Homotopy spherical space formsÐa numerical bound for homotopy types

Homotopy spherical space formsÐa numerical bound for homotopy types

Marek GolasinÂski and Daciberg Lima GoncËalves

(Received July 27, 1999) (Revised July 5, 2000)

Abstract. Let G be a ®nite group. We show that for a ®xed nV1 the set of homotopy typesof orbit spacesof all free G-actionson homotopy…2n 1†-spheres is

®nite and bounded by the order of some quotient group associated with G. In particular, we deduce that there are at most two homotopy types of lens spaces de- termined by all freeZ=p^m-actionson homotopy 3-sphereswhenpisan odd prime, and only one homotopy type of those spaces provided that 4ap 1. There isalso only one homotopy type of lens spaces of dimension 2n 1 determined by allZ=2^m-free actions provided that n isodd.

Introduction

A ®nite group hasthe periodic homology if it actsfreely on a ®nite homology sphere [1, Chap. III] and [3, 4]. On the other hand, Swan [10]

showed that any ®nite group with periodic homology of period lactsfreely on a ®nite CW complex of the homotopy type of an …nl 1†-sphere for some positive integer n. ThisCW complex may not have the homotopy type of a closed manifold. The symmetric group S₃ hasa periodic homology but does not act freely on any manifold which hasthe homotopy type of a sphere due to the result of Milnor [8]: Every element of order two must be in the center.

The study of free actions of a group on homotopy spheres is related to the study of their orbit spaces. In the case of the cyclic group Z=2 of order two, the orbit space has the homotopy type of the real projective space and the problem of classifying manifolds of this homotopy type has been studied extensively in [7, 11]. An old result of homotopy type theory (see e.g. [9]) says that, up to homotopy, the set of lens spaces exhausts all the homotopy types of orbit spaces of free actions of the cyclic group Z=mof order m on homotopy …2n 1†-spheres. Thus the classi®cation of all the Z=m-actionson those homotopy spheres is equivalent to the classi®cation of manifolds of the homotopy type of lens spaces studied by Browder in [2]. The set of homotopy

2000 Mathematics Subject Classi®cation. Primary 55M35, 55P15; secondary 57S17.

Key words and phrases: automorphism group, CW complex, freeG-action, homotopy sphere, lens space, orbit space.

types of lens spaces is evidently ®nite and their homotopy classi®cation has been presented in [5, 9]. The main purpose of this paper is to present in Corollary 1.3 a proof of the following result to estimate the number of homotopy types of the orbit spaces.

Let G be a ®nite group with H²ⁿ…G;Z† ˆZ=jGjfor a ®xed nV1,wherejGj is the order of the group G. Then the set of homotopy types of orbit spaces of all free G-actions on all homotopy …2n 1†-spheres is ®nite and bounded by the order of the quotient group Aut…Z=jGj†=fGj;jAAutGg of the automorphism group Aut…Z=jGj†, where j is the induced automorphism on the cohomology H²ⁿ…G;Z† ˆZ=jGj by j in the automorphism group AutG.

In particular, under some assumptions on n and m, we calculate in Corollary 2.3 the number of homotopy typesof lensspacesgiven by freeZ=p^m- actionson homotopy …2n 1†-spheres and we estimate in Corollary 2.5 the number of homotopy typesof orbit spacesgiven by free actionsof the generalized quaternion group Q₂m‡2 on homotopy …4n 1†-spheres.

The paper is divided into two sections. In § 1 we use tom Dieck's theorem [6, p. 126] to prove our main result. Then in Proposition 1.5 we deduce that the group of homotopy classes of homotopy equivalences of the orbit space of a free G-action on a homotopy sphere is isomorphic to the subgroup of AutG which consists of all the automorphisms j with j1 G1 …modjGj† provided that jGj>2.

In § 2 we examine the quotient group Aut…Z=jGj†=fGj;jAAutGg for a cyclic group Gof ®nite order. In particular, there are at most two homotopy typesof lensspacesdetermined by all the free Z=p^m-actionson homotopy 3- spheres when p is an odd prime, and only one homotopy type of those spaces provided that 4ap 1. There is also only one homotopy type of lens spaces of dimension 2n 1, determined by all the Z=2^m-free actionsprovided thatn is odd. We prove that the number of homotopy typesof the orbit spacesof free actionsof the quaternion groupQ₈ on homotopy 3-spheres is bounded by 2.

At the end, we deal with the group of homotopy typesof self-homotopy equivalencesof a lensspace given by a free Z=p-action on a homotopy …2n 1†-sphere, provided that p isa prime.

The authorsare extremely grateful to the referee for carefully reading the original manuscript and his many valuable suggestions. The ®rst author acknowledgesthe SaÄo Paulo University hospitality and the FAPESP-SaÄo Paulo (Brasil) ®nancial support during his visit when the work has been done.

1. Main result

An n-dimensional CW complex Xwith the homotopy type of an n-sphere Sⁿ iscalled a homotopy n-sphere. An orientation for X consists of a choice of

a generator z…X† in the cohomology group Hⁿ…X;Z† ˆZ. Suppose that a

®nite group G of order jGj actsfreely on X. Then by [4], the group G has periodic cohomology groupsof period n‡1 with H^n‡1…G;Z† ˆZ=jGj. Given two oriented homotopy n-spheres X¹ and X² with free G-actions, we can associate to each G-map f :X¹!X² a degree d…f†, de®ned by fz…X²† ˆ d…f†z…X¹†, where f isthe induced map on cohomology groups. Throughout thissection G will be a ®nite group acting freely and cellularly on an odd dimensional homotopy sphere X. Then all the hypotheses of Theorem 4.11 in [6, p. 126] are satis®ed for homotopy spheres and it can be formulated as follows.

Theorem 1.1. Suppose that X¹ and X² are oriented homotopy n-spheres with free cellular actions of a ®nite group G. Then the following holds:

(1) The set ‰X¹;X²Š_G of G-homotopy types of all G-maps X¹!X² is not empty.

(2) Let f :X¹!X² be a G-map. Suppose that d1d…f† …modjGj†. Then there exists a G-map h:X¹!X² such that d…h† ˆd.

(3) Suppose that f0;f1:X¹!X² are G-maps. Then d…f₀†1d…f₁† …modjGj†.

(4) Two G-maps f₀;f₁:X¹!X² are G-homotopic if and only if d…f₀† ˆd…f₁†.

From Theorem 1.1 it followsthat for a G-mapf :X¹ !X² of homotopy …2n 1†-spheres with free G-actionsthere isa G-map g:X²!X¹ with d…gf†

ˆd…g†d…f†11 …modjGj†. Hence the degree d…f†determinesa unit in Z=jGj and then an automorphism of the cyclic group Z=jGj. Since H²ⁿ…G;Z† ˆ Z=jGj, we have the induced map of automorphism groups AutG! Aut…H²ⁿ…G;Z†† ˆAut…Z=jGj†. For an automorphism jof the groupG, letj be itsimage in Aut…Z=jGj†. Observe that j can be identi®ed with a unit of the ring Z=jGj so in the sequel we may also identify j with some positive integer less than the orderjGjof the groupG, which isinvertible in this ring.

Write X_g for a homotopy …2n 1†-sphere X with a free cellular G- action g:GX !X and let X=g be the associated orbit space. By the Serre spectral sequence argument H^k…X=g;Z† ˆH^k…G;Z† for 0Uk<2n 1, H^{2n 1}…X=g;Z† ˆZ and H^k…X=g;Z† ˆ0 for k>2n 1. Consequently, with any map between G-orbit spaces we can also associate its degree. Fix an arbitrary free cellular G-action g₀ on a homotopy …2n 1†-sphere X⁰. Then by Theorem 1.1, there isa G-map fg:Xg!X_g⁰₀.

Theorem 1.2. Two orbit spaces X¹=g₁ and X²=g₂ of homotopy …2n 1†- spheres X¹ and X² have the same homotopy type if and only if d…f_g1†1 Gjd…f_g2† …modjGj†for somej in the group AutG and G-maps f_g1:X_g¹₁!X_g⁰₀ and fg2 :X_g²₂ !X_g⁰₀.

Proof. Suppose that for two orbit spaces X¹=g₁ and X²=g₂ there is an automorphism j of the group G such that d…f_g1†1 Gjd…f_g2† …modjGj†.

Then, in view of Theorem 1.1, there isa G-map f_g⁰₂ :X_g⁰₀!X_g²₂ such that d…f_g⁰₂fg₁† ˆd…f_g⁰₂†d…f_g1†1 Gj …modjGj†.

Let …X¹=g₁†_{2n 1} be the …2n 1†-stage of the Postnikov tower of X¹=g₁ and consider the canonical maps …X¹=g₁†_{2n 1} !K…G;1† to the Eilenberg-MacLane space K…G;1† and X¹=g₁! …X¹=g₁†_{2n 1}. The map K…j ¹;1†:K…G;1† ! K…G;1† determined by the automorphism j ¹ yieldsa map j:…X¹=g₁†_{2n 1}! …X¹=g₁†_{2n 1} with a homotopy commutative square

…X¹=g₁†_{2n 1} ƒƒƒƒ!^j …X¹=g₁†_{2n 1}

??

?y

??

?y

K…G;1† ƒƒƒƒ!^{K…j 1;1†} K…G;1†:

Since the homotopy ®ber of the map X¹=g₁! …X¹=g₁†_{2n 1} is …2n 2†- connected, by obstruction theory the mapj:…X¹=g₁†_{2n 1}! …X¹=g₁†_{2n 1} admits a map F:X¹=g₁!X¹=g₁ such that the diagram

X¹=g₁ ƒƒƒ!^F X¹=g₁

??

?y

??

?y

…X¹=g₁†_{2n 1} ƒƒƒ!^j …X¹=g₁†_{2n 1}

commutesup to homotopy. Ifx₀ isa base point in X¹=g₁ andF~:X¹!X¹ is a cover of the mapF, thenF…gx† ˆ~ j ¹…g†F…x†~ for all g in the groupG andx in the space X¹. Thismeansthat F~:X_g¹0

1!X_g¹₁ isa G-map with d…F†~ 1 …j† ¹ …modjGj†, where g₁⁰ :GX¹!X¹ isthe G-action such that g₁⁰…g;x† ˆ j…g†x for all gAG andxAX¹. For the composite G-map f_g⁰₂f_g1F~:X_g¹0

1!X_g²₂, we get that d…f_g⁰₂fg₁F†~ 1 G1 …modjGj†. Then Theorem 1.1 yieldsa G-map h:X_g¹0

1 !X_g²₂ with d…h† ˆG1 which in view of [9] inducesa homotopy equivalence X¹=g₁!X²=g₂ of the orbit spaces, since the spaces X¹=g₁ and X¹=g₁⁰ are homeomorphic.

Suppose now that f :X¹=g₁!X²=g₂ isa homotopy equivalence and ®x a base point x0 in the orbit space X¹=g₁. Then we get an automorphism jˆ p₁…f† of the group G, determined by the fundamental group functor p₁. Let f~:X¹!X² be a cover of the map f andF~:X¹ !X¹ the map associated to the automorphism j ¹ in the light of the ®rst part of this proof. Then f~…gx†

ˆj…g†f~…x† andF…gx† ˆ~ j ¹…g†F…x†~ for allgAG andxAX¹, so the composite

f~F~:X_g¹₁!X_g²₂ isa G-map with d…f~F†~ 1 G…j ¹† …modjGj†. Finally, in the light of Theorem 1.1 we get that d…f_g1†1 G…j ¹†d…f_g2† …modjGj† and the

proof iscomplete. r

In particular, if two orbit spaces X¹=g₁ and X²=g₂ of homotopy …2n 1†- spheres X¹ andX² are homeomorphic, then d…f_g1†1 Gjd…f_g2† …modjGj† for some jAAutG and G-maps f_g1:X_g¹₁!X_g⁰₀ and f_g2:X_g²₂!X_g⁰₀.

Write K_G^{2n 1} for the set of all homeomorphism classes ‰X=gŠ of orbit spaces X=gof all homotopy …2n 1†-spheres X with respect to all free cellular G-actions. Then the map

D²ⁿ_G ¹:K_G^{2n 1}!Aut…Z=jGj†=fGj;jAAutGg

given by D_G^{2n 1}…‰X=gŠ† ˆ ‰d…f_g†Š, where ‰d…f_g†Šis the class of the automorphism in Aut…Z=jGj† determined by the integer d…f_g†, iswell-de®ned. Here we note that the equality d…f_g† ˆ 1 holdsif f_g isthe identity map of a homotopy …2n 1†-sphere to itself with the opposite orientation. Let K_G^{2n 1}=_F be the quotient set of K_G^{2n 1} by the homotopy relation F. Then we can state

Corollary 1.3. The mapD_G^{2n 1}:K_G^{2n 1}!Aut…Z=jGj†=fGj;jAAutGg induces an injection Dg_G^{2n 1}:K_G^{2n 1}=_F!Aut…Z=jGj†=fGj; jAAutGg. Conse- quently, the setK_G^{2n 1}=Fis ®nite and bounded by the order of the quotient group Aut…Z=jGj†=fGj;jAAutGg.

If G isthe cyclic group Z=m of order m, then by [5, 9] the map Dg²ⁿ_G ¹ is surjective on the homotopy types of the lens spaces. Thus we deduce from Theorem 1.2 one of the classical results of homotopy theory.

Corollary 1.4. Given a free cellular action gof the cyclic group Z=m on a homotopy …2n 1†-sphere X, the orbit space X=g is homotopy equivalent to a lens space.

Let E‰X¹=g₁;X²=g₂Š denote the set of homotopy classes of homotopy equivalences f :X¹=g₁!X²=g₂. Then from the proof of Theorem 1.2, we may deduce the following result which is a generalization of the result in [5, p. 96] stated for cyclic groups only.

Proposition 1.5. If jGj>2 and orbit spaces X¹=g₁, X²=g₂ of homotopy spheres are homotopy equivalent, then there is a bijection from the set E‰X¹=g₁;X²=g₂Š to the subgroup of the group AutG consisting of all auto- morphisms j such that j1 G1 …modjGj†. In particular, the group E‰X=g;

X=gŠis isomorphic to that subgroup of the group AutG, for the orbit space X=g of a homotopy sphere X.

Proof. Let f :X¹=g₁!X²=g₂ be a homotopy equivalence, ®x a base

point inX¹=g₁ and letjˆp₁…f†be the automorphism of the group Ginduced by the fundamental group functor p1. Then d…f† ˆG1 and by the Serre spectral sequence argument j1d…f† ˆG1 …modjGj†. De®ne l…‰fŠ† ˆ p1…f† for ‰fŠAE‰X¹=g₁;X²=g₂Š. We show that l givesthe required bijection from the set E‰X¹=g₁;X²=g₂Š.

Consider two homotopy equivalences f;g:X¹=g₁!X²=g₂ with p1…f† ˆ p₁…g† ˆj. If f~;g~:X¹!X² are coversof f and g, respectively then f~F;~ g~F~:X_g¹₁!X_g²₂ are G-maps, where F~:X¹!X¹ is a map associated with the automorphism j ¹ asin the proof of Theorem 1.2. In view of Theorem 1.1, d…f†…j† ¹1d…f~F†~ 1d…~gF†~ 1d…g†…j† ¹ …modjGj†. Consequently, d…f†1d…g† …modjGj†, hence d…f† ˆd…g† since d…f†;d…g† ˆG1 and jGj>2.

Finally, in the light of [9], the maps f;g:X¹=g₁!X²=g₂ are homotopic.

Let now c be an automorphism of the group G such that c1 G1 …modjGj† and f :X¹=g₁!X²=g₂ be a ®xed homotopy equivalence with jˆ p₁…f†. Then, by the proof of Theorem 1.2, there isa map W:X¹=g₁! X¹=g₁ associated with the automorphism oˆj ¹c and d…W† ˆG1, so it is a homotopy equivalence by [9]. But p₁…fW† ˆc and the proof is

complete. r

Remark 1.6. It iswell-known that the group E‰X=g;X=gŠ isisomorphic to the cyclic group Z=2 provided that jGjU2 for any homotopy sphere X.

In the next section we calculate the group E‰X=g;X=gŠ, when Gisa cyclic group Z=p with p an odd prime and X a homotopy …2n 1†-sphere.

2. Estimation of the number of orbit spaces

Let g2 be a generator of the cohomology groupH²…Z=m;Z† ˆZ=m. The results of [4, Chap. XII, § 11] show that …g₂†ⁿ generates H²ⁿ…Z=m;Z† ˆZ=m.

Then, for any jAAut…Z=m†, the induced automorphism j of the group H²ⁿ…Z=m;Z† ˆZ=m isdetermined by the power kⁿ, for some k with …k;m† ˆ 1, that is kA…Z=m†.

Let now Q4mˆ fx;y;x^mˆy²;xyxˆyg be the generalized quaternion group of order 4mand g₄ a generator of H⁴…Q_4m;Z† ˆZ=4m. Then by [10], the element …g4†ⁿ generates H⁴ⁿ…Q4m;Z† ˆZ=4m, and for any j in the group AutQ_4m the induced automorphism j of the group H⁴ⁿ…Q_4m;Z† ˆZ=4m is determined by the power k²ⁿ, for some k with …k;4m† ˆ1.

The automorphism group Aut…Z=m† isisomorphic to the unit group …Z=m† of the ringZ=m. Therefore, we are led to compute the quotient group …Z=m†=fGxⁿ;xA…Z=m†g. If mˆp₁^k1 p_s^ks isthe prime factorization of an integer mV1 with k_iV1 for iˆ1;. . .;s, then it iswell-known (see e.g. [12, Chapter IV]) that …Z=m†ˆ …Z=p₁^k1† …Z=p_s^ks†. Moreover, …Z=p^l†ˆ

Z=…p 1† Z=p^{l 1} for p an odd prime or l<3 and …Z=2^l†ˆZ=2Z=2^{l 2} forlV3 with generators, multiplication by 1 giving the element of order two, and multiplication by 5 or 3 giving the element of order 2^{l 2}.

In the group Z=…p 1† Z=p^{m 1}, if p isan odd prime, …p 1†=2 isthe only element of order 2, which corresponds to 1 in the group …Z=p^m† by the isomorphism above. Therefore, the equation 1ˆxⁿ hasa solution in the multiplicative group…Z=p^m† if and only if …p 1†=2 isdivisible by the integer n in the additive group Z=…p 1†. Note that …2k‡1†…p 1†=21…p 1†=2 …mod…p 1††. On the other hand 10xⁿ for any element x in the group …Z=2^m† if n is an even integer. Thus, we will get the proposition below and we may concentrate in the sequel on the quotient group …Z=p^m†=fGxⁿ; xA…Z=p^m†g, for positive integers m;n and a prime p.

Proposition 2.1. If mˆp₁^k1 p_s^ks is the prime factorization of an integer mV1, then for any integer nV1 there is an extension

0! …Z=2†^t! …Z=m†=fGxⁿ;xA…Z=m†g

! …Z=p₁^k1†=fGxⁿ;xA…Z=p₁^k1†g …Z=p_s^ps†=fGxⁿ;xA…Z=p_s^ks†g

!0;

where t is determined as follows:

(1) If 1Afxⁿ;xAZ=p_i^kig for any iˆ1;. . .;s, then tˆ0;

(2) Otherwise tˆs 1 afi; 1Afxⁿ;xAZ=p_i^kigg. In particular, when n is an odd integer, 1Afxⁿ;xAZ=p_i^kig for any iˆ1;. . .;s and hence tˆ0;

when n is an even integer, t‡1 is the number of odd primes p_i which appears in the prime factorization of m and …p_i 1†=2 is not divisible by n in the group Z=…p_i 1†.

Proof. Of course, we have the extension

0! fGxⁿ;xA…Z=p₁^k1†g fGxⁿ;xA…Z=p_s^ks†g=fGxⁿ;xA…Z=m†g

! …Z=m†=fGxⁿ;xA…Z=m†g ! …Z=p₁^k1†=fGxⁿ;xA…Z=p₁^k1†g …Z=p_s^ks†=fGxⁿ;xA…Z=p_s^ks†g !0:

The square of every element in fGxⁿ;xA…Z=p₁^k1†g fGxⁿ;xA …Z=p_s^ks†g liesin fxⁿ;xA…Z=m†g, because …Gxⁿ†²ˆ …x²†ⁿ. So, it follows that the group

fGxⁿ;xA…Z=p₁^k1†g fGxⁿ;xA…Z=p_s^ks†g=fGxⁿ;xA…Z=m†g is isomorphic to a direct sum of Z=2's. Certainly, tˆs 1 in the case (1).

In the case (2), 1Afxⁿ;xA…Z=m†g; we can replace Gxⁿ withxⁿ for…Z=m†

and for …Z=p_i^ki† with 1Afxⁿ;xA…Z=p_i^ki†g. Then it isnot hard to see the

result. r

Of course, we can assume that nˆq₁^t1 q_l^tl and p 1ˆq₁^u1 q_l^ul, where q1;. . .;ql are di¨erent primes, t1;. . .;tl;u1;. . .;ul are non-negative integersand t_i or u_i ispositive for all iˆ1;. . .;l.

Proposition 2.2. Let m;n be positive integers, p a prime and q1;. . .;ql

primes from the factorization above. Then the quotient group …Z=p^m†= fxⁿ;xA…Z=p^m†g is isomorphic to one of the following groups:

(1) Z=q^min…t₁ ^1;u1† Z=q^min…t_l ^l;ul†, if p02 and p0q₁;. . .;q_l;

(2) Z=q^min…t₁ ^1;u1† Z=q^min…t_{i0 1}^{i0 1;ui0 1†}Z=p^{min…m 1;ti0†}Z=q^min…t_i0‡1^{i0‡1;ui0‡1†} Z=q^min…t_l ^l;ul†, if p02 and pˆq_i0 for some 1Ui₀Ul;

(3) the trivial group E, if pˆ2 and n is odd;

(4) Z=2Z=2^{min…ti0;m 2†}, if pˆ2 and qi0 ˆ2 for some 1Ui0Ul.

Proof. Let A be a ®nite additive abelian group and n a positive integer.

Then, for the endomorphism n:A!A given by the multiplication by n, the quotient group A=Imn isisomorphic to the kernel Kern, where Imn is the image of n. We consider the endomorphism of the multiplicative group …Z=p^m†given by taking then-th power. But the group…Z=p^m†isisomorphic to the additive group Z=…p 1† Z=p^{m 1}, for p an odd prime or m<3 and to the group Z=2Z=2^{m 2} for pˆ2 and mV3. Thus the result follows.

Remark that ui0 ˆ0 in (ii). r

Consequently, the following holds.

Corollary 2.3. Let m;n and p be as in Proposition 2.2 and let Tp…m;n†

denote the number of homotopy types of lens spaces given by free Z=p^m-actions on homotopy …2n 1†-spheres.

(1) If p is an odd prime and …p 1†=2 is divisible by n in the group Z=…p 1†, or pˆ2 and n is an odd integer, then T_p…m;n† is equal to:

( i ) q^min…t₁ ^1;u1† q^min…t_l ^l;ul†, if p02 and p0q1;. . .;ql;

( ii ) q^min…t₁ ^1;u1† q^min…t_{i0 1}^{i0 1;ui0 1†}p^{min…m 1;ti0†}q^min…t_i0‡1^{i0‡1;ui0‡1†} q^min…t_l ^l;ul†, if p02 and pˆq_i0 for some 1Ui₀Ul;

(iii) 1, if pˆ2.

(2) Otherwise Tp…m;n† is equal to:

( i ) 2 ¹q^min…t₁ ^1;u1† q^min…t_l ^l;ul†, if p02 and p0q₁;. . .;q_l;

( ii ) 2 ¹q^min…t₁ ^1;u1† q^min…t_{i0 1}^{i0 1;ui0 1†}p^{min…m 1;ti0†}q^min…t_i0‡1^{i0‡1;uio‡1†} q^min…t_l ^l;ul†, if p02 and pˆqi0 for some 1Ui0Ul;

(iii) 2^{min…ti0;m 2†}, if pˆ2, mV3 and q_i0ˆ2;

(iv) 1, if pˆ2, mU2 and qi0ˆ2.

In particular, we get

Corollary 2.4. Let p be a prime and m a positive integer. Then the number of homotopy types of lens spaces given by free Z=p^m-actions on a homotopy 3-sphere is equal to:

(1) 2, if either p 1 is divisible by 4 or pˆ2 and mV2;

(2) 1, if either p 1 is divisible by2 but not divisible by 4 or pˆ2 and mˆ1.

Moreover, for the generalized quaternion group Q₂^m‡2 one can deduce Corollary 2.5. The number of homotopy types of orbit spaces given by free Q₂^m‡2-actions on homotopy …4n 1†-spheres is bounded by 2 for any m>0, provided that n is an odd positive integer.

Proof. In the light of [10, Proposition 8.1], for any j in the group AutQ₂^m‡2, the induced automorphism j of H⁴ⁿ…Q₂^m‡2;Z† ˆZ=2^m‡2 isde- termined by the power k²ⁿ for some k with …k;2^m‡2† ˆ1. The result follows

from the same argument as for Corollary 2.3. r

Finally, let L be a lensspace determined by a Z=p-free action on a homotopy …2n 1†-sphere with p an odd prime. Then, by Proposition 1.5, the group E‰L;LŠ of homotopy typesof all itsself-homotopy equivalencesis isomorphic to the subgroup of…Z=p† consisting of all elementskwithkⁿ1 G1 …modp†. But …Z=p†ˆZ=…p 1†, so the corresponding subgroup of Z=

…p 1† consisting of all elements l with nlˆ0 or nlˆ …p 1†=2. If d ˆ …n;p 1† is the greatest common divisor of the integers n and p 1, then the subgroup Gd of Z=…p 1† generated by …p 1†=d consists of all elements l in the group Z=…p 1† with nlˆ0. Moreover, if there isa solution l₀ in the group Z=…p 1† of the equation nlˆ …p 1†=2, then the set of all solution of that equation in the group Z=…p 1† isequal to the coset l₀‡G_d. We can conclude the paper with the following result deduced from the consideration above.

Proposition2.6. Let L be a lens space determined by a freeZ=p-action on a homotopy …2n 1†-sphere with an odd prime p. Then the group E‰L;LŠ of homotopy classes of all its self-homotopy equivalences is isomorphic either to the subgroup G_d of the cyclic group Z=…p 1† or to the subgroup G_dU…l₀‡G_d†, provided that there is a solution l₀ of the equation nlˆ …p 1†=d in the group Z=…p 1†.

References

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Marek GolasinÂski

Faculty of Mathematics and Informatics Nicholas Copernicus University Chopina 12/18, 87-100 TorunÂ, Poland

e-mail: [email protected] Daciberg Lima GoncËalves Department of Mathematics-IME

University of SaÄo Paulo

Caixa Postal 66.281-AG. Cidade de SaÄo Paulo 05315-970 SaÄo Paulo, Brasil

e-mail: [email protected]