HOMOTOPY CLASSES OF SELF-MAPS AND INDUCED HOMOMORPHISMS OF HOMOTOPY GROUPS MARTIN ARKOWITZ, HIDEAKI ~OSHIMA AND JEFFREY STROM Abstract.For a based space X, we consider the group E#n(X) of all self ho* *motopy classes ff of X such that ff# = id: ssi(X) ! ssi(X), for all i n, where* * n 1, and the group E (X) of all ff such that ff = id. Analogously, we study the semig* *roups Z#n(X) and Z (X) defined by replacing `id' by `0' above. There is a chain of con* *tainments of the E-groups and the Z-semigroups, and we discuss examples for which the * *containment is proper. We then obtain various conditions on X which ensure that the E* *-groups and the Z-semigroups are equal. When X is a group-like space, we derive lower* * bounds for the order of these groups and their localizations. In the last section we* * make specific calculations for the E-groups and Z-groups of certain low dimensional Lie* * groups. 1.Introduction Let X and Y be topological spaces with base point. A major objective of homotop* *y theory is to investigate and understand the set [X, Y ] of homotopy classes of based m* *aps from X to Y . Typically, one restricts the spaces in order to put more structure on the s* *ets [X, Y ]. In this paper we consider the case X = Y , so that there is a binary operation in * *[X, X] obtained from composing homotopy classes. We study certain subgroups and subsemigroups o* *f [X, X]. More specifically, we consider the monoid [X, X] and its group of units E(X). T* *hen E(X) is the group of homotopy classes of homotopy equivalences X ! X. Define subgroups * *E (X) and E#n(X) of E(X) by E (X) = {ff 2 E(X), ff = id} and E#n(X) = {ff 2 E(X), ff# = id: ssi(X) ! ssi(X), for all i },n where is the loop-space functor, ff# is the induced homomorphism of homotopy * *groups and idis the identity homomorphism. Furthermore, we allow n = 1, that is, E#1 (X) = {ff 2 E(X), ff# = id: ssi(X) ! ssi(X), for all}i. 1 2 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom In addition, if X is a CW-complex of dimension n, define the subgroup E# (X) of* * E(X) by E# (X) = E#n(X). Then there is a chain of subgroups of E(X): (1.1) E (X) E#1 (X) E# (X). We refer to them collectively as E-groups. They have been studied extensively (* *for some of the references, see [AC , AL1, AL2, AM , AS1, AS3, D, DZ, FM , M1, M4, M5, O3, * *P, Ts]). We also define subsets of [X, Y ] by Z (X, Y )= {ff 2 [X, Y ], ff = 0} and Z#n(X, Y )= {ff 2 [X, Y ], ff# = 0 : ssi(X) ! ssi(Y ) for all i}, n where n 1. When X = Y we obtain subsemigroups of [X, X] by setting Z (X) = Z (X, X) and Z#n(X) = Z#n(X, X). When X is a CW-complex of dimension n, we defi* *ne Z# (X) = Z#n(X). Then we have a chain of subsemigroups of [X, X]: (1.2) Z (X) Z#1 (X) Z# (X). These semigroups have also been widely studied [AMS , AS1, AS2, M1, M2, M3]. We* * refer to them collectively as Z-semigroups. It is natural to ask if there are spaces X for which containments in (1.1) an* *d (1.2) are proper. Several known results show that three of the four inclusions can be pro* *per, and we complete the answer to this question by giving an example in Proposition 2.1 wh* *ich shows that the fourth inclusion can be proper. The full result is stated as Propositi* *on 2.1. Most of the spaces which serve as examples are finite complexes. However, the only know* *n space X for which E (X) ( E#1 (X) is an infinite-dimensional complex. The obvious analogy b* *etween the Z-groups and the corresponding E-groups and the fact that there is a finite* * complex X with Z (X) ( Z#1 (X) have led us to make the following conjecture, which also a* *ppears in [P, p. 680]. Conjecture 1.1. There is a finite-dimensional CW-complex X such that E (X) ( E#* *1 (X). When the space X is group-like, this analogy is more precise, for there is a * *bijection between the Z-groups and the corresponding E-groups (Proposition 3.1). On the other han* *d, finite- dimensional group-like spaces are strongly related to products of odd-dimension* *al spheres, for which E = E#1 (Corollary 2.8). Thus we make the following additional conjectu* *re, which is supported by the calculations of x4. Homotopy Classes of Self-Maps 3 Conjecture 1.2. If X is a finite-dimensional group-like space, then E (X) = E#1* * (X). After discussing the examples mentioned above, we consider in x2 the general * *problem of determining when the E-groups are all equal and when the Z-semigroups are al* *l equal. We first present an alternate characterization of E (X) and Z (X). Our main re* *sult in x2 (Theorem 2.13) is that the E-groups of X are equal and the Z-semigroups are * *equal if X is a product of spheres and projective spaces. We then consider localization* * and show that the E-groups and Z-semigroups are equal for a rational space and for a gro* *up-like space localized at a regular prime. In x3 we obtain lower bounds for the order of the* * localizations of the E-groups and Z-groups of a group-like space. We further specialize to Li* *e groups in x4 and give conditions which are equivalent to the triviality of an E-group or * *a Z-group. We conclude the paper by explicitly calculating these groups for low-dimensional L* *ie groups such as U(2), S1 x SO(3), SU(3), Sp(2), S3 x SO(3) and SO(4). We end this section by describing our notation and assumptions. Each space i* *s to be connected, based and have the homotopy type of a based CW-complex. All maps and* * homo- topies are to preserve the base point, which is denoted *. We do not distinguis* *h notationally between a map and its homotopy class. A nilpotent space is one such that the fu* *ndamental group is nilpotent and which acts nilpotently on the higher homotopy groups [HM* *R , p. 62]. The identity map of X is denoted idXor simply idand the constant map is 0 : X !* * Y . A space X is a co-H-space if there is a map OE : X ! X _ X (the wedge of X with i* *tself) whose composition with each of the two projections X _X ! X is idX. If f : X ! Y and * *g : Y ! X are such that gf = idXthen X is a retract of Y ; g is called a retraction of f * *and f is called a section of g. A space is group-like if it satisfies all the axioms of a group u* *p to homotopy [W , p. 118]. A map f : X ! Y induces functions f* : [A, X] ! [A, Y ] and f* : [Y, B* *] ! [X, B] by composition, for all A and B. The homomorphism of homotopy groups ssn(X) ! s* *sn(Y ) induced by f is denoted f# or f#n. The standard notation of homotopy theory wil* *l be used: ` ' for same homotopy type, ` ' for (reduced) suspension, ` ' for loop-space, `* *_' for wedge and `^' for smashed product. The natural isomorphism between [ X, Y ] and [X, * *Y ] is called the adjoint isomorphism. Finally nilG denotes the nilpotency (class) of the gro* *up G, and `~=' denotes isomorphism of groups or Lie groups. Acknowledgment The last two authors would like to express their appreciation t* *o the Department of Mathematics at Dartmouth College for its hospitality during July * *2004. 4 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom 2.Equality of the E-groups and Z-semigroups We begin this section by discussing CW-complexes X for which the inequalities i* *n (1.1) and (1.2) are strict containments. We then turn our attention to proving results wh* *ich guarantee that the containments of (1.1) and (1.2) are actually equalities. The bulk of t* *his work is done in subsection 2.2, where we are concerned with results that are valid for gener* *al spaces. These results can be localized, and in the third subsection we study in more detail s* *paces that have been localized; of particular interest here are some special properties of grou* *p-like spaces. 2.1. Proper Containment. Our first result gives examples showing that the conta* *inments in (1.1) and (1.2) can be proper. All of these except (4) are already known. Proposition 2.1. (1)There is an infinite-dimensional CW-complex X such that E (X) ( E#1 (X). (2)There is a finite complex X such that E#1 (X) ( E# (X). (3)There is a finite complex X such that Z (X) ( Z#1 (X). (4)There is a finite complex X such that Z#1 (X) ( Z# (X). All of these spaces can be chosen to be simply-connected. Proof.For (1), the example can be found in [FM ]; an example for (2) is given i* *n [AM , Cor. 4.13 or Prop. 6.3]; and examples for (3) can be found in [AS1] or [AS2, p. 395]. We now turn our attention to part (4). Let 2 : Sn-1 ! Sn-1 be the map of degr* *ee 2, M = Sn-1 [2 en a Moore space of type (Z2, n - 1) and X = Sn _ M, where n > 3. L* *et q : M ! Sn collapse Sn-1 to a point. Let p2 : X ! M be the projection and i1 : * *Sn ! X the injection, and set ff = i1qp2 : X ! X. We will show that ff#r = 0 for all r* * n, and that ff#n+1 6= 0. For the first assertion, it suffices to show that q#r = 0 : ssr(M) ! ssr(Sn) * *for r n. For r < n, this is obvious because ssr(Sn) = 0. For r = n, we have ssn(M) = Z2 by t* *he Hurewicz theorem, so q#n is a homomorphism Z2 ! Z, which must also be trivial. We show ff#n+1 6= 0 by showing q#n+1 6= 0. The Blakers-Massey Theorem [W , p.* * 368], applied to the cofiber sequence Sn-1 i!M q!Sn-1 yields the exact sequence q#n+1 2#n ssn+1(M)____//_ssn+1(Sn) ~=ssn(Sn-1)//_ssn(Sn-1) Since ssn(Sn-1) = Z2, and 2#n is multiplication by 2, we have Ker 2#n = ssn(Sn-* *1). Thus q#n+1 is onto Z2, and hence is non-zero. Therefore ff =2Z#1 (X). Homotopy Classes of Self-Maps 5 2.2. General Results. In this subsection, we find conditions that are sufficien* *t to guarantee that the inclusions in (1.1) and (1.2) are equalities. These results will be ap* *plied to products of spheres and projective spaces and will play a role in the calculations of x4. We begin with a simple, but useful, observation. Remark 2.2. Let OE : W ! X, where W is a wedge of spheres with dim(W) N. Then (1)if f 2 E#N (X) then f O OE = OE and (2)if f 2 Z#N (X) then f O OE = 0. Now we give an alternate characterization of E and Z . Proposition 2.3. For any two spaces X and Y , E (X) = {ff 2 E(X), ff* = id: [ A, X] ! [ A, X] for every space}A and Z (X, Y )= {ff 2 [X, Y ], ff* = 0 : [ A, X] ! [ A, Y ] for every space}A. Proof.We only prove the statement for E , since the proof for Z is analogous. * * Let ff 2 E (X) and fi 2 [ A, X]. Let ^fi: A ! X be the adjoint of fi. Then ( ff) O ^fi=* * ^fi, since ff = id. Taking the adjoint gives ff O fi = fi. Next let ff 2 E(X) be such that ff* = id: [ A, X] ! [ A, X] for every space A* *. Let A = X and let p : X ! X be the canonical map. Then ff O p = p. By taking adj* *oints we obtain ff = id X. Using Proposition 2.3, we can give short proofs (and some easy generalization* *s) of some results of Pave~si'c on E which were originally proved by spectral sequence ar* *guments [P]. The first of these concerns co-H-spaces. The second one, Corollary 2.8, concern* *s products of spheres. The third one deals with rational spaces, and appears below in Proposi* *tion 2.17. Corollary 2.4. [P, Cor. 3.1] If X is a co-H-space, then E (X) = {id} and Z (X) * *= {0}. Proof.We will just prove the result for E ; the proof for Z is analogous. Sin* *ce X is a co-H-space, X is a retract of some suspension A [Hi, p. 209]. Therefore there* * are maps i : X ! A and r : A ! X such that r O i = idX. If f 2 E (X), then f O r = f*(* *r) = r by Proposition 2.3. Applying i to both sides, we get f = idX. Remark 2.5. One possible approach to Conjecture 1.1 would be to find a finite c* *o-H-space X such that E#1 (X) 6= {id} (cf. [P, p. 680]). 6 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom Next we consider when E# = E and Z# = Z for certain products of spaces. For* * this we will make use of the following lemma, which may be of independent interest. Lemma 2.6. Let f, g : A1x . .x.Ar ! X be two maps, let j : A1_ . ._.Ar ,! A1x .* * .x.Ar be the canonical inclusion and consider the conditions (1)f O j = g O j and (2) f = g. Then (1) implies (2). Furthermore, if each Aiis a co-H-space, then (2) implies * *(1). Proof.Assume (1) and let V = A1 _ . ._.An and P = A1 x . .x.An. Consider the h* *o- momorphism j* : [ P, V ] ! [ P, P] and the canonical map p 2 [ P, P]. Since* * j* is easily seen to be onto, there is ` 2 [ P, V ] such that j O ` = p. By taking a* *djoints we get j O ^`= ^p= id P. From hypothesis (1) we have ( f) O ( j) = ( g) O ( j), so f = f O id P= ( f O j) O ^`= ( g O j) O ^`= g. Thus (2) holds. Now assume that each Aiis a co-H-space and that f = g. We claim that f|Ai =* * g|Ai for each i. For this we use the commutative diagram ji f= g JAiJ__________//_ P__________//_ X ______ _|_______________|_______________|__ s _|_______________|_______ |p _fflffl|_ji______fflffl|_____f, gfflffl| Ai _____________//P_____________//X where jiis the inclusion of the ithfactor. The section s exists because Aiis a * *co-H-space [Hi, p. 209]. Now f O ji= p O f O jiO s = p O g O jiO s = g O ji, which proves (1). Lemma 2.6 has several corollaries. The first is well-known. Corollary 2.7. If q : A1x . .x.An ! A1^ . .^.An is the projection, then q = 0. Proof.Since q O j = 0 = 0 O j, Lemma 2.6 shows that q ' (0) = 0. Pave~si'c [P, Thm. 3.6] has proved E (Sk x Sl) = E# (Sk x Sl). Using Lemma 2.* *6 we can easily generalize this to arbitrary finite products of spheres. Homotopy Classes of Self-Maps 7 Corollary 2.8. Let X = Sk1x . .x.Skr and write N = max{k1, . .,.kr}. Then E (X) = E#1 (X) = E# (X) = E#N (X) and Z (X) = Z#1 (X) = Z# (X) = Z#N (X). Proof.We will only prove E (X) = E#N (X). Let j : Sk1_ . ._.Skr,! X. If f 2 E#N* * (X), then f O j = j = idO j by Remark 2.2. Therefore Lemma 2.6 shows that f ' (id)* * = id. In order to further generalize Corollary 2.8, we give a definition. Definition 2.9. For a space X, we say ss*(X) is spherically generated in dimens* *ions N if there is a wedge of spheres W with dim(W) N and a map OE : W ! X such * *that OE# : ssk(W) ! ssk(X) is surjective for all k. Proposition 2.10. Let X and Y be spaces. Assume that ss*(X) is spherically gene* *rated in dimensions N and that ss*(Y ) is spherically generated in dimensions M. Then (1)E#1 (X) = E#N (X) and Z#1 (X) = Z#N (X), (2)ss*(X x Y ) is spherically generated in dimensions max(M, N) and (3)ss*(Sn) is spherically generated in dimensions n, and ss*(FPn-1) is sph* *erically generated in dimensions nd - 1, where d = 1, 2 or 4 according as F = R,* * C or H. Proof.We let f 2 E#N (X) and ff 2 ssn(X) for some n and show that f O ff = ff. * *Since ss*(X) is spherically generated in dimensions N, there is an wedge of spheres W of d* *imension N and a map OE : W ! X such that OE#i is surjective for all i. Thus ff = OE# * *(fi) for some fi 2 ssn(W). Using Remark 2.2, we have f O OE = OE because dim(W) N. Therefore f# (ff) = f# (OE# (fi)) = (f O OE)# (fi) = OE# (fi) = ff, which completes the proof of the first assertion of (1). The proof of the secon* *d assertion is similar. For (2), observe that if W and V are wedges of spheres and W ! X and V ! Y ar* *e maps which induce surjections on homotopy groups, then the composite W _V ! W xV ! X* * xY is also surjective on homotopy groups. In (3), only the statements about the projective spaces require proof. We beg* *in with the fibration sequence s___________________________________________* *____ _______________________________________________* *_____________________________________________________________________ p _____________________________0p . ._.0_//_ Snd-1___//_ FPn-1___//_Sd-1___//_Snd-1___//FPn-1. 8 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom where s is a section of FPn-1 ! Sd-1. Then we have the well-known homotopy equ* *ivalence Snd-1x Sd-1 FPn-1 determined by p and s [G ], and so ssk(FPn-1) ~=ssk-1( FPn-1) ~=Im( p# ) Im(s# ). Now let j : Sd ,! FPn-1 be the inclusion of the lowest dimensional cell. Since * *j#d, and hence ( j)#d-1, is surjective, there is a lift ~ in the diagram m66Sd m m m ~mm ||j m m fflffl| Sd-1_____s_____// FPn-1. From this it follows that Im(s# ) Im( j# ), and so ssk-1( FPn-1) ~=Im( p# ) +* * Im( j# ) (the sum may not be direct). Therefore ssk(FPn-1) ~=Im(p# ) + Im(j# ), and henc* *e the map (j, p) : Sd _ Snd-1! FPn-1 determined by j and p is surjective on all homotopy * *groups. Remark 2.11. There are other interesting spaces X for which ss*(X) is spherical* *ly generated in dimensions N for some N. One important case occurs when X is the orbit spa* *ce Sn=G of a free action of a finite group G on Sn, (n 1). Examples of such actions c* *an be found in [Wo , Chap. 6]. Another important case consists of spaces of the form S4n+3=* *N(S1), where N(S1) is the normalizer of S1 in S3 and the action is the restriction of the st* *andard one of S3 on S4n+3[Br, Chap. III]. Now we turn our attention to analogous properties of the following collection* * of spaces: A = {X | X a finite or countably infinite wedge}of.spheres Proposition 2.12. Let X and Y be spaces such that X, Y 2 A. Then (1)Z (X) = Z#1 (X) and E (X) = E#1 (X) (cf. [AS1, Prop. 5.1]), (2)X x Y 2 A and (3)Sn, FPn 2 A for all n 1 and F = R, C or H. W Proof.(1) Suppose ~ : Sni ! X is a homotopy equivalence, and let f 2 Z#1 (X* *). Write p : X ! X for the canonical map. Then f O p O ~ = 0, and so f O p = 0. * *By taking adjoints we find that f = 0. The second assertion of (1) is similarly proved. (2) Clearly (X x Y ) = (X) x (Y ). For any spaces A and B, we have (A x B) A _ B _ (A ^ B) by [Hi, 11.10] and so ( (X) x (Y )) (X) _ (Y ) _ ( (X) ^ (Y )). Homotopy Classes of Self-Maps 9 Then (2) follows. (3) Since Sn has the homotopy type of a wedge of spheres according to [J], * *Sn 2 A. Also FPn Sd-1x S(n+1)d-1, where d = 1, 2 or 4, according as F = R, C or H, * *as noted earlier. The result follows from the decomposition for (A x B) used in (2). We now put all of these results together. Theorem 2.13. Let X = X1xX2x. .x.Xr where each Xiis either a sphere or a projec* *tive space FPn with F = R, C or H and r, n 1. Let N = 3 + max{dim(Xi)}. Then E (X) = E#1 (X) = E#N (X) and Z (X) = Z#1 (X) = Z#N (X). Proof.Since each Xi is in A by Proposition 2.12(3), so is the product X by Prop* *osition 2.12(2). Now Proposition 2.12(1) shows that E (X) = E#1 (X) and Z (X) = Z#1 (X). Analogously, each ss*(Xi) is spherically generated in dimensions dim(Xi) + * *3 by Propo- sition 2.10(3), so ss*(X) is spherically generated in dimensions N by Proposi* *tion 2.10(2). Hence Proposition 2.10(1) applies to show that E#1 (X) = E#N (X) and Z#1 (X) = * *Z#N (X). Remark 2.14. Generally, N dim(X), where N the number in the previous theorem.* * This is the case, for example, if either there are at least 4 spaces in the product,* * or if there are at least 2 factors with dimension at least 3. Furthermore, the term `3 + max{d* *im(Xi)}' can be replaced with `1 + max{dim(Xi)}' if no Xi is equal to HPn; it can be rep* *laced with `0 + max{dim(Xi)}' if none of the factors is HPn or CPn. The latter special case of this remark will be used in the following sections* *, so we state it as a separate corollary. Corollary 2.15. If each Xi is either a sphere or a real projective space, then * *the product X = X1x . .x.Xr (r 1) satisfies E (X) = E#1 (X) = E# (X) and Z (X) = Z#1 (X) = Z# (X). 10 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom 2.3. Localized Spaces. We now turn to results that are specific to p-local spac* *es or to rational spaces (for details on localization, see [HMR ]). We write A ! A(0)fo* *r rationalization of groups or spaces and A ! A(p)for localization of groups or spaces at a prime* * p. To begin, we observe that many of the above results are true p-locally. Remark 2.16. The proofs of all of the results of x2.2 are valid for p-local spa* *ces, where p is either a prime number or zero. Proposition 2.17. (Cf. [P, Cor. 3.2]) If X is the rationalization of a finite n* *ilpotent CW- complex, then E (X) = E#1 (X) and Z (X) = Z#1 (X). Proof.In fact, since X is the suspension of a rational space, it is homotopy * *equivalent to a wedge of rational spheres [He, p. 167]. Therefore, we apply the rational versio* *n of Proposition 2.12, and the proposition is proved. We next consider group-like spaces, which enjoy special localization properti* *es. In par- ticular, since group-like spaces are nilpotent, they can be localized. Furtherm* *ore, if a finite complex X is group-like, then for any sufficiently large prime number p, X(p) * *Sk1(p)x. .x.Skr(p) [K , p. 73]. Such primes are called regular for X. Corollary 2.18. Let X be a finite group-like complex of dimension N and p a reg* *ular prime for X. Then E (X(p)) = E#N (X(p)) and Z (X(p)) = Z#N (X(p)). Proof.If p is regular, then X(p) Sk1(p)x . .x.Skr(p). Now apply (the p-local * *version of) Corollary 2.8. We conclude this section with a general discussion of localization and the E-* *groups and Z-semigroups. Let p denote either 0 or a prime number and let X be a finite ni* *lpotent CW-complex of dimension N. Then the groups E# (X) and E#1 (X) are nilpotent [DZ* * ], so they may be localized. On the other hand, localization of spaces defines a homo* *morphism E# (X) ! E#N (X(p)). It is known (see [M4 ] and [M1 , Thm. 2.7]) that these hom* *omorphisms are in fact p-localization homomorphisms. Furthermore, if X is an H0-space (i.e* *., X(0)is an H-space), then E#1 (X) ! E#1 (X(p)) is also p-localization [M1 , Cor. 2.10]. If, in addition, X is a group-like space, then the sets Z (X), Z#1 (X) and Z#* * (X) have a nilpotent group structure obtained from the additive nilpotent group [X, X] [W * *, p. 464], and so they may be localized. It has been proved that the natural maps Z# (X) ! Z#N (X(p)) and Z#1 (X) ! Z#1 (X(p)) Homotopy Classes of Self-Maps 11 are also p-localization homomorphisms [M1 , Lem. 1.6 and Cor. 1.7]. This discussion suggests the following questions. Question 2.19. Let X be a finite nilpotent complex, and let p be a prime number* * or zero. (1)Is the natural map E (X) ! E (X(p)) p-localization? (2)If X is also group-like, is the natural map Z (X) ! Z (X(p)) p-localizati* *on? 3.Grouplike Spaces In this section we consider group-like spaces in more detail. In the first sub* *section, we establish a close link between the Z-groups and the E-groups for group-like spa* *ces. We then use commutator subgroups to give lower bounds for the order of some of these gr* *oups and their localizations. Throughout this section, X denotes a finite group-like complex and so X is a * *nilpotent space and the additive group [X, X] is nilpotent. 3.1. Z and E for Grouplike Spaces. As was mentioned in the introduction, there * *is a strong analogy between the E-groups and the Z-groups when X is group-like. In t* *his case, : [X, X] ! [ X, X] is a homomorphism of groups. Thus Z (X) = -1(0) = Ker i* *s a subgroup of [X, X] and E (X) = -1(id) is a coset of Ker of the group [X, X]. * *Similarly, we obtain homomorphisms M rn : [X, X] ! Hom (ssk(X), ssk(X)) k n and we deduce that Z#n(X) = ker(rn) is a subgroup and E#n(X) is a coset of Z#n(* *X). Furthermore, since rm factors through rn for m < n and rn factors through for* * all n 1, we have the following proposition. Proposition 3.1. If X is group-like, then the function ff 7! id+ ff defines bij* *ections and n making the diagram " O " Z (X) O_________//_Z#n(X)__________//Z#m (X) || n || ||m fflffl|" fflffl|" fflffl| E (X) O_________//_E#n(X)O_________//E#m (X) commutative for each m n 1. 12 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom Remark 3.2. Proposition 3.1 has been observed by several people (e.g., [M1 , p.* * 51] and [AS3, p. 693]). It follows from Theorem 4.3(2) that : Z (X) ! E (X) is not n* *ecessarily a homomorphism. 3.2. Grouplike Spaces and Commutators. Commutators feature prominently in this * *sec- tion, so we adopt the notation H(X) = [X, X] in order to avoid possible confusi* *on of [X, X] with the commutator subgroup of X, when X is a topological group. We begin by establishing our notation for commutators. If is a group and if* * A, B , then we write [A, B] for the subgroup of generated by all commutators [a, b] * *= a-1b-1ab with a 2 A and b 2 B. The lower central series of is the sequence of subgroups = (1) (2) . . . (k) (k+1) . . . defined by setting (1)= and (i)= [ , (i-1)] for i 2. If k is the smalles* *t integer such that (k+1)= {id} for some k, then is nilpotent with nilpotency k, written ni* *l = k. In what follows, we write |S| to denote the number of elements of a set S. Lemma 3.3. Let be a nilpotent group with nil = k, and let p be a prime numbe* *r or 0. Then (1)| (2)| 2k-1; (2)If is a p-local group, then | (2)| pk-1; (3)If is a finitely-generated infinite nilpotent group, then (p)is infini* *te. Proof.(1) Since the case k = 1 is trivial, we assume k 2. Then we have (1)% (2)% . .%. (k)% (k+1)= 1. By [HMR , Cor. 2.6 and Thm. 2.7], the groups (i)= (i+1)are nontrivial for 2 * * i k. It follows that | (i)= (i+1)| 2 for 2 i k, and so | (2)| 2k-1. Assertion (2) follows from the same argument because the nontrivial groups (* *i)= (i+1) are p-local and so they must have at least p elements each. We prove (3) by induction on the nilpotency of . If nil = 1, then is abel* *ian. Hence (p)= Z(p), and so (p)is infinite. Suppose that the result is true for all* * groups with nilpotency < k. Let nil = k and let (k)= [ , (k-1)] and consider the exact * *sequence 1 ! (k)! ! = (k)! 1, with (k)and = (k)both finitely-generated and of nilp* *otency at most k - 1. This gives rise to an exact sequence 1 ! ( (k))(p)! (p)! ( = (k* *))(p)! 1 Homotopy Classes of Self-Maps 13 of p-localized groups [HMR , p. 12]. If is infinite, either (k)or = (k)is * *infinite. By the inductive hypothesis, at least one of ( (k))(p)or ( = (k))(p)must be infinite a* *nd thus (p)is infinite. Our next lemma establishes the link between commutators and Z (G) (and theref* *ore, by Proposition 3.1, with E (G)). Lemma 3.4. If X is a group-like space, then H(X)(2) Z (X). Proof.If f, g 2 H(X), we have [f, g] = [ f, g] = 0, because X is homotopy-co* *mmutative. Therefore, [f, g] 2 Z (X), and since H(X)(2)is the subgroup generated by these * *commuta- tors, we have H(X)(2) Ker( ) = Z (X). Proposition 3.5. Let X be a finite group-like complex, and p be a prime number.* * Then (1)(a) |Z (X)(p)| pnilH(X)(p)-1; (b) |E (X)(p)| pnilH(X)(p)-1; (c)|Z (X)| 2nilH(X)-1. (2)|Z (X(p))| pnilH(X)(p)-1. Proof.(1)(a) We have H(X)(2) Z (X) by Lemma 3.4, and so by [HMR , Ch. I, Thm.* * 2.7], (H(X)(p))(2)= (H(X)(2))(p) (Z (X))(p). Then by Lemma 3.3(2), we have pnilH(X)(* *p)-1 |(H(X)(p))(2)| |Z (X)(p)|. (1)(b) This does not follow immediately from (1)(a) since there is only a bij* *ection, not necessarily an isomorphism, between E (X) and Z (X). If Z (X) is infinite, so i* *s E (X) by Proposition 3.1. Therefore by Lemma 3.3(3), E (X)(p)is infinite, and the in* *equality holds. If Z (X) is finite, so is E (X), and both nilpotent groups have the sam* *e order. But the p-localization of a finite, nilpotent group is its unique p-Sylow subgr* *oup. Therefore |E (X)(p)| = |Z (X)(p)|, and the result now follows from (1)(a). (1)(c) We assume without loss of generality that Z (X) is finite. The result * *then follows form Lemma 3.3(1). (2) Since H(X)(p)~=H(X(p)) by [BK , Prop. 5.3], we have |Z (X(p))| |(H(X)(p* *))(2)| pnilH(X)(p)-1by Lemmas 3.4 and 3.3. Remark 3.6. (1) By (1.2) and subsection 2.3, the results of Proposition 3.5 imm* *ediately provide lower bounds for the order of: Z#1 (X)(p)~=Z#1 (X(p)), Z# (X)(p)and Z#1* * (X). By (1.1) and subsection 2.3, this is also true with E replacing Z. 14 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom (2) The inequalities in Proposition 3.5(1)(a) and (1)(b) can be strict: H(Sp(* *2))(3)is com- mutative by [MO , Thm. 2] and E (Sp(2))(3)~=Z (Sp(2))(3)~=Z3 by Theorem 4.3. 4.Lie Groups In this final section we specialize further and study Lie groups. Our first th* *eorem gives the equivalence of several statements including the triviality of the Z-semigro* *ups and the E-groups. We then give explicit computations for some low-dimensional Lie group* *s. Throughout this section, G denotes a compact Lie group. This implies, in part* *icular, that G is a finite nilpotent complex. We shall need the following lemma in Theorem 4.2. Lemma 4.1. If X1 and X2 2 {S1, S3, SO(3)} and q : X1x X2 ! X1^ X2 is the projec* *tion inducing q* : [X1^ X2, X1x X2] ! [X1x X2, X1x X2], then Z (X1x X2) = Z#1 (X1x X2) = Z# (X1x X2) = Im(q*) ~=[X1^ X2, X1x X2]. Proof.First of all, the exact sequence q* i* 0 ____//_[X1^ X2, X1x X2]__//_H(X1x X2)____//_[X1_ X2, X1x X2]__//_0 shows that Im(q*) ~=[X1^ X2, X1x X2]. Since SO(3) ~=RP3, we know Z (X1x X2) = Z#1 (X1x X2) = Z# (X1x X2) by Corollary 2.15. Since q = 0 by Corollary 2.7, we clearly have Im(q*) Z (X* *1x X2). For the reverse containment, let f 2 Z# (X1x X2). We claim that f is in Im(q** *). In every case, X1 x X2 has one of S1 x S1, S1 x S3 or S3 x S3 as a covering space (depen* *ding on dim(X1x X2)); fix a covering map p : Sa x Sb ! X1x X2. Since f#1 = 0, there is * *a lift ^fin the diagram Sa x Sb ^f ll55 ll l |p| l l fflffl| X1x X2 ____f_____//_X1x X2. Since p is a covering, p# : ss*(Sa x Sb) ! ss*(X1x X2) is injective, and so ^f#* *n= 0 for all n. We next show that ^f|X1_X2 = 0. It is well-known that [RP3, S1] ~=H1(RP3; Z) * *= 0 and [RP3, S3] ~=H3(RP3; Z) ~=Z, generated by the quotient map RP3 ! RP3=RP2 = S3. T* *his Homotopy Classes of Self-Maps 15 map, and all of its nonzero multiples, is nontrivial on ss3. Therefore, ^f|Xi =* * 0 if X RP3. Clearly, if Xi is a sphere (i = 1 or 2 or both), then ^f|Xi = 0. Therefore f 2* * Im(q*) as claimed. The next theorem shows the equivalence of several statements about Lie groups* * G, most of which have been previously proved elsewhere. We denote the torus of dimensio* *n n by Tn, n 1, and let T0 be the trivial group. Theorem 4.2. The following statements are equivalent: (1)Z (G) = {0} (or, equivalently, E (G) = {id}). (2)Z#1 (G) = {0} (or, equivalently, E#1 (G) = {id}). (3)Z# (G) = {0} (or, equivalently, E# (G) = {id}). (4)The left distibutive law holds in H(G): a O (b + c) = a O b + a O c for a* *, b, c 2 H(G). (5)G is isomorphic to one of Tn (n 0), S3 or SO(3). (6)H(G) is commutative and G is not isomorphic to Tn x S3(n = 1, 2). Proof.In [KO , Thm. 1.1], it is proved that (2), (4), (5) and (6) are equivalen* *t. Obviously (3) implies (2) and (2) implies (1). Therefore it suffices to show that (5) imp* *lies (3) and (1) implies (5). To prove that (5) implies (3), let G be Tn, S3 or SO(3), and let k be 1 or 3 * *accord- ing to whether or not G is Tn. Then the map f 7! f#k induces isomorphisms H(G)* * ~= Hom(ssk(G), ssk(G)) (see [KO , Prop. 4.1] for SO(3)). Hence Z#k(G) = {0}. Since* * Z# (Tn) = Z#1(Tn), we have Z# (G) = {0}. We conclude by showing that (1) implies (5), that is, if G 6~=Tn, S3 or SO(3)* *, then E (G) 6= {id}. First suppose G 6~=Tn, Tm xS3(m = 1, 2), S3 orSO(3). Then H(G) is not com* *mutative by [KO , Thm. 1.1], so that E (G) 6= {id} by Proposition 3.5(1b). Next suppose * *that G ~= Tm x S3 with m = 1, 2. The following square is commutative i0* E(S1 x S3)____________//_E(S1 x S1 x S3) || || fflffl| i00* fflffl| E( (S1 x S3))__________//E( (S1 x S1 x S3)), where i0*and i00*are the injective homomorphisms defined by i0*(f) = idS1x f an* *d i00*(f) = id S1x f. Hence to show E (Tm x S3) nontrivial for m = 1, 2, it suffices to pr* *ove that 16 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom E (S1 x S3) 6= {id}. But |E (S1 x S3)| = |Z (S1 x S3)| = |ss4(S1 x S3)| = 2 by * *Lemma 4.1, and so the proof is complete. We conclude with some concrete calculations for low-dimensional Lie Groups. Theorem 4.3. (1)If G is isomorphic to one of S1 x S3, U(2), S1 x SO(3), SU(3), Sp(2) or S* *3 x S3, then E (G) = E#1 (G) = E# (G) ~=Z (G) = Z#1 (G) = Z# (G), and this common group is explicitly given as follows: Z2 ifG ~=S1 x S3 orU(2), Z2 Z2 ifG ~=S1 x SO(3), Z12 ifG ~=SU(3), Z12 Z12 ifG ~=S3 x S3, Z120 ifG ~=Sp(2). (2)If G is isomorphic to S3 x SO(3) or SO(4), then E (G) = E#1 (G) = E# (G)~= M4 (Z3)2 and Z (G) = Z#1 (G) = Z# (G)~= (Z4)4 (Z3)2, where M4 is the noncommutative group of order 28 defined in [O3 ]. Proof.First we make the elementary observation that the groups in question depe* *nd only on the homotopy type of G, and not on its structure as a Lie group. (1) We begin by considering the groups S1xS3 and U(2). Since they are homeomo* *rphic, it suffices to prove the result for S1xS3. Lemma 4.1 shows that the Z-groups are a* *ll isomorphic to ss4(S3x S1) ~=ss4(S3) ~=Z2. The result for the E-groups now follows from Pro* *position 3.1. Now consider G = S1 x SO(3). Since SO(3) is homeomorphic to RP3, we have E (G* *) = E#1 (G) = E# (G) and Z (G) = Z#1 (G) = Z# (G) by Corollary 2.15. By Lemma 4.1, * *these latter groups are isomorphic to q*[S1 ^ RP3, S1 x RP3] where q : S1 x RP3 ! S1 * *^ RP3 is the quotient map. It is known that q*[S1 ^ RP3, S1 x RP3] ~=Z2 Z2 by [O3 , Pro* *p. 3.1]. To complete the proof for G = S1 x SO(3) it suffices to show that : Z (G) ! E (* *G) is a homomorphism. This follows from the methods in the proof of [O3 , Prop. 3.1] an* *d is similar to the proof below that : Z (S3 x S3) ! E (S3 x S3) is a homomorphism. For G = SU (3), let q : G ! S8 be the quotient map obtained by collapsing the* * 7- skeleton. Recall from [MT ] that ss8(G) ~=Z12 and from [MO ] that q* : ss8(G) !* * H(G) is a monomorphism whose image is generated by a commutator, and so Im(q*) Z (G).* * On the other hand, it follows from [MO , Thm.5.1] that Z#1 (G) Im(q*), and so Z * *(G) = Homotopy Classes of Self-Maps 17 Z#1 (G) = Im(q*) ~=Z12. Also, by [M2 ], Z#1 (G) = Z# (G). Hence E (G) = E#1 (G)* * = E# (G). By [MO , Prop.7.2(2)], the composite of q* 1 ss8(G)_~__//_Im(q*) = Z#1 (G)_//_E#1 (G) = is an isomorphism of groups. Hence E#1 (G) ~=ss8(G) and the assertion for G = * *SU(3) follows. Let G = Sp(2), and consider the quotient map q : G ! S10. It is known that q** * : ss10(G) ! H(G) is injective, that Z# (G) = Z#1 (G) [M2] and that Z#1 (G) = Im(q*). Thus w* *e have Z (G) Z#1 (G) = Z# (G) = Im(q*) ~=ss10(G), and so the statement about the Z-groups will be verified once we show that Im(q* **) Z (G). The reduced diagonal d : G ! G ^ G is the composition of the diagonal : G ! G* * x G and the quotient map p : G x G ! G ^ G. Since p = 0 by Corollary 2.7, it follows t* *hat d = 0. Let fl = fl0O q, where fl0 is a generator of ss10(G) ~=Z120. By the proof of [M* *2 , Thm. 3.3], fl factors as d _ G _____//G ^ G___//_G for some map _. Since d = 0, we have fl 2 Z (G), and so Im(q*) Z (G). For * *the E-groups, we now have that E (G) = E#1 (G) = E# (G). By [MO , Prop. 7.2(1)], th* *e bijection 1 : Z#1 (G) ! E#1 (G) is a homomorphism, and so it is an isomorphism. Hence * *the assertion for G = Sp(2) follows. Let G = S3 x S3. By Lemma 4.1, Z (G) = Z#1 (G) = Z# (G) and E (G) = E#1 (G) = E# (G). Furthermore, Z (G) = Im(q*) ~=ss6(G) ~=Z12 Z12, where q : S3xS3 ! S3^S3* * is the quotient map. To complete the proof we show that the bijection 1 : Z#1 (G) ! E* *#1 (G) defined by 1 (ff) = id+ ff is a homomorphism. For k = 1, 2, let ik : S3 ! S3 x* * S3 and pk : S3xS3 ! S3 be the standard inclusions and projections, respectively. Write c : * *S3xS3 ! S3 for the commutator map; then c = [p1, p2]. It is known that Z#1 (G) ~=Z12 Z12a* *nd that ffk = ik O c are generators [?]. We first prove ffk O (id+ ffl) = ffk for k, l * *= 1, 2. Now c O (id+ ff1)=[p1, p2] O (id+ i1O [p1, p2]) = [p1O (id+ i1O [p1, p2]), p2O (id+ i1O [p1, p2])] = [p1+ [p1, p2], p2] which we write in multiplicative notation as [p1[p1, p2], p2]. But [p1[p1, p2], p2] = p1[[p1, p2], p2]p-11[p1, p2] = [p1, p2] = c. 18 Martin Arkowitz, Hideaki ~Oshima and Jeffrey Strom and so c O (id+ ff1) = c. Similarly, c O (id+ ff1) = c. It follows that ffk O* * (id+ ffl) = ik O c O (id+ ffl) = ik O c = ffk and, more generally, ffk O (id+ ffl)n = ffk f* *or any n 0. We now prove by induction on n 0 that ( (ffk))n = (nffk). The result is trivial* * for n = 0 or n = 1. For the inductive step, we have ( (ffk))n= (ffk) O (ffk)n-1 = (id+ ffk) O (ffk)n-1 = (id+ (n - 1)ffk) + ffk(id+ ffk)n-1 = (id+ (n - 1)ffk) + ffk = id+ nffk = (nffk). Hence, for any nonnegative integers n1, n2, we have (ff1)n1O (ff2)n2= (1 + n1ff1) O (1 + n2ff2) = 1 + n2ff2+ n1ff1O (1 + n2ff2) = 1 + n2ff2+ n1(ff1O (1 + ff2)n2) = 1 + n2ff2+ n1ff1 = (n2ff2+ n1ff1). Similarly (ff2)n2O (ff1)n1= (n1ff1+ n2ff2), and so (ff1)n1O (ff2)n2= (n2ff2+ n1ff1) = (n1ff1+ n2ff2) = (ff2)n2O (ff1)n1 since Z#1 (G) ~=Z12 Z12is an abelian group. Now we have ((m1ff1+ m2ff2) + (n1ff1+ n2ff2))= (ff1)m1+n1O (ff2)m2+n2 = (ff1)m1 O (ff1)n1O (ff2)m2 O (ff2)n2 = (ff1)m1 O (ff2)m2 O (ff1)n1O (ff2)n2 = (m1ff1+ m2ff2) O (n1ff1+ n2ff2), which proves that is a homomorphism. Homotopy Classes of Self-Maps 19 (2) The groups S3 x SO(3) and SO(4) are homeomorphic to S3 x RP3, so it suffi* *ces to verify the statement (2) for the space S3 x RP3. In fact, we know that Z (S3 x RP3) = Z#1 (S3 x RP3) = Z# (S3 x RP3) ~=[S3 ^ RP3, S3 x RP3] by Lemma 4.1. To identify this latter group we calculate [S3 ^ RP3, S3 x RP3]~=[S3 ^ RP3, S3] x [S3 ^ RP3, RP3] ~= ((Z=12) (Z=4)2) x ((Z=12) (Z=4)2) by [O2 ]. This completes the proof for Z. Therefore, by Proposition 3.1, we know that E (G) = E#1 (G) = E# (G). 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