based self-homotopy equivalences of X, which becomes a group whose multiplication is induced from composite of maps.

The group EðXÞ is called the group of self-homotopy equivalences of X.

Definition 2. If K is a CW complex and X ¼KU_feⁿ with dimKa n2, we define the homomorphism l: ~jjðpnðKÞÞ !EðXÞ by

lð~jjgÞ ¼‘ ð14jj~gÞ m⁰:X ^m!

0

X4S^{n 14jjg~}!X4X ^‘!X for gApnðKÞ, where ~jj:K!X denotes an inclusion, m⁰:X!X4Sⁿ the co- action map given by pinching the hemisphere of the top cell eⁿ and ‘ is a folding map.

If y:X !^F X is a homotopy equivalence, it follows from the cellular approximation Theorem that the restriction yj_K also defines a self-homotopy equivalence on K. So we can define the homomorphsim f:EðXÞ !EðKÞ by the restriction fðyÞ ¼yj_K for yAEðXÞ.

Proposition 3.1. If mb1 is an odd integer, there is an exact sequence p₆ðXmÞ !^l EðXmÞ !^f EðLmÞ !1;

where we take Z₂¼ fG1g and EðLmÞGZ₂.

Proof. Because jðp6ðLmÞÞ ¼p6ðXmÞ, the assertion easily follows from the Barcus-Barratt Theorem [[1], Theorem 6.1] and [[11], Corollary 4.8]. r 4. An action of EðXmÞ on p₇ðXmÞ

For CW complexes X and Y, we write XFY if there is a homotopy equivalence X !^F Y. Let MðjÞ denote the mapping cone defined by

MðjÞ ¼X_mU_je⁸ for jAp₇ðXmÞ: ð5Þ

Recall the following well-known result.

Lemma 4.1 (Homotopy Theorem). Let K be a simply-connected CW complex and let X , Y denote the CW complexes defined by X ¼KU_feⁿ and Y ¼KU_geⁿ, where dimKan2, nb4 and f;gApn1ðKÞ. Then there is a homotopy equivalence XFY if and only if there is a homotopy equivalence yAEðKÞ such that y f ¼Gg.

Theorem4.2 ([6]). Let mb1 be an odd integer. Then Mis an m-twisted CP⁴ if and only if there is some element gAp₇ðLmÞ such that MFMðjÞ ¼ XmU_je⁸, where j¼Gj_mþjðgÞ.

Proof. This follows from [[6], Theorem 4.5] and the homotopy exact

sequence of the pair ðXm;L_mÞ. r

So it is useful to consider the left EðXmÞ action on p7ðXmÞ given by the composite of maps, EðXmÞ p₇ðXmÞCðy;jÞ 7!yjAp₇ðXmÞ.

Lemma 4.3. Let mb1 be an integer and jAp₇ðXmÞ be an element such that j1ðjÞ ¼a ½b_m;i_rþeb_mh₅⁰ for some ða;eÞAZZ=2. Then

m⁰ðjÞ ¼ jXjþa½j6;jX ji þe j6h₆;

where Xm!^jX Xm4S^{6 j6} S⁶ denote the corresponding inclusions, m⁰:Xm! Xm4S⁶ is a co-action map and m⁰ :p7ðXmÞ !p7ðXm4S⁶Þ is the induced homomorphism.

Proof. This follows from [[12], Lemma 2.2]. r Corollary 4.4. If mb1 be an odd integer, the following equalities hold:

( i ) m⁰ðj_mÞ ¼ jXj_mþ ½j6;jX ji þj6h₆. ( ii ) m⁰ðjð½bm;iðh₂ÞÞÞ ¼ jX jð½bm;iðh₂ÞÞ.

(iii) If m10 ðmod 3Þ, m⁰ðjðiðh₂oÞÞÞ ¼ j_X jðiðh₂oÞÞ.

Proof. The assertions follows from (3) and Lemma 4.3. r Definition 3. Let mb1 be an odd integer and let l: jðp6ðLmÞÞ ! EðXmÞ denote the homomorphism defined in Definition 2. Then define the homotopy equivalence yk AEðXmÞ by

y_k¼lððkjðf_msÞÞÞ for each kAZ=m:

ð6Þ

Similarly, when m10 ðmod 3Þ, we define y_l⁰AEðXmÞ by y_l⁰¼lðl jðiðh₂oÞÞÞ for each lAZ=3:

ð7Þ

Proposition 4.5. If mb1 is an odd integer, the following equalities hold for any ðk;lÞAZ=mZ=3:

( i ) ykj_m¼j_mþk jð½bm;iðh₂ÞÞ, yk jð½bm;iðh₂ÞÞ ¼ jð½bm;iðh₂ÞÞ:

(ii) If m10 ðmod 3Þ,

y_l⁰j_m¼j_m; y_l⁰jð½bm;iðh₂ÞÞ ¼ jð½bm;iðh₂ÞÞ; y_l⁰ jðfmomÞ ¼ykjðfmomÞ ¼ jðfmomÞ:

Proof. (i) It follows from Corollary 4.4 that we have y_k jð½bm;iðh₂Þ ¼‘ ð14ðkjðf_msÞÞÞ m⁰ð½bm;iðh₂ÞÞ

¼‘ ð14ðkj f_msÞÞ j_X jð½bm;iðh₂ÞÞ

¼ jð½bm;iðh₂ÞÞ: Since f_msh₆¼0 (by [6]), we also obtain

ykj_m¼‘ ð14ðkjðfmsÞÞÞ m⁰ðj_mÞ

¼‘ ð14ðkjðfmsÞÞÞ ðjXj_mþ ½j6;jX ji þj6h₆Þ

¼j_mþ ½kjðfmsÞ;ji þk jðfmsh₆Þ

¼j_mþkjð½fms;iÞ

¼j_mþkjð½bm;iðh₂ÞÞ ðby Corollary 2:5Þ:

(ii) Because the proof is similar to that of (i), we only give the proof of the first equality. This follows from

y_l⁰j_m¼‘ ð14ðl jðiðh₂oÞÞÞ m⁰ðj_mÞ

¼‘ ð14ðl jðiðh₂oÞÞÞ ðj_Xj_mþ ½j₆;j_X ji þ j₆h₆Þ

¼j_mþ ½l jðiðh₂oÞÞ;ji þljðiðh₂oh₆ÞÞ

¼j_mþl jðið½h₂o;i₂ÞÞ ðby iðh₂oh₆Þ ¼0Þ

¼j_m ðby ½h₂o;i₂ ¼0 ðby ½3ÞÞ: r Definition4. Letmb1 be an odd integer. Then we denote by M0 the m-twisted CP⁴ defined by

M0¼Mðj_mÞ ¼XmU_j

me⁸: ð8Þ

Moreover, when m10 ðmod 3Þ, we denote by M1 and M1 the m-twisted CP⁴’s defined by

M_e¼Mðj_mþejðf_mo_mÞÞ ¼X_mU_j

mþejðfmomÞe⁸ ðfor e¼G1Þ: ð9Þ

Theorem 4.6. If m11 ðmod 2Þ and mD0 ðmod 3Þ, M⁴_m¼ f½M0g.

Proof. We note that M_m⁴0 q by Theorem 1.2. Now let M be anym- twisted CP⁴. It su‰ces to show that MFM₀. Since MðjÞFMðjÞ by Lemma 4.1, it follows from Theorem 4.2 that there exists some kAZ=m such that MFMðj_mþk jð½bm;iðh₂ÞÞÞ. Then because

ykj_m¼j_mþkjð½bm;iðh₂ÞÞ ðby Proposition 4:5Þ;

M₀¼Mðj_mÞFMðj_mþk jð½bm;iðh₂ÞÞÞFM. r Corollary 4.7. Let mb1 be an odd integer.

( i ) If mD0 ðmod 3Þ, there is an exact sequence 0!p6ðXmÞ !^l EðXmÞ !^f Z₂!1;

where p₆ðXmÞGZ=m.

(ii) If m10 ðmod 3Þ, there is an exact sequence

0!Z=mlG_m!^l EðXmÞ !^f Z₂!1;

where Gm¼Z=3 or Gm¼0.

Proof. (i) It su‰ces to show that l:Z=mjðf_msÞ ¼p₆ðXmÞ !EðXmÞ is injective. If we write yk ¼lðk jðfmsÞÞ (as in (6)), it follows from Proposition 4.5 that y_kj_m0y_lj if k0lAZ=m. Hence, y_k0y_l if k0lA Z=m, and l is injective.

(ii) The same proof as that of (i) shows that lj_Z=mjðfmsÞ:Z=m jðfmsÞ !EðXmÞ is injective. Because p6ðXmÞ ¼Z=m jðfmsÞlZ=3 jðiðh₂oÞÞ, Kerl¼Z=3jðiðh₂oÞÞ or Kerl¼0. Then (ii) follows

from Proposition 3.1. r

Proposition4.8. If mb3 is an odd integer with m10 ðmod 3Þand M is an m-twisted CP⁴, then MFM0 or MFM1 or MFM1 holds.

Proof. Becausep7ðXmÞ ¼Zj_mlZ=mjð½bm;iðh₂ÞÞlZ=3jðfmoÞ, by Theorem 4.2, there is a homotopy equivalence

MFMðj_mþk jð½bm;iðh₂ÞÞ þljðf_mo_mÞÞ ¼M_k;l for some kAZ=m and lAZ=3. Then by Proposition 4.5,

yk ðj_mþljðfmomÞÞ ¼ykj_mþlyk jðfmomÞ

¼j_mþkjð½bm;iðh₂ÞÞ þl jðfmomÞ:

Hence, there is a homotopy equivalence Mk;lFM0;l. Then because

M0;l¼

M₀ if l¼0AZ=3;

M₁ if l¼1AZ=3;

M1 if l¼ 1¼2AZ=3;

8>

<

>:

the assertion follows. r

5. The case m10 ðmod 3Þ

From now on, we assume that mb3 is an odd integer with m10 ðmod 3Þ, and consider the EðXmÞ-action on p₇ðXmÞ.

We remark that (by Corollary 4.7) there is a homotopy equivalence y~

yAEðXmÞ such that yyj~_Lm ¼h₁ represents the generator of EðLmÞGZ₂. In this case, because EðXmÞ is generated by fyk;y_l⁰;yy~:kAZ=m;lAZ=3g and the actions of yk’s or those of y_l⁰’s are given in Proposition 4.5, it remains to consider the action ofyy~on p₇ðXmÞ. For this purpose, we recall self-homotopy equivalences h1 and yy.~ First, recall h1. Because

ði2Þ ðmh₂Þ ¼mðh₂þ ½i2;i2 Hðh₂ÞÞ ð10Þ

¼mðh₂þ ð2h₂Þ i3Þ ¼mh₂

by [[10]; page 537, (8.12)], there is a map h₁ :L_m!L_m such that the following diagram is homotopy commutative:

S^{3 mh}!² S^{2 i}! Lm ^qm! S⁴

ⁱ²

??

?y ^h1

??

?y

S^{3 mh}!² S^{2 i}! Lm ^qm! S⁴: ð11Þ

Then the following is known:

Lemma 5.1 ([11]). Let mb1 be an odd integer.

( i ) h1 AEðLmÞ and EðLmÞ ¼ fh1;idLmg ¼hh1jh²₁ ¼idLmiGZ₂. ( ii ) The degree of h₁ on S² is 1 and the that of it on e⁴ is þ1.

(iii) h1bm¼ bm.

If m11 ðmod 2Þ, it follows from Proposition 3.1 that there exists a homotopy equivalence yy~AEðXmÞ such that

y~

yj_Lm ¼h1: ð12Þ

We note that yy~also defines a self-homotopy equivalence on ðXm;L_mÞ, and we write it as the same letter yy.~

Lemma 5.2. Let mb3 be an odd integer with m10 ðmod 3Þ.

( i ) yy~b_m¼ b_m.

(ii) yy~j_m1j_m mod Im j.

Proof. (i) If x_2k AH^2kðXm;ZÞGZ denotes the corresponding generator (k¼1;2;3), x2x4¼mx6. Hence, the degree ofyy~on the top celle⁶ on Xm is 1 (by (ii) of Lemma 5.1). So it follows from the commutative diagram

Zb_m¼p₆ðXm;L_mÞ ^h!

G H₆ðXm;L_m;ZÞ _G^j1 H₆ðXm;ZÞGZ

y~ y

??

?y

??

?y

??

?y^ð1Þ Zb_m¼p6ðXm;LmÞ ^h!

G H6ðXm;Lm;ZÞ _G^j1 H6ðXm;ZÞGZ that we have yy~b_m¼ b_m.

(ii) Consider the induced homomorphism j1ðXmÞ !p7ðXm;LmÞ. Then because

j1ðyy~j_mÞ ¼yy~ðj1ðj_mÞÞ ¼yy~ð½b_m;i_rþb_mh₅⁰Þ ðby ð3ÞÞ

¼ ½yy~b_m;h1i_rþyy~b_mh₅⁰

¼ ½b_m;i_rþ ðb_mÞ Eh₄⁰ ðby ðiÞ;ð11ÞÞ

¼ ½b_m;i_rþb_mh₅⁰ ¼ j1ðj_mÞ;

the assertion (ii) follows form the exact sequence of the pair ðXm;LmÞ. r Lemma 5.3. There exists a homotoy equivalence hPAEðP⁴ðmÞÞ such that h₁ f_m¼ f_mh_P with h_Pj_S3¼i₃.

Proof. Consider the fibration sequence,

P⁴ðmÞ4S^{5 ðfm;b}!^mÞL_mⁱ!KðZ;2Þ:

Since ðfm;bmÞ is a 2-connective covering of Lm, we may assume that the map i:L_m!KðZ;2Þ represents the oriented generator of ½Lm;KðZ;2ÞG H²ðLm;ZÞGZ. Now we define the involution v:Z!Z by vðnÞ ¼ n. It induces a self-homotopy equivalence vv~AEðKðZ;2ÞÞ. Here, because h1j_S2¼ i2, h₁:H²ðLm;ZÞ !^G H²ðLm;ZÞ is given by h₁ðxÞ ¼ x for xAH²ðLm;ZÞG Z. Hence, ~vvi¼ih1 (up to homotopy). So there exists a self-homotopy equivalence ~hh₁AEðP⁴ðmÞ4S⁵Þ such that the diagram

P⁴ðmÞ4S^{5 ðfm;b}!^mÞ Lm ⁱ! KðZ;2Þ

~h h1

??

?y^{F h1}

??

?y^{F vv~}

??

?y^F P⁴ðmÞ4S^{5 ðfm;b}!^mÞ Lm ⁱ! KðZ;2Þ

is homotopy commutative, where horizontal sequences are fibration sequences.

Now we define the map hP:P⁴ðmÞ !P⁴ðmÞ by hP¼pP~hh1iP, where i_P:P⁴ðmÞ !P⁴ðmÞ4S⁵ and p_P:P⁴ðmÞ4S⁵!P⁴ðmÞ denote the natural in- clusion and the natural projection, respectively. By chasing the diagram, we can see h_p AEðP⁴ðmÞÞ and that f_mh_P¼h₁ f_m.

On the other hand, it follows from the diagram (2) that fmj_S3¼h₂. Hence, using h₁j_S2¼ i2 and ði2Þ h₂¼h₂, we can choose the map h_P such

that hPj_S3¼i3. r

Lemma 5.4. If m is an odd integer with m10 ðmod 3Þ, we may choose the homotopy equivalence yy~AEðXmÞ such that

y~

yj_m¼j_mþlm jðfmomÞ for some lmAZ=3:

ð13Þ

Proof. It follows from Lemma 5.2 that there exists a pair ðk;lmÞA Z=mZ=3 such that, yy~j_m¼j_mþk jð½bm;iðh₂ÞÞ þl_mjðf_mo_mÞ.

If we take c¼ykyy~AEðXmÞ, by Proposition 4.5, we have cj_m¼ykyy~j_m

¼yk ðj_mþk jð½bm;iðhÞÞ þlmjðfmomÞÞ

¼y_kj_mþky_k jð½bm;iðh₂ÞÞ þl_my_k jðf_mo_mÞ

¼j_mk jð½bm;iðh₂ÞÞ þk jð½bm;iðh₂ÞÞ þl_mjðf_mo_mÞ

¼j_mþl_mjðf_mo_mÞ:

Then because cj_Lm ¼h₁, we can change the generator c7!yy, and we may~ assume that yy~AEðXmÞ satisfies the equality (13). r

Lemma 5.5. Let mb3 be an integer such that m10 ðmod 3Þ.

( i ) yy~ jð½bm;iðh₂ÞÞ ¼ jð½bm;iðh₂ÞÞ.

(ii) yy~ jðf_mo_mÞ ¼ jðf_mo_mÞ.

Proof. (i) Since yy~ j¼ jh₁ (by (12)), we have y~

y jð½bm;iðh₂ÞÞ ¼ jðh1 ½bm;iðh₂ÞÞ ¼ jð½h1b_m;h₁ih₂Þ

¼ jð½bm;i ði2Þ h₂Þ ðby Lemma 5:1 and ð11ÞÞ

¼ jð½bm;iðh₂ÞÞ ðby ð10ÞÞ:

(ii) Since yy~ j¼ jh1, it su‰ces to prove that h1 fmom¼ fmom. Then it follows from Lemma 5.3 that we have h₁ f_mo_m¼ f_mh_Po_m¼

fm ði3Þ om¼ fmom. r

Lemma 5.6. Let mb3 be an integer such that m10 ðmod 3Þ.

( i ) If lm00AZ=3, ½M0 ¼ ½M1 ¼ ½M1 in M_m⁴.

(ii) If l_m¼0AZ=3, ½M00½M1, ½M10½M1 and ½M10½M0 in M_m⁴. Remark. So we can determine the set M⁴_m completely if we know whether lm¼0 or not. In fact, lm¼0 holds and this will be proved in Theorem 5.8.

Proof. (i) If lm00AZ=3, lm¼G1. Then because y~

yj_m¼j_mGjðfmomÞ ðby ð13ÞÞ; and y~

y ðj_mGjðf_mo_mÞÞ ¼j_mHjðf_mo_mÞ; (

it follows from Lemma 4.1 that we obtain ½M0 ¼ ½M1 ¼ ½M1AM⁴_m. (ii) If l_m¼0, by using (13) and Proposition 4.5, we have

yj_m0Gðj_mGjðfmomÞÞ;

y ððj_mþjðfmomÞÞ0Gðj_mjðfmomÞÞ

for any yAEðXmÞ. Hence, by Lemma 4.1, ½M00½M1, ½M10½M1 and

½M10½M0 in M_m⁴. r

Lemma 5.7. Let mb3 be an odd integer with m10 ðmod 3Þ and let M be an m-twisted CP⁴.

( i ) There is a homotopy equivalence

M=S²FS⁴4S⁶U_ge⁸¼Nðn0Þ for some n0AZ=12;

where g¼i4n4þi6h₆þn0i4EoAp7ðS⁴4S⁶Þ and il:S^l!S⁴4S⁶ ðl¼4;6Þ denotes the corresponding inclusion.

(ii) In this case, P¹:Z=3GH⁴ðM;Z=3Þ !H⁸ðM;Z=3ÞGZ=3 is iso- morphism if n0D0 ðmod 3Þ and it is trivial if n010 ðmod 3Þ.

Proof. (i) Since Sq²:H⁴ðM;Z=2Þ !H⁶ðM;Z=2Þ is trivial by [[6], Proposition 4.1], there is a homotopy equivalence M=S²FS⁴4S⁶U_ge⁸¼N_g for some gAp7ðS⁴4S⁶Þ ¼Zi4n4lZ=12i4EolZ=2i6h₆.

Since NgFN_g, without loss of generalities, we may suppose that g¼ai₄n₄þn₀i₄Eoþei₆h₆ ðab0AZ;n₀AZ=12;eAZ=2Þ: If x2lAH^2lðM;ZÞGZ denotes the corresponding generator (l¼2;4), since M is an m-twisted CP⁴, the equality x₄x₄¼Gx₈ holds. Hence, by the solution of Hopf invariant one problem, we have a¼1. Moreover, it follows from [[6], Lemma 4.2] that Sq²:Z=2¼H⁶ðM;ZÞ !^G H⁸ðM;Z=2Þ ¼Z=2 is an iso- morphism. Then if q:M!M=S²FNg denotes the pinch map, because N_g=S⁴FS⁶U_eh

6e⁸, it follows from the commutative diagram H⁶ðM;Z=2Þ ^q

G H⁶ðNg;Z=2Þ _G H⁶ðS⁶U_eh

6e⁸;Z=2ÞGZ=2

Sq²

??

?y^{G Sq2}

??

?y ^Sq2

??

?y H⁸ðM;Z=2Þ ^q

G H⁸ðNg;Z=2Þ _G H⁸ðS⁶U_eh

6e⁸;Z=2ÞGZ=2 that we obtain e¼1. Therefore, (i) is proved.

(ii) By (i) we may assume that M=S² ¼Nðn0Þ. It follows from the solution of mod 3 Hopf invariant one problem that

P¹:Z=3GH⁴ðNðn0Þ;Z=3Þ !H⁸ðNðn0Þ;Z=3ÞGZ=3

is an isomorphism if n₀D0 ðmod 3Þ, and it is trivial if n₀10 ðmod 3Þ. Then the assertion (ii) follows from the following commutative diagram.

H⁴ðM;Z=3Þ ^P!

1

H⁸ðM;Z=3Þ

q

x?

??^{G q}

x?

??^G

Z=3GH⁴ðNðn0Þ;Z=3Þ ^P1! H⁸ðNðn0Þ;Z=3ÞGZ=3: r

Theorem 5.8. Let mb3 be an odd integer with m10 ðmod 3Þ.

( i ) ½M00½M1, ½M00½M1 and ½M10½M1 in M_m⁴. ( ii ) M_m⁴ ¼ f½M0;½M1;½M1g.

(iii) If we choose the free generator j_mAp7ðXmÞ suitably, P¹:Z=3GH⁴ðMe;Z=3Þ !H⁸ðMe;Z=3ÞGZ=3 is an isomorphism if e¼G1 and it is trivial if e¼0.

Proof. (i) It follows from Lemma 5.7 that there is a homotopy equivalence

M₀=S²FS⁴4S⁶U_ge⁸¼Nðn0Þ for some n₀AZ=12;

where g¼i₄n₄þn₀i₄Eoþi₆h₆Ap₇ðS⁴4S⁶Þ.

Now we recall the definition of fM1;M1;M0g; MG1¼ Mðj_mGjðfmomÞÞ and M0¼Mðj_mÞ. If we consider the induced homo- morphism

q_m⁰ :Z=3o_m¼p₇ðP⁴ðmÞÞ !p₇ðS⁴Þ ¼Zn₄lZ=12Eo;

because q_m⁰ ðomÞ ¼4Eo¼Ea₁ð3Þ (by Lemma 2.1), we may assume that there are homotopy equivalences

M₁=S²FNðk0þ4Þ and M₁=S²FNðk04Þ:

ð14Þ

Because k₀G41k₀G1 ðmod 3Þ, one of N¼ fk0;k₀4;k₀þ4g is zero mod 3 and the other two numbers of N are both non-zero mod 3.

Hence, by Lemma 5.7, there is some e₀Af0;1;1g such that P¹ :Z=3G H⁴ðMe;Z=3Þ !H⁸ðMe;Z=3ÞGZ=3 is trivial if e¼e0 and an isomorphism if eAf0;1;1g and e0e₀.

So ½Me00½Me1 in M_m⁴ if e00e1Af0;1;1g. Then by Lemma 5.6,

½M00½M1, ½M10½M1, ½M10½M0 in M⁴_m.

(ii) The assertion (ii) follows from Proposition 4.8 and (i).

(iii) We note that p₇ðXmÞ ¼Zj_mlZ=3jðf_mo_mÞlZ=m ½bm;iðh₂Þ.

Then if we change the free basej_m byj_m7!j_mþe0jðfmomÞ, the assertion

(iii) is also satisfied. r

Now we can complete the proof of Theorem 1.3.

Proof of Theorem 1.3. The assertion (i) follows from Theorem 4.6, and the assertions (ii), (iii) follow from Theorem 5.8. r

Finally we compute the action of yy~ on p₇ðXmÞ explicitly.

Theorem 5.9. Let mb3 be an odd integer with m10 ðmod 3Þ. Then the left action of yy~on p₇ðXmÞ is determined by the following:

y~

y jð½bm;iðh₂ÞÞ ¼ jð½bm;iðh₂ÞÞ;

y~

y jðfmomÞ ¼ jðfmomÞ; y~

yj_m¼j_m: 8>

<

>:

Proof. It follows from Theorem 5.8 and Lemma 5.6 that l_m¼0.

Hence, the assertion follows from Lemma 5.5 and (13). r Acknowledgments

The author would like to take this opportunity to thank Professor Juno Mukai and Professor Mikiya Masuda for their numerous helpful suggestions and advices concerning the homotopy groups of Moore spaces and the to- pology of twisted complex projective spaces.

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Okayama Univ.

Kohhei Yamaguchi

Department of Information Mathematics University of Electro-Communications

Chofu, Tokyo 182-8585 Japan E-mail: kohhei@im.uec.ac.jp