EQUIVARIANT HOMOTOPY EQUIVALENCES AND A FORGETFUL MAP KOUZOU TSUKIYAMA Abstract.We consider the forgetful map from the group of equivariant self equivalences to the group of non-equivariant self equivalences. A suffi* *cient condition for this forgetful map being a monomorphism is obtained. Sever* *al examples are given. 1.Introduction Let (P; q; B; G) be a principal G-bundle over B. Then G acts on P freely and P=G = B. Let autG (P ) be the space of unbased G-equivariant homotopy equiva- lences from P to P of the total space P of the given principal G-bundle. We work in the category of connected CW-complexes. The path component of autG (P ) forms a group where the multiplication is giv* *en by the composition of maps. That is, we define FG (P ) = ss0(autG (P )): Two maps f; g in autG (P ) are in the same path component if there is a G-equiv* *ariant homotopy between f and g. We call this group the group of unbased G-equivariant self equivalences. On the other side, we consider the space of unbased self homotopy equivalences from P to P , which is denoted by aut(P ). Two maps are in the same path com- ponent if there is a homotopy between them. By the same consideration as above, we define F(P ) = ss0(aut(P )): We call this group the group of unbased self homotopy equivalences. The author has been interested in the natural map which forgets a G-action on P , that is, a forgetful homomorphism (1.1) F : FG (P ) ! F(P ) and raised the following problem in 1988 (see [5, p. 206, Problem 13]): When is the homomorphism F of (1.1)a monomorphism? This problem seems to be difficult, because the generators of FG (P ) are not known in general, even if we have the group structure of FG (P ) (see [3], [7]). An example where F is not a monomorphism, is given by [8] in 1996 as in the following. ____________ 1991 Mathematics Subject Classification. Primary 55P10; Secondary 55R10. Key words and phrases. equivariant self homotopy equivalence, self homotopy * *equivalence, bundle map theory, principal G-bundle. 1 2 KOUZOU TSUKIYAMA Let (G=T; q; BT; G) be a principal G-bundle, where G is a compact connected Lie group, which is not a torus and T is a maximal torus of G. It is shown that FG (G=T ) is an infinite group. But F(G=T ) is a finite group. So F : FG (G=T ) ! F(G=T ) cannot be a monomorphism. On the contrary, many examples where F might be a monomorphism, could be found in [3], [7] by the calculations of the group of G-equivariant self equiva* *lences. For example, let (EG ; q; BG ; G) be a universal principal G-bundle, then FG * *(EG ) = 1 (see [7, Example 3.1]). So FG (EG ) ! F(EG ) is a monomorphism. In this note we consider the sufficient condition that the homomorphism F of (1.1)is a monomorphism and several examples are given. 2.Bundle map theory Let f be a G-equivariant map from P to P . Then one has the induced map on f on B such that qf = fq. P --f--!P ? ? q?y ?yq f B ----! B One has naturally the following map : autG (P ) ! aut(B); (f) = f: This construction determines a Serre fibration with fibre the space IG (P ) of * *unbased bundle equivalences over B (cf. [3], [4], [7]): (2.1) IG (P ) ! autG (P ) ! aut(B) By the Gottlieb's theorem [1], (2.2) ss0(IG (P )) = ss1(map(B; BG ); k); where k : B ! BG is a classifying map and BG is a classifying space. So we have the following exact sequence of groups and homomorphisms by (2.1) and (2.2)(see [3], [4], [7]) (2.3) ss1(map(B; BG ); k) ! FG (P ) ! Fk(B) ! 0; where Fk(B) = {f 2 F(B); kf ' k}. 3. K(ss; n)-action Let Rq(P ) be the group of homotopy classes of q-retracting equivalences, tha* *t is, an element f of F(P ) for which there is an element f of F(B) satisfying qf ' fq (see [6, p.645]). Rq(P ) is a subgroup of F(P ) and F (FG (P )) is a subgroup * *of Rq(P ). EQUIVARIANT HOMOTOPY EQUIVALENCES AND A FORGETFUL MAP 3 If the structure group of the principal G-bundle (P; q; B; G) is an Eilenberg- MacLane space K(ss; n) = G(n 1) P ----! EK(ss;n) ? ? (3.1) q?y ?y B --k--! BK(ss;n); then BK(ss;n)= K(ss; n + 1) and ss1(map(B; BK(ss;n)); k) = ss1(map(B; K(ss; n + 1)); *) = Hn(B; ss): Therefore if Hn(B; ss) = 0, we have the following diagram by (3.1) and (2.3) FK(ss;n)(P_)___-Rq(PF) | 0 jj3 ~|= F j | j |? j Fk(B) ________F(B);-i where i is an inclusion and F 0: Fk(B) ! Rq(P ). Theorem 3.2. Let (P; q; B; K(ss; n)) be a principal K(ss; n)-bundle, we assume Hn(B; ss) = 0. The forgetful map F : FK(ss;n)(P ) ! Rq(P ) F(P ) is a monomor- phism, if a map G : Rq(P ) ! F(B) which commutes the following diagram (3.3) exists. FK(ss;n)(P_)___-Rq(PF) | 0 jj3 | (3.3) ~|= F j |G | j | |? j |? Fk(B) ________F(B)-i Proof.Since GF 0= i and i is a monomorphism(inclusion), F 0is a monomorphism._ FK(ss;n)(P ) is isomorphic to Fk(B). So F is a monomorphism. |__| Let E(B) be the group of based self homotopy equivalences of B, and we de- note by E# (B) the subgroup of E(B) consisting of classes that induce the ident* *ity automorphism of all homotopy groups. Corollary 3.4.Let (Pk; q; B; K(ss; n)) be a principal K(ss; n)-bundle, where ss* *i(B) = 0(i n), ss is a free abelian group and E# (B) = 1. Then F : FK(ss;n)(Pk) ! F(Pk) is a monomorphism for any k. Proof.By the homotopy exact sequence of the given principal bundle, we have ssi(B) = ssi(Pk)(i n + 2), ssn+1(B) = ssn+1(P ) ss0 (ss0 is a free subgroup o* *f ss) and ssi(B) = 0(i n). Hn(B; ss) = 0, since ssi(B) = 0(i n). Since E# (B) = 1, the induced map on the base space B is determined uniquely for the given self homotopy equivalence in Rq(P ). So a map G : Rq(P ) ! F(B) which commutes__ the diagram (3.3)exists. |__| 4 KOUZOU TSUKIYAMA 4.Examples Example 4.1. For the trivial bundle (T nxB; q; B; T n), where T nis an n-dimens* *ional torus (n 1) and ss1(B) = 0, F : FTn(T nx B) ! F(T nx B) is a monomorphism. Proof.For a given homotopy equivalence f : T nx B ! T nx B, we define f : B ! B by qfi(i : B ! S1 x B; i(b) = (*; b)). Since ss1(B) = 0, f : B ! B induces an isomorphism on homotopy groups. So f is a homotopy equivalence. Therefore G : Rq(T nx B) = F(T nx B) ! F(B) exists. Since K(Zn; 1) = T nand __ H1(B; Zn) = 0 by assumption. The result follows by Theorem 3.2. |__| Example 4.2. For the trivial bundle (K(ss; n) x B; q; B; K(ss; n)) with ssi(B) * *= 0 (i n), F : FK(ss;n)(K(ss; n) x B) ! F(K(ss; n) x B) is a monomorphism. Proof.Since ssi(B) = 0 (i n), Hn(B; ss) = 0. The proof is similar to_example_4* *.1. |__| Example 4.3. Let (Pk; q; B; T n) be a principal T n-bundle (n 1) with ss1(B) =* * 0 and E# (B) = 1. For example, take B = P m(C)(m 1) (complex projective space) (see [2, p.32]). Then F : FTn(Pk) ! F(Pk) is a monomorphism for any k. Proof.This is obtained from Corollary 3.4 for n=1. |___| Example 4.4. Let (P; q; B; K(ss; n)) be a principal K(ss; n)-bundle with ssi(B)* * = 0 (i n) and Hn(B; ss) = 0. Then F : FK(ss;n)(P ) ! F(P ) is a monomorphism. Proof.Since ssi(B) = 0(i n), it is easy to see that a map G : F(P ) ! F(B) exists by the elementary homotopy theory (e.g. Postnikov tower) and the diagram_ (3.3)is commutative. So the result is obtained from Theorem 3.2. |_* *_| Problem 1. For any S1-bundle (Pk; q; B; S1) with ss1(B) = 0, is the forgetful * *map F : FS1(Pk) ! F(Pk) a monomorphism? References [1]D. H. Gottlieb, Applications of bundle map theory, Trans. Amer. Math. Soc. * *171, pp. 23-50 (1972). [2]P. J. Kahn, Self-equivalences of (n-1)-connected 2n-manifolds, Math. Ann. 1* *80, pp. 26-47 (1969). [3]H. Oshima and K. Tsukiyama, On the group of equivariant self equivalences o* *f free actions, Publ. Res. Inst. Math. Sci. Kyoto Univ. 22, pp. 905-923 (1986). [4]H. Oshima and K. Tsukiyama, Bundle map theory in the category of weak Hausd* *orff k- spaces, Mem. Fac. Sci. Eng. Shimane Univ. 31, pp. 27-55 (1998). EQUIVARIANT HOMOTOPY EQUIVALENCES AND A FORGETFUL MAP 5 [5]R. A. Piccinini, Groups of self-equivalences and related topics, Springer Le* *cture Notes in Math. 1425, (1988). [6]J. W. Rutter, Self-equivalences and principal morphisms, Proc. London Math. * *Soc. 20, pp. 644-658 (1970). [7]K. Tsukiyama, Equivariant self equivalences of principal fibre bundles, Math* *. Proc. Camb. Phil. Soc. 98, pp. 87-92 (1985). [8]K. Tsukiyama, An example of self homotopy equivalences, J. Math. Soc. Japan * *48, pp. 317-319 (1996). Department of Mathematics, Shimane University, Matsue, Shimane 690-8504, Japan E-mail address: tukiyama@edu.shimane-u.ac.jp