Finite group actions on homotopy complex projective spaces

Math. Z. 199, 133-151 (1988) Mathematische Zeitschrift

�9 Springer-Verlag 1988

Finite Group Actions on Homotopy Complex Projective Spaces

Mark Hughes Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA

1. Introduction

The basic question which motivates our work is: What do the symmetry proper- ties of a smooth manifold imply about its differentiable structure? In particular, what restrictions does the existence of a group action on a manifold place on its differentiable structure?

A good class of manifolds to study are those oriented, closed, 2n-dimensional smooth manifolds which are homotopy equivalent to complex projective space, CPn; hereafter called homotopy cpn's. It is known that for n=>3, there are infinitely many differentiably distinct homotopy CP"'s.

A basic conjecture due to Petrie [P2] states that if X is a homotopy ~ P " which admits a smooth SX-action, then the Pontryagin class of X is of standard form, i.e., P(X)=(1 +x2) ~§ for a generator x~H2(X). Together with a result of Sullivan [Su] this would imply that, given n, at most finitely many homotopy CP~'s admit an S~-action. We note that if a homotopy ~ P " admits an SX-action with the property that the (isotropy) representations obtained by restricting the tangent bundle to fixed points are the same as those which occur for some linear action on II;P", then Petrie's conjecture holds. See [Ha] and [P2].

This paper is concerned with the question of which finite groups can act on infinitely many differentiably distinct homotopy (EP"'s. In light of the above remark, we are also interested in seeing how these actions compare with linear actions. Our results are as follows.

Theorem 2. Every odd order group acts smoothly and effectively on infinitely many differentiably distinct homotopy (12 P"'s, for some n.

Theorem 2 is a consequence of Theorem 1 which provides a sufficient condition for the vanishing of equivariant surgery obstructions in a setting which produces G-manifolds which are G-homotopy equivalent to CP" with a particular G-action. (A key idea involved here is that the G-normal maps are each designed to possess a G-equivariant, orientation reversing involution, thereby ensuring that the appropriate equivariant signatures vanish.) If A is a complex representation of finite group G, let P(A) denote the space of complex lines in A. Then P(A) is just ~ P " with the associated linear G-action. (Here n = d i m c A - 1 . ) Given an integer m>2, we consider the representation

134 M. Hughes

A=2mr(G) , where r(G) is the complex regular representation of G. We will give P(A) a particular G x 2gz-action in Sect. 5. Theorem 1. Let G be an odd order group and let A=2mr(G) , with m>2. Let

tI~KOa(P(A)) be such that t/=resGq' for some tI'sKO~• with JOa• and suppose that t I satisfies Condition (*). Then, there exists a G-homotopy CP", X, , and a G-homotopy equivalence F: X,--*P(A) such that T X , = F*( TP(A) + tl). Here, n= 2m l G I - 1.

Condition (.) shall be explained further along in this paper. It involves the representations obtained by restricting TP(A)+ t l to fixed points and the G-fiber homotopy equivalence which exists since J O t • ~(t l ' )= O.

Considerations similar to those involved with Theorems 1 and 2 yield Theo- rem 3. Theorem 3. Let G be a finite abelian group with 2-Sylow subgroup which is either cyclic o r ~2 X ~'2, or let G be a finite product of 292's. Then G acts smoothly and effectively on infinitely many differentiably distinct homotopy C P"'s, for some n.

Finally, in Theorem 4, we compare these actions with linear actions on CP". We show that, in most cases, if G is as in Theorem 2 or 3, then the actions constructed can be designed to possess isotropy representations which do not occur for any linear action of G on ~ P " with the same fixed point data. We mention that producing this type of exotic action requires using normal maps which are more flexible than the transverse linear type.

Previous work has been done by several authors concerning the existence of cyclic group actions on differentiably non-standard homotopy CP"'s. For the Z 2 case, see [Pl] , [DMS], and [M1]. The existence of infinitely many differentiably distinct homotopy IEP"'s admitting smooth 2g, actions (p prime) was first demonstrated in [MT]. Also see [DM]. For the case of 7Z,,,, with m odd, we refer the reader to Tsai's work [T]. We mention that the examples in [MT] and IT] are non-isotropy linear. Recently, Sampaio [Sa] has considered the dihedral group case. We mention that certain types of cyclic group actions on homotopy tEP 's are known to force the Pontryagin class to be standard. See [DMSu].

This paper is organized as follows. Section 2 sets up notation and explains some of the basic results we will need. Section 3 summarizes the features of equivariant surgery theory which apply to our problem. Section 4 concerns the work of Dovermann and Rothenberg on the torsion obstruction. Sections 5, 6, and 7 provide the proofs of Theorems 1, 2, and 3, respectively. Finally, Sect. 8 deals with the question of isotropy linearity.

! would like to conclude this introduction by expressing my sincere thanks to Professors Ted Petrie, Heiner Dovermann, and Ian Hambleton, whose help has greatly aided me in my understanding of this problem. Much of this work was done at Rutgers University.

2. Background Information

Let G be a finite group and X a G-manifold. The following notions and notation shall be useful. Given a subgroup H _ G, the fixed point set of H is denoted

Finite Group Actions on Homotopy Complex Projective Spaces 135

X H. For xeX, Gx={geGlgx=x} is called the isotropy group of x, and we set Iso (X) = {Gx Ix eX}. The singular set of X is X s = {x ~ X [ G~ + 1 }, where 1 denotes the trivial group. As usual, X/G denotes the orbit space. The G~-representation TXJ~ induced by the differential dgl, is called an isotropy representation.

Representations play an important role in this work. We let R(G) (resp, RO(G)) denote the complex (resp. real) representation ring of G. For example, R(Zm)=7Z[t]/(t m- 1) and R(S1)=~[t, t -1] . (See [Se].) The set of 1-dimensional complex representations of G form a group G under tensor product, which is called the character group. (We mention that if G is abelian, then G ~ G, and in general, [ G I - [G : G'], where G' is the commutator subgroup.) An important example of a representatio n is the complex regular representation, denoted

r(G). The regular representation can be written as r (G)= ~ ni 01, where the i = l

sum is over all irreducible representations 0i of G and n i = d i m c 0 ~. (Here c is the number of conjugacy classes of G.) One of the key ideas in the solution of the surgery problem which follows is to properly exploit the symmetry inher- ent in the regular representation. Of primary importance is the following:

Fact 2.1. Given H~_G, resHr(G)= [ G/H[ r(H), where resrt denotes restriction to H.

Next, we consider notions involving G-vector bundles. (See [PR] or [A].)

Definition 2.2. Let q + and q_ be G-vector bundles over a G-manifold X. Assume that given H _~ G and x ~ X/t, we have dim 0/+ [~)r~ = dim 0/- [x)r~. Then (~: t/+ --* q_ is a G-fiber homotopy equivalence if it is a proper, fiber preserving G-map such that, given H o G and x e X H, the map (~ol~)t~: (q+l~)~r___,(~/_[~)H has degree 1 when extended to one point compactifications.

We say that G-vector bundles t/+ and q _ over X are stably G-fiber homotopy equivalent or J-equivalent if there exists a G-fiber homotopy equivalence between q + | V and t/_ O_V, where V= X x V for some G-representation V. The set of J-equivalence classes of G-vector bundles over X is a commutative semi- group. The Grothendieck construction provides the groups JG(X) and JOG(X ) (when considering real bundles). There are homomorphisms JG: KG(X)~JG(X) and JOt: KOG(X)--+JOG(X) such that if tl=tl+--tl_ lies in the kernel, then

+ and q_ are J-equivalent. See [-PR]. Now, let's consider actions on IEP". As mentioned in the introduction, given

a G-representation A, P(A) denotes (EP" with the induced linear action. We can also describe P(A) as S(A| t)/S 1, where t is the standard 1-dimensional Sl-representation, and S(-) denotes the unit sphere of the indicated G x Sl-representation (using a G x S 1 invariant metric). In the linear case, use

of homogeneous coordinates facilitates the analysis. For instance, if A = 0 1 + . . . +0 ,+~ is a sum of 1-dimensional complex representations, g~G, and z = [Zl : ... : z, + 1] ~P (A), then g. z = [0 ~ (g) z 1 : ... : 0 , + 1 (g) z, + 1I-

ra Proposition 2.3. Let A= ~ n~O~, where each 0~ is an irreducible complex G-

e = l representation.

136 M . H u g h e s

a) P(A) ~= I_I P(n~O~)= H (EP "~-1, where H denotes disjoint { ~ : d i m r = 1} (~:dimazr = 1}

union. b) Let peP(A) ~. There is an i and a k such that both the point p and the

point pi= [0: ... : 1: ... : 0] (1 in the fh coordinate) lie in P(nkOk). Then the isotropy representation

TP(A)ip= TP(A)Iv~ =(nk-- 1) 1~+ ~ n~O~Ok- 1, a = l a ~ k

where 1G denotes the irreducible trivial complex G-representation.

Proof Part a) follows easily from looking at the action in homogeneous coordinates. Part b) relies on the fact that if Y is a G-manifold with p and q lying in the same path component of Ya, then TY[p=TYhq as G-representations. ([PR], Ch. 1) The rest of part b) is elementary and follows by using the coordi-

nate chart around Pi, [ Z l : " '" : Zn+ 1] }--)'(ZI/Zi, " " , Z~Zi, . . . . Zn+ 1/zi)e~, to find the differential dg]~ c Q.E.D.

We end this section by showing how Sl-representations can be used to construct G-vector bundles over P(A). Given a complex Sl-representation V, we form the cartesian product S(A| t )x V. Recalling the right Sl-action on S(A | t), we define an action on S(A | t) via t.(a, v)=(a.t -1, t.v) for teS 1. The orbit space is denoted S(A| t)x sl V - I / and we denote the points of ~" by [a, v], for aeS(A| and veV. Then V is a G-vector bundle over P(A) with action g- [a, v] = [g-a, v].

If f : V ~ W is an Sl-map between Sl-representations, then the G-map f : 9"~ 17V can be formed by taking Id x f : S(A | t) x V--* S(A | t) x W and passing to orbit spaces.

3. Equivariant Surgery Theory

In this section, we summarize the features of equivariant (or G-) surgery which apply to our problem. We use G-surgery as developed in [PR] and [DP]. Equi- variant surgery provides us with a method for constructing G-manifolds which are G-homotopy equivalent to a given G-manifold Y. (A G-homotopy is a homotopy {ft}]=o such that each f is a G-map.) The fixed point data on our model Yis preserved up to homotopy. Two major steps are involved in the construction of such a G-manifold.

1. We build a G-normal map (X,f, b) with target manifold Y This can be thought of as an approximation to a G-homotopy equivalence.

2. We determine whether or not we can add G-handles to X in such a way that X is converted to a G-manifold X' and f to f ' where f ' : X ' ~ Y is a G-homotopy equivalence. That is, we determine whether or not G-surgery to a G-homotopy equivalence is possible on the normal map constructed in step 1.

Before we elaborate on this, we need some definitions.


Definition 3.1. A G-manifold X is said to satisfy the gap hypothesis if: a) Given H _ G, each fixed point set component X~/~_ X H has the same

dimension. b) Whenever L = H and X L 4= X H, then dim X L < �89 dim X H.

Next, given any irreducible real G-representation 0, we can define too: RO(G) ~ Z by setting m e ( V ) equal to the multiplicity of 0 in the virtual representation V. Again, for an irreducible real G-representation ~, we set D o =Hom~(O, ~), the ring of N-linear G-endomorphisms of ~. Schur's Lemma implies that D o is a division algebra and we denote dim~D o by d o.

Definition 3.2. Given V~RO(G), let S(V) be the set of real irreducible G-representations 0 such that m e ( V ) 4: O.

Definition 3.3. A virtual G-vector bundle r l e K O ~ ( X ) is called stable if, for each x ~ X and each irreducible Gx-representation OeS(q[x), we have ml~(q]x) < d o mo(rl Ix), where 1~ denotes the trivial 1-dimensional real Gx-representation. A G-manifold is called stable if its tangent bundle is stable.

Now, we can give:

Definition 3.4. The triple (X , f , b) is a G-normal map with target Yif: a) X is a smooth, closed G-manifold which is stable and satisfies the gap

hypothesis. Given HeIso(X) , we have that rcl(Xff)=0 and d i m X f f > 5 for all components X~ _~ X H.

b) Y is a smooth, closed G-manifold with I so(X)= Iso(Y). c) f : X ~ Y is a smooth G-map which, given H M s o ( X ) = I s o ( Y ) , induces

a bijection y: ~o(X H) ~ rCo(yH). We also require that deg( f lx~)= 1 for all fixed point components X~r~ ~ x H .

d) b: s T X ~ s f * ( T Y + rl) is a stable G-vector bundle isomorphism, for some

rlEKO~(Y).

According to [PR], Ch. 11, we can construct a G-normal map ( X , f b) with target Y provided that we have a G-fiber homotopy equivalence co: t/+ ~ t/_ over Y such that:

a) Y is a smooth, closed G-manifold which satisfies the gap hypothesis. Given H e Is o (Y), we have 7z, (Y~)= 0 and dim y n > 5 for all components Y~_~ y/t.

b) Iso (~ +) _ Iso (Y). c) T Y + r I is a stable bundle over Y. d) For each H~Iso(Y) and each component Y~___ yH the following holds.

Let ye Y~. For each real H-representation OeS(rl_ly) we have

dim Y~ = m~, ( T Y [y) <= d o m e (( T Y + r I + -- r I _ )ly) + d o - 1.

This last statement is called the "transversality condit ion" and it plays a very important role in our constructions. (See [PR] and [P 31.) If the transversality condition is satisfied, then there are no obstructions to moving co by a proper G-homotopy to a G-map h which is transverse to Ycr l_ . Here, Y is thought of as the zero section. We then set X = h - I ( Y ) , f=hJx , and b is con-

structed from r l = r l + - r l _ ~ K O G ( Y ). More precisely, for He_G, we set X H

138 M. Hughes

= ( f n ) - l(y//). Zero and one dimensional equivariant surgery provide the bijection ~z o (X Lr) --~ ~o (ytt) and ensure the simple connectivity condition on the fixed point set components.

Once we have constructed our G-normal map (X, f b) with target Y,, we proceed to step 2, that is, we must determine whether or not surgery up to a G-homotopy equivalence is possible.

It is a fact that f : X ~ Y is a G-homotopy equivalence <=> f n : X n__, y n is an ordinary homotopy equivalence for all H ~ G. ([JS]) Therefore, we need to construct a G-manifold X,7 and a G-map F: X, ~ Y such that F a is a homotopy equivalence for all H c G. At this point, we introduce the equivariant surgery obstructions which are defined inductively as follows. Let H___ G be such that fL : X L ~ yL is a homotopy equivalence for all LD H. Take a component X~ of X H, set f~n=fn]x~, and let Go be the subgroup of G which leaves X~ invariant. Then Xff is a GJH= W(e)-manifold. There is an obstruction ~r~r(f~) with the property that o-~(ff) = 0 ~ - f f : Xff ~ Yf can be converted (by adding W(~)-handles) into a homotopy equivalence ( f f ) ' : (X ~) '~ Y~. The surgery is done leaving ~ X L unchanged. The obstruction lies in the Wall group

/2~(7/[W(c~)], w~), where n = dim Yf and w~: W ( e ) ~ ~2 iS the orientation homomorphism of the W(e)-action on Yf. If an(f f)= 0, then we can perform surgery on any component X~ in the orbit G.X~. (Note that in this case W(/~) and W(e) are isomorphic.) Next, define crti(f) to be I-lo-~(J~ n) where there is one factor for each orbit of the G-action on ~o (XR) -

Therefore, o-H(f)=0 ~- we can convert f : X ~ Y to f ' : X ' ~ Y' where (f ,)r : (X,)L~ yL is a homotopy equivalence for all L~H. If ~rH(f)=0 for all H~_G, then (X,f, b) can be converted to a G-normal map (X,, F, B) such that F: X, ~ Y

is a G-homotopy equivalence and TX, = F* ( TY + tl) (in K O G( Y)). Clearly, we would like to have a criterion which guarantees the vanishing

of these surgery obstructions. Such a criterion exists for the groups we are considering. First, let G be a finite abelian group with 2-Sylow subgroup which is either cyclic or ~2 O 7~2 or let G be a finite product of Z2's. Assume that G preserves orientation and that for all H~_ G, every component of y n has dimension ~ 2 rood4.

Take o-~(f~)= ~/2,(2~ [W(~)], 1) and set W(~)= re. Consider the Rothenberg exact sequence [Sh]:

... ~/2,(#[~z] , i ) ~ / ~ , ( Z [n] , i ) ~ ,H " (Z 2; Wh(~)) ~ . . . ,

where/2( . ) denotes the group of obstructions to surgery to a simple homotopy equivalence, and Wh(~) denotes the Whitehead group of r~. The cohomology group H"(292; Wh(~))is defined as:

{ae Wh(~): a = ( - 1)"~*}/{~ + ( - 1)"~*: ~s Wh(~)},

where �9 denotes the conjugation involution. Now, the conjugation involution on the Whitehead group of a finite abelian group is trivial [B1], so for n even, H"(~2; Wh(rc))= Wh(~)/2 WhOc). It is a fact that for the groups we are consider-


ing, the Whitehead group has no 2-torsion, and hence for n odd, H"(2~ 2 ; Wh(x))= O.

Now, if e==0, then a is the image of a unique element of /2 , (Z[x] , 1) which we shall denote a s. At this point, we consider the multisignature homomorphism Sign: /2, (2g Ex], l) -> R (x ). Now, Sign(as)=Sign(rc, Xf f ) --Sign(x,Y~), where Sign(x, .) is the equivariant signature of Atiyah-Singer (see [AS] and [W1]). The Arf invariant of classical surgery provides an isomorphism c: Ker(Sign)

Z2. (See [B2].) As c(a s) depends only on the initial G-fiber homotopy equiva- H lence, we denote it by c(co~n). (Here co~ = (co I,+ lye)/t-)

Therefore, we will have a = 0 <:~ 1) e~(o-)=0 in Wh(rc)/2Wh(rc), 2) Sign(x, X~)=Sign(x, In), and 3) c(coff)=0 in 2g 2. The element e=(a) is called the torsion invariant and is the subject of Sect. 4.

Now, let G be an odd order group and keep a and x as above (for some subgroup H_~ G). This time we consider the exact sequence

. . . > H " ( 2 g 2; W h ' (re)) ---> . . . .

where/2(-) denotes the "weakly simple" obstruction group and Wh'(x) is the torsion free part of Wh(Tc). (See [W2].) Here ~=/~oe~, where /~: H"(292; Wh(x ) ) ~ H" (292; Wh'(x)) is induced by the projection W h ( x ) ~ W h'(x).

Since G (and hence x) is of odd order, we have that/2n(TZ, [~], 1)= I2, (2g Ix], 1) ([W2]). Once again, the Arf invariant provides an isomorphism between the kernel of the multisignature and 2g 2. Reasoning as before, we see that o-= 0 <=~ 1) c~;(~)=0 in H"(292; Wh'(x)), 2) Sign(~z, Xff) = Sign(~, Y~), and 3) c(coff)=0.

Now, the conjugation involution on Wh'(rc) is trivial and hence, for n even, we have H"(292; Wh'(x))= Wh'(x)/2 Wh'(x). (For n odd, the cohomology groups vanish.)

Note that if ~ (a ) is a multiple of 2, then e'~(a) will be also. Hence if e~(~) is a multiple of 2, then c~ (0)= 0 and ~;(a)= 0.

Therefore, if G is any of the groups that we are considering, we will have o-=0 if 1) e~(a) is a multiple of 2, 2) Sign(x, Xff) = Sign(zc, Y~), and 3) c(coff)= 0.

4. Isonormal Maps and the Torsion Invariant

Here we summarize the results of Dovermann [D] and Dovermann-Rothenberg [DR] which will be of use to us. Let G be a finite group and recall that superscript s denotes the singular set.

Definition 4.1. A G-normal map (X,f, b) with target manifold Y is called an isonormal map if there are manifold regular neighborhoods N X s and N Y s of X s and ys, f induces a degree 1 map of triads (X, N X s ,X \ in t (NXs) ) -* (Y, N Y s, Y~int (NYS)), and the induced map N fS: ( N X s, ~ N X ~) ~ ( N Y s, ONY ~) is a G-homotopy equivalence of pairs.

Now, suppose that we have constructed a G-normal map (X,f , b) from a G-fiber homotopy equivalence co: q+ ~ q_ over Y Further, suppose that ( X , f b) is an isonormal map. As indicated in Sect. 3, we can then define the surgery

140 M. Hughes

obstruction a 1 ( f ) e / ) , ( ~ [ G ] , 1), where n = dim Y In this situation, Dovermann has given a formula for

aa(a~(f))eH~(Z2; Wh(G)).

He shows (Cor. 4.4 in [D]) that

c~(al ( f ) )= [ + TzI,(~o)],

where : I , denotes the component lying in Wh(G) of the generalized Whitehead torsion

ze W"~(G)= @ Wh(N~(H)/H) H ~ G

and [-] denotes the equivalence class as defined in Sect. 3. Here T denotes a conjugation type involution on Wh(G) based on the orientation homomorphism w and the sign ___ depends upon the mod2 fiber dimension of q+.

In [DR], a formula for z(m) is given. A thorough description would involve a large amount of new material and notation. Briefly, they compute ~(co) = [Y]. To), where [Y] denotes the class of Y in a generalized Burnside ring which is based on the fixed point component structure of g and T,o is a function which associates to each fixed point set component y U an element in the generalized Whitehead group of H. The multiplication is based on the fact that the generalized Whitehead group is a Frobenius module over the Burnside ring.

In Sect. 4 of [D1, we find the following result:

Proposition 4.2. :(c9) is a multiple of 2 if either of the following conditions are met:

a) ]GI is odd and the Euler characteristic of each fixed point set component is even.

b) e)= 2e5 and tl + - =20_+, where ~: ~1+ -~ 0- is a G-fiber homotopy equivalence.

Indeed, if a) holds then 21 [Y] in the generalized Burnside ring and hence 2Iv(co). If b) holds, then Theorem 6.12 of [DR] implies that 2Iv(co ). Now, 2[z(co) =~ 2]_+T:i,(co ) and hence if either a) or b) holds then e~(a:( f ) ) is a multiple of 2.

This computation depended on the assumption that ( X , f b) was an isonor- real map. There is a condition on a G-normal map which guarantees that we can convert it by G-surgery to an isonormal map. This is also due to Dovermann.

Definition 4.3. A G-normal map ( X , f b) with target Y is called adjusted if fs : X S ~ YS is a G-homotopy equivalence.

Theorem A in [D] tells us that G-surgery can be used to convert an adjusted G-normal map into an isonormal map.

5. Proof of Theorem 1

Let G be an odd order group, m an integer ~2, and let A=2mr(G) . If V is a complex G-representation, then we denote its conjugate representation by


V (V is just V with the complex conjugate structure). Now, if ~ is an irreducible

complex G-representa t ion , then so is ~. This implies tha t r(G)= r(G) and hence

A=mr(G)+mr(G). This makes it clear tha t complex conjuga t ion induces a G- equivar ian t involu t ion on P(A) and hence P(A) m a y be considered a G x Zz-mani fo ld . F o r reasons which will become apparent , we design this involu-

t ion ~o to be free. F o r instance, if G = 2g 3 and m = 1, we can write A = 1 + t + t 2 2r~i

+ 1 + t + ~2 = 1 + t + t 2 + 1 + t 2 + t. (Here, G = ( e ~ - ) and t = 112 with G act ing by complex mult ipl icat ion.) Then, given z = [zl "z2" z3" z , " zs "z6] ~P(A), we have

(p(z) = [ - ~ " - - z 6 �9 - z 5 "za "z3 :z2]. Not ice tha t 9 is a free, G-equivariant , orien-

ta t ion reversing involut ion on P(A). Recall f rom Sect. 2 tha t if tl'=tl'+--q:eKO~• is such tha t

JOo• then there exists a G x2g2-fiber h o m o t o p y equivalence co': ~/'+ | O V for an app rop r i a t e trivial G x2g2-vector bundle _V. Let 4'_+

' ' r ' =t/+G_V, t/_+ =resGt/_+, and co= esGco. Now, given H~_G, we have resnA=2m[G/Hlr(I-I)=m[G/Hlr(I-I)

+ m lG/Hlr(I-I ) and hence an / - / -d i f f eomorph i sm

: P (res n A) ~ P (m[ G/H[r (H) + m l G/HIr (H)).

We choose �9 so tha t the points of P ( 2 m l ~ ) remain fixed; i.e., points of the fo rm

[z~: ...: z2,,: 0: . . . : 0: zl~l+ t �9 . . . : zl~i+~: 0: . . . : 0].

The c o m p o n e n t s of P(A) n conta in ing these points is denoted P(A)~. The results of Sect. 2 show us tha t �9 sets up a bijection between the c o m p o n e n t s of P(A) n and g [ P(2m[G/H[O~). (For instance, P(A)~ cor responds to P(2m[G/Hlln).)

q,~eft Iftl- 1

So, we write P(A)n= LI P(A)~. T o simplify nota t ion, we let q~r denote (t/b.~a)g) n ~ = 0

and let cog denote (co IP<A)g) u. We can now present Condi t ion ( .) which places restrict ions on co and 4'_+.

Condit ion ( . ) : a) Let H~_G and let p~P(A)~cP(A) n. If c ~ 0 , then 1H is not a sub H-

represen ta t ion of 4'_+ ]p. b) Given H___ G, the Ar f invar iant c(co~ r) vanishes. c) The real fiber d imens ion of ~'+ < 4 m. We can n o w give the s ta tement and p r o o f of T h e o r e m 1.

Theorem 1. Let G be an odd order group and let A=2mr(G) with m>2. Let

rI~KO~(P(A)) be such that t / = r e s ~ / ' for some q'6KO~• with JOa• and suppose that t 1 satisfies Condition (*). 7hen, there exists a G-homotopy q2P", X,, and a G-homotopy equivalence F: X , ~ P ( A ) such that TXn = F* (TY+ tl). /-/ere, n = 2 m f G I -- 1.

Proof Given q EKO~(P(A)) satisfying the required propert ies , we shall first construct a G-norma l m a p (X,f, b) with target manifo ld P(A). Our set-up shall

142 M. Hughes

be such that all the surgery obstructions vanish, enabling us to construct (X,, F, B) with F: X , ~ P(A) a G-homotopy equivalence, and TX,= F*(TP(A) +~).

To build the G-normal map (X,f, b) we shall first build a G xZ2-normal map also denoted (X,f, b). Restricting this normal map to G will provide us with a G-normal map endowed with an extra bit of symmetry which will be used to ensure that the surgery obstructions vanish.

The G x 2gz-normal map is to be constructed from the G x Ze-fiber homotopy equivalence co': ~'+ ~ ~'_. We must verify the four conditions delineated in Sect. 3.

Let's first determine the (G x 292) p = Gv isotropy representations {TP(A)Ip: peP(A)}. (Recall that 292= {1, ~o} acts freely.) Take peP(A) and set H = G v. Then, there is an ~ e {0 . . . . . I /~[ -1} such that p e P(A)ff, the component corresponding to P(2m[G/H[~9~) under q~. Since ~b is an H-diffeomorphism, we can use Proposition 2-3 to see that, as an H-representation,

TP(A) Iv = TP(resn A) l; = TP(2mlG/HIr(H))[o{p)

=(2mlG/Hl -1 ) ln+ ~ 2 m [ G / H l ~ l +~2mlG/Ulnt~Ot~ 1, o ~d~

where na = dirn~0a. Then,

TP(A) Ip=(2mlG/HI- 1)ln+ 2m[G/HIp(H),

where p(H)=r(H)- ln . (This uses the fact that if 0a is an irreducible complex H-representation, then 0p | ~ - 1 will be also.) We can also see that given H_~ G and p e P (A) n, TP (A) I p = (2 m ] G/H ] - 1) 1 et + 2 m [ G/H I P (H) as an H-representation. Note that this is independent of which component of P(A) n that p lies in.

Now, let's verify the four conditions needed to build our G • 2g2-normal map.

a) P(A) is a smooth, closed G xZ2-manifold. Given Helso(P(A)) and P (A)ff ___ P (A) t4, we have that P (A)f is diffeomorphic to P (2 m[ G/H [ ~ ) = IF. P"~, where nH=2m[G/H[-1. Then n0(P(A)~)=0 and d imP(A)~>5 as m__>2. It remains to show:

Claim 5.1. P(A) satisfies the Gap Hypothesis. (Definition 3.1)

Proof. First of all, it's clear that, given H ~ G, each fixed point set component P (A)n~ ~_ P (A) u has dimension 2 (2 m [ G/H [ - 1) = 4 m [ G/H[ -- 2. So, all (H-)components have the same dimension. Now, take L~H. Of course, the dimension of each component of P(A) L is 4m[G/L[--2. To show that dimP(A) L <�89 n, we need 4m[G/LI-2<�89 So, we

need IG/L[<llG/H]+ 1. Now, L ~ H ~ I G / L [ = ]G/HI and I L / H [ > 2 ~ � 8 9 4 m" [ L/H [ =

1 1 = > [L/H]" Therefore, [ G/L] < �89 [ G/HI < �89 [ G/H[ + 4mm as required. (Note that since

2g 2 acts freely, this is all we need to show.) Q.E.D.


b) Note that Iso(P(A)) = {all subgroups of G}. Indeed, let H_~ G and consider a component P ( A ) ~ _ P ( A ) n. If there is no point peP(A)~ with isotropy group H, then K = 0 Gp strictly contains H and we have P(A)y=P(A)g for some

pEP(A)~

/L But this is impossible due to dimension considerations. (See Proof of Claim 5.1.)

Clearly, 2 2 acts freely on 4'+, so every element of Iso(~'+) must be a subgroup of G. Hence, Is o (~'+) c Iso (P (A)).

c) We must show that TP(A)+4 ' is a stable bundle over P(A). Therefore, given peP(A) with (G x 2g2)p=Gp=H, we must show that m l~(TP(A)+4'Iv) < d o mo(TP(A ) + ~' [p) for each real H-representation ~b eS(TP(A) + 4' [p). Since these calculations are based on real representations, we consider the realification

(TP (A) + 4' Ip)~ = ((2 raiN~HI- 1) 1 n + 2 m IG/HIp (H) + 4'+ 1 , - _~'-Iv)~

= (4 m l G/HI - 2) lr~ + 4 m[ G/HI PR (H) + (4'+ I , )~- (4'-I,)~,

where p~(H)=y'n~,O (the sum being over all non-trivial irreducible real H- representations) and n o = k dimR0. We have used the fact that (p (H,))~ = 2pR(H). Clearly, the inequality holds for ~b = 1~. For ~b :t: lr,, note that n o > 1 and d o = 2 since [G I is odd (see [Se] P. 108). So, we must check that

4 m [ G/HI - 2 < 8 m n e I G/HI + 2 m O ((4+ Iv)R) -- 2 m o ((42 [p)~),

or, equivalently,

O<--_4mno]G/H] + 2me ((4'+ Iv)a)-- 2me ((~2 Ip)~) + 2.

Now, Condition (.) tells us that

dim ((~; Iv)~0 < 4 m

and therefore that m e ((~;]p)~) < 2 m. Thus 2 m e ((42 Iv)~) < 4 m and the required inequality always holds.

d) It remains to show that the transversality condition is satisfied. Note that since there is a G x 2g2-fiber homotopy equivalence between 4'+ and ~'_, we must have that ml~(~'lp)=0 for all peP(A). Note also that OeS(4;[p) implies that O ~ S (TP (A) + 4' I,), because m O (TP (A) + 4'+ I p) --> 4 m n 0, while m e (~2 Iv) < 2 m. (Here we are considering G,-representations. Our use of the regular representation ensures that all irreducible @-representations appear in TP (A)Ip-) Therefore, transversality follows from the stability of TP(A)+ 4'. Indeed, take H~_G and peP(A)~c_P(A) n. If 0eS(~LIp), then OeS(TP(A)-t-~'Ip) and then by stability, m~ (TP(A) + 4' 1~) < do me(TP(A) + ~' I~)- But this means that

m ~ (TP(A)[p) = ma~(TP(A)+ 4' ]p)

<= d e m e ( TP (A) + ~' Iv) --< de me ( TP (A) + ~' ]p) + d o - 1,

thus establishing the transversality condition. So, we can move c9' to a G x7Z2-map h which is transverse to P(A) allowing us to construct the G x Z2-manifold X as indicated.

Therefore, we can construct our G-normal map (X,f , b) with target P(A).

Remark 5.2. Let H ~_ G. Note that part a) of Condition (*) implies that, if e ~= 0, H fixes only the zero sections of 4'_+[P(m~. So, the method of constructing G-

144 M. Hughes

normal maps allows us to assume that f ~ is a W(a)= GJH-diffeomorphism for all c~=t=0. To get the bijection no(Xn)--+no(P(A)n), we do 0-dimensional W(0)x2E2-surgery on (hn)-l(p(A)g)=xg. By doing 1-dimensional W(0) x Zz-surgery on x g we get the required simple connectivity. (To see why P(A)g and x g admit 2Ez-actions see Claim 5.3.)

Now we proceed to consider the equivariant surgery obstructions associated to our G-normal map. (It is important to keep in mind that we are attempting G-surgery as opposed to G x 2gz-surgery. ) Given H ~ G, we need to show that aH(f)=I~au(f~) vanishes. Since f f i is a W(a)-diffeomorphism for c~:t=0 (Remark5.2), it suffices to show that a u ( f ~ ) = 0 . Recall that dimP(A)g =4m[ G/H[-2 = 2 n n and s o a H ( f ~ ) ~ I ~ z n n ( I [ W ( O ) ] , 1).

Before doing stepwise surgery, we make the following observation:

Claim 5.3. Sign(W(0), x g ) = 0 for all H_~ G.

Proof The key fact used here is that if M is a G-manifold admitting a G- equivariant, orientation reversing diffeomorphism, then Sign(G,M)=0. As Remark 5.2 indicates, P(A)g and Xo u can be viewed as W(0)x~z-manifolds. Indeed, q0g=q0 ]P(A)g is a W(0)-equivariant, orientation reversing involution on P(A)g. (To see that ~0 leaves P(A)g invariant, take p l = [ l : 0 : ...: 0]~P(A)~. Then ~0(pl)= [0: ... : 0: - 1: 0: ... : 0] with - 1 in the ([ G[ + 1) th place. Clearly, qo(pa)eP(A)g, and so P(A)~oC~q)(P(A)g)+o. By G-equivariance, qo(P(A)g) =P(A)~ for some cc Since the components of P(A) H are disjoint, we have that

(p(P(A)g) = P(A)g.) Now, recall that tIeK'OG(P(A)) was chosen so that the involution q~ lifted to involutions ~ + : q + ~ q + in such a way that the G-fiber homotopy equivalence co: q + @_V--* t/_ @_V w a s 71 2 = { 1, 0_+ }-equivariant. The G-equivariance of the lifted involutions implies that ~og lifts to qg in such a way that cog is W(0) x Z2-equivariant. Therefore, by construction, Xg also admits a W(0)- equivariant involution which can be shown to be orientation reversing using the 2~2-equivariance of f f . Our initial comment then implies that Sign(W(0), X0~)=0. Q.E.D.

Now, let's consider the surgery obstructions. (Note that G preserves orientation.) We first encounter o-G(f0~)~/)2,~(~[1], 1). This obstruction is detected (See [PR], Ch. 3.12) by the Arf invariant c(coo ~) which vanishes by Condition (,).

Let H be a subgroup which is maximal with respect to the following property: By doing surgery, we have converted (X,f b) to a G-normal map (X',f', b') such that (f,)L: (x,)L~p(A)L is a homotopy equivalence for all L~H. Then, as explained above, a n ( f ) = an(fdr~) is defined. Using the exact sequence argument outlined in Sect. 3, the following three steps show that an(fdn)=0. Let zc denote W(0).

a) c~(an(f~t~)) is a multiple of 2. At this point, we consider the n-normal map (X'oU, fd ~, b'o') with target manifold P(A)g. Now, n acts on P(A)g with singular set (P(A)g) ~ which is a union of elements of ILl no(P(A)L). According

L ~ H

to the inductive assumption, we have that (fdtr)~: (X~n)~ ~(p(A)g)~ is a ~-homotopy equivalence. Thus, this n-normal map is adjusted in the sense of Sect. 4.


This means that the results of Dovermann and Dovermann-Rothenberg apply. Notice that if Y is a fixed point component of the zc-action on P(A)~, then y _ ~ p , L , for some L ~ H . (As before, nL=2mIG/L]--l .) In particular, every such component has even Euler characteristic. In addition, ~z has odd order and hence Proposition 4.2 implies that e~(o-H(fd~/)) is a multiple of 2 as desired.

b) Sign(To, X~z)= Sign(n, P(A)g). Since the middle dimensional cohomology of P(A)~ vanishes, we have that Sign(re, P(A)g)= 0. Now, X~) ~ is obtained from Xo ~ by ~-surgery and the involution on X~ may not be preserved in the process. However, the equivariant signature does not change under surgery as it is a G-normal cobordism invariant. Therefore, Sign(re, X~ z) = Sign(re, X0 ~) = 0 by Claim 5.3.

c) c(co~)=O by Condition (*). Therefore, a /d fal l)=0. As this argument holds for any subgroup of G, we

have that all the surgery obstructions vanish. So, we can construct a G-normal map (X,, F,B), where F: X , ~ P ( A ) is a G-homotopy equivalence and TX, =F*(TP(A)+tl) .


The goal of this section is to prove:

Theorem 2. Every odd order group acts smoothly and effectively on infinitely many differentiably distinct homotopy (E P"'s, for some n.

Actually, given G and m > 2, we shall construct an infinite number of differentiably distinct G-homotopy tEP"'s, with n = 2 m[ G[ - 1. We shall prove Theorem 2

by finding an infinite number of elements {th}~KO~(P(A)) which satisfy the hypotheses of Theorem 1. They shall be chosen so that, by calculating Pontryagin classes, we can show that the resulting G-homotopy II;P"'s, {X,~}, are pairwise non-diffeomorphic.

At this point, we would like to present an Sl-map which will be very important in our constructions. (See [MeP], P. 74) Let p, q be relatively prime integers and a, b integers such that - a p + b q = l . Let t ~ be the 1-dimensional S 1-representation where t~S 1 acts on IE by t-z = tlz (complex multiplication). Define f : tP+tq---~t+t pq by f ( z l , -a b Z2)=(ZlZz, Z~+Z~). It can be shown that f is a proper Sl-map such that the degree o f f + is 1, where f + is the extension of f to 1-point compactifications.

Proof of Theorem 2. Let (~, p, q) be a triple of integers such that: a) (p, q) = 1 ; b) y, p and q are prime to I G [; and c) 7 is even.

Consider the G-vector bundles r l+=S(A| Xs,(t~p+t~q) and q_ =S(A | t )x sl(t~+t~Pq). Note that the above S~-map gives an S~-map f : V+ = t ~p + t ~q ~ V_ = t ~ + t ~pq. As indicated in Sect. 2, f induces a G-map f : t/+ ~ q _. We set co = f and note that since 7, P, and q are prime to ]G], co is a G-fiber homotopy equivalence. (See [PR], Ch. 3.12, or [MP].) So, we have ~/~,p,q=q+

--t l_ ~KOa(P(A)) with JOG(tl~,p,q)= O. r ~ A Next, we need to show that rl~,p,q=resGq~,p,q, for some tl~,p,qeKOr215

with JOG • z~ (t/'~,v,q) = 0. Define q)': S (A | t) ~ S (A | t) to be the obvious G-map

146 M. Hughes

covering the involution q~: P ( A ) ~ P ( A ) . Notice that ~• q_+--+q_+ defined by ~• [a, (vl, v2)] = [~0'(a), (vl, v2)] are involutions which cover ~0 and make tl+_ into G x Z2-vector bundles. (Choosing 7 to be even ensures that ~5~ = 1.) It is easy to check that coo~+ =(~_ ~ Hence, we see that q) lifts to ~,p,q in such a way that co can be viewed as a G x ~g2-fiber homotopy equivalence.

It remains to check that Condition (.) is satisfied. To check part a), we first take H _ G and recall our H-diffeomorphism q~: P(res uA)

P (2 m] G/H[ r (H)). Defining

~ • : ~• =S(resnA) x sl V• -~ 0• =S(2m[a/HIr(H)) x sl V•

by ~• v]=l-~b(a), v], we obtain further H-diffeomorphisms. Now, given y s P (A) n, we have �9 (y) e P(2 m I G/H I O~) for some ~. The H-equivariance of ~ + implies that ~• as/-/-representations. So, we need to know the H- representations 0 • I~ for x e P (2 m I G/H [ 0~) with ~ + 0 (i.e., ~ + 1 n). A calculation shows that fl+l~=(O~)~P+(0~)~q and 0 _ [ x = ( 0 ~ ) ' + ( 0 J pq. The fact that 7, P, and q are prime to the order of G implies that lt~ does not occur as a sub-representation of either of these representations, hence demonstrating the validity of part a).

Part b) of Condition (.) follows from a result due to Masuda [M2] and independently to Schultz I-S]. It shows that the fiber homotopy equivalences that we are considering have vanishing Arf invariant if ~ is even.

Part c) of Condition (.) follows immediately, since the real fiber dimension of q_+ is 4 and m> 2. (Notice that we have not needed to stabilize in order to get a G-fiber homotopy equivalence.)

Therefore, Theorem 1 allows us to construct X, ..... and F: X~ ..... ~ P ( A ) , a G-homotopy equivalence such that TX, ..... = F* (TP(A) + tl~,p,q ).

Let's use our knowledge of the tangent bundle TX, ..... to compute the Pon- tryagin class P(X , ..... ). Let c~ be a generator of I-I2(X, ..... ). Using the facts that S (A | t) x sl t b = (S (A | t) x sl t) b and that P1 (S (A | t) x s~ t) = ~2, we compute:

P(q+)=(1 -I-72p2c~a)(1-}-y2q2~2) and P(t /_)=(1 + 7 2 ~ 2 ) ( 1 - t - 7 2 p a q 2 ~ 2 ) .

Now, since P(X. ..... )- P (t/_) = P (P (A))- P (t/+), we get that

P(X . ..... )=( l + xZ)"+ a(1+ y2pZx2)(l + 72 qZx2)(l + TZ)- l (1+ TZp2 qZx2) -1

From this, we compute the first Pontryagin class,

P, (X. ..... ) = ((n + 1) + 7 2 (p2 _ 1)(q2 _ 1))X 2.

It is easy to see that by varying 7, P, and q, we obtain infinitely many differentiably distinct homotopy CP"'s.


This section is devoted to the proof of Theorem 3, which we now restate:

Theorem 3. Let G be a finite abelian group such that its 2-Sylow subgroup is either cyclic or 2g 2 G 7/2, or let G be a finite product of •2's. 7hen G acts smoothly and effectively on infinitely many differentiably distinct homotopy ll2 P"'s, for some n.


Proof. Let G be as above and let m > 2 . As before, we consider P(A), where A=2mr(G). We shall need to single out characters of order 2 and so we write

T I~l- 1 r ( G ) = 1 ~ + ~ hi+ ~ O~, where the hi's are the characters of order 2. Then

i = 1 ~ = T + I

Tis equal to the n u m b e r of e lements of G of order 2. We define the G-equivar iant , o r ien ta t ion reversing involut ion q~" P(A) --* P(A) as in T h e o r e m 1.

Once again, we shall use a G x 2gz-fiber h o m o t o p y equivalence to cons t ruc t a G x Z 2 - n o r m a l m a p (X,f , b) which u p o n restr ict ion will p rov ide us with a G-norma l map . This t ime we cons t ruc t a fiber h o m o t o p y equivalence which is a mult iple of 2. This feature, in conjunct ion with the involut ion qo, ensures tha t all surgery obs t ruc t ions vanish.

Let p and q be integers which are pr ime to ]G[ and have (p, q ) = 1. As in T h e o r e m 2, we cons t ruc t the G x 7Z2-ma p co: q + = S(A | t) x s : ( t2p-t- t 2 q ) ~ 17 = S(A | t) x s~ ( t2 + t2Pq)" (Here 7 = 2. Since it 's even, 9 lifts to 11 + mak ing o2 G x 2g2-equivariant. )

Given H _~ G, we have

P(A) '= P(2mI G/HIln)u I I P(2mI G/HIhOu L[ P(2mI G/H[ ~ )

= P(A)~ u UP(A)," u H P (A)~". i a

(In wha t follows, R o m a n letter subscripts will refer to characters of order 2, while G r e e k letter subscripts will refer to characters of higher order.) I f yEP(A)~uI_IP(A)`", then q+ [y=2-1 u as H-represen ta t ions , because (h i )2J=ln

i

for any j . As in T h e o r e m 2, ifyeP(A)~, then t/+ [y= 2p 2q ~ + ~ and ~/_ ]y = ~2 + tfi2pq. Since p and q are pr ime to ] G [ and the order of O, > 2 for all e, we have:

co u = o~ o u [ I {~ u L[ c%: r/+ ]Z(A)o~ U L[ r/+ Ie(a)ff w [21 P (A)~ n i a i

q -]etA)# W ]J[ ~/_ [P(A)f U I_I P (A)f . i

(Here we are viewing L[P(A)~ as a par t of the zero sections of tl+.) These

r emarks show us tha t co is a G x 2g2-fiber h o m o t o p y equivalence over P(A). Now, let 's set ~ + = 2 t/+ and f2 = 2 co. Then s ~ + ---, ~_ is also a G x 7/2-fiber

h o m o t o p y equivalence. I t is (2 f rom which we choose to build the G x 2gz-normal m a p (X,f, b). As with T h e o r e m 1, a check mus t be m a d e tha t the four condi t ions f rom Sect. 3 are satisfied. These details are similar in na ture to those presented in T h e o r e m 1 and are t h e r e b y omit ted.

Therefore, we can build our G-norma l m a p (X,f, b) with target P(A) as desired. We can m a k e an obse rva t ion similar to R e m a r k 5.2. This time, X g -=(h H)- l(P(A)no) and {X~=(ht~) - I (P(A)~)} are the G x •2-manifolds, while {Xy = (h H)- 1 (P(A)y)} are G-manifolds.

148 M. Hughes

Now, let's consider the obstructions to doing G-surgery. Given H__G_ G, recall that H acts trivially o n ~++]p(A)~tji]P(a)in, while on ~+-[Ue(Al".' only the zero section

i

is fixed. Therefore, as in Remark 5.2, we can assume that f ~ is a G/H-diffeomorphism for each c~. Hence, at each stage of the surgery constructions, we

0" H T ~ will only have to consider the obstructions an(fo u) and { n(fi )}i= 1, where T H is the number of elements of H of order 2. (Of course, T n can be 0.)

Proposition 7.1. Sign(G/H, Xo~)=0= Sign(G/H, X~), for all i and for all H~_ G. (Note that here W(O) = W(i) = G/H.)

Proof We have already seen that Sign(G/H, Xon)=0 in Claim 5.3. Recall that, given i, P(A)I u= P(2m]G/Hlhi), where hi has order 2 in/~. If O EG is such that resn~k has order 2 i n /1 , then resnO=resnO=(resnO)-l=resnO. This implies that qo restricts to P(A)~. Note that this restriction endows P(A)~ with a G/H- equivariant, orientation reversing involution, ~0~. As with Claim 5.3, q~ lifts to ~+]P(A)~" in such a way that s is G/Hx~z-equivariant. So, X~ admits a G/H2equi~cariant, orientation reversing involution and hence Sign(G/H, X~) =0. Q.E.D.

Now, the Arf invariant of two times a fiber homotopy equivalence vanishes ([PR], Ch. 3.12), so the first obstructions that we encounter aa(fo~), {a~(f/G)}T2~ T, all vanish. (We could also use the Masuda-Schultz result to see that the Arf invariant vanishes.)

Let H _ G be maximal with respect to the property that by doing surgery, we have converted ( X , f b) to (X',f ' , b') with (if)L: (X,)L~p(A)L a homotopy equivalence for all L2H. Then, the obstructions a tc,H~ , , each lying in I2',,~(Z[G/H], 1) are defined. Now, fix i t{0, 1 . . . . . Tn}.

a) c~am(a~(fi'n)) is a multiple of 2. Consider the G/H-normal map ,H tH ,H H (Xi ,fi , bi ) with target P(A)i �9 As in Theorem 1, this normal map is adjusted

and the results of Sect. 4 apply. Now, since we have started with a G-fiber homotopy equivalence f2 = 2 co: ~ + = 2 t/+ --. ~_ = 2 q_, Proposition 4.2 implies that m �9 aom(an(fl )) is a multiple of two (and hence vanishes).

b) Sign(G/H, X'in)= Sign(G/H, P(A)~). Both of these signatures vanish. This uses Claim 7.1 and the same argument which is found in the proof of Theorem 1.

c) c(O/~)=0. Again, we can either use the fact that twice a fiber homotopy equivalence has vanishing Arf invariant or use the Masuda-Schultz result.

Therefore, we see that all surgery obstructions vanish. Writing ~p,q= 4+

-~_eKOG(P(A)), we construct a G-normal map (Xr162 F, B), where F: Xr P(A) is a G-homotopy equivalence and TXr = F* (TP(A) + ~p,q). As in Theorem 2, we compute the first Pontryagin class of Xr to be

P1 (Xr = ((n + 1) + 8 (p2 -- 1)(q 2 -- 1))x 2,

where x ~ H 2 (X~p,q) is a generator and n = 2 m [ G [ - 1. Clearly, we can vary p and q so that, for any m>2 , we obtain infinitely

many differentiably distinct homotopy CW's. Q.E.D.


8. Isotropy Representations

In this section, we compare the actions constructed in previous sections with linear actions on CP". This comparison is carried out by an examination of the isotropy representations induced by the action.

Definition 8.1. Let G be a finite group and let X be a homotopy IEP n with G-action. We say that X is isotropy linear if there exists a complex G-representation B such that for all/-/___ G, the sets of real/-/-representations {TX ]p: p EX ~} and {(TP(B)I~)R: qeP(B) n} are equal. (Note that these sets are finite as the H fixed set has finitely many components.)

Notice that all of the actions we have constructed are G-homotopy linear in that each of our homotopy IEP"'s is G-homotopy equivalent to P(A).

Theorem 4. Assume that either: a) [ G] is odd and G is not of exponent 3, or b) G is finite abelian with 2-Sylow subgroup equal to Z2n , for some n >= 4. Then we can construct infinitely many differentiably distinct G-homotopy C P"'s

which are not isotropy linear.

Proof. First consider case a) and suppose that [G[ has a prime factor r>5 . Cauchy's Theorem tells us that there is an element heG of order r. Let H = ( h ) ~2g r. We construct the G-homotopy linear IEP~'s exactly as in Theorem 2 with one difference. That is, we choose 7, P, and q subject to the previous conditions and further stipulate that p, q~g ___1 modr. (We can still construct infinitely many C P"'s; consider Dirichlet's Theorem, for instance.) Let X = X,,,p,q be a G-homotopy linear II2P ~ so constructed.

Suppose that X is isotropy linear with linear model P(B). With H as above, note that given x E X n, we have

TXIx=(4mIG/HI--2) I~+4mIG/I-1Ip~(H)+(7+ - 7 - [ r )R ,

where y=F(x) and F: X ~P(A) is the G-homotopy equivalence constructed above. (See Sect. 5.) This implies that the multiplicity of 1R in TP(B)lz is the same for all z~P(B) n and hence that each component of P(B) n has dimension 2nH=4mIG/H]-2. This in turn implies that resn B = ~ (nH+l)~kj

=2mIG/H[r(H) (since dimcB=2mlGI). Hence, given zeP(B) It, o:n

(TP (B) 1~)~ = (4 m [ G/H [ -- 2) 1R + 4 m l G/HI Pr~ (H).

Therefore, to show that X is not isotropy linear, it suffices to find a point y~P(A) n with (7+ -7-1y)R~e0 in RO(H). Take any y~P(A)ff, with c~=0, then

implies that 0~Y=O~ p and/or ~kf~=O~ ~ Since O~e/~, a cyclic group of order r, this happens

,* r and/or 7q=-+_ymodr~:~p--t- lmodr and/or

q - _+ 1 rood r

150 M. Hughes

(since (~, r )= 1). However, this is contrary to our assumption, and we see that indeed X is not isotropy linear.

I f [G[ does not have a prime factor >5, then G is a 3-group. Since it's not of exponent 3, there must be an element g of order > 9. Say that g has order 3 k for some k>2 . Then H = <g> -----2g3k is a subgroup of G. Let O~e/t be of order 3 k. As before, construct G-homotopy (EPn's, choosing 7, P, q as in Theorem 2 and such that p, q ~ ___lmod3 k. Taking y~P(A)f, (rl+-~l_ly)a is easily seen to be non-vanishing in RO(H) implying, as above, that X is not isotropy linear.

Next, we consider case b). Suppose that G contains H=~E2~, with n>4 . We choose p and q as in Theorem 3, subject to the further condition that p, q ~ +1 mod2 n-1. Take yeP(A)n~=P(2m[G/H[~9~,), where ~O~e/1 has order 2L Then

2p 2q 2 (,7+-r#_l,)~=(g,~ + ~ - 0 ~ - 0 ~ ' % + 0 in RO(H) because 2p, 2 q ~ +2mod2" . The arguments in case a) show that the G-homotopy ll2Pn's built using Theorem 3 with p and q as above will be non-isotropy linear. Q.E.D.

References

[A] [As]

[BI]

[B2]

[D] [DM] [DMS]

[DMSu]

[DR]

[Ha] [JS] [M1]

[M 2]

[Me]

[MT]

[MeP] I-P1]

Atiyah, M.F.: K-Theory. New York: W.A. Benjamin, Inc. 1967 Atiyah, M.F., Singer, I.M.: The Index of Elliptic Operators III. Ann. Math. 87, 546-604 (1968) Bak, A.: The Involution on Whitehead Torsion. General Topology and its Applications 7, 201-206 (1977) Bak, A.: The Computation of Surgery Groups of Finite Groups with 2-Hyperelementary Subgroups. Lecture Notes in Math. vol. 551, pp. 384-409. Berlin Heidelberg New York: Springer 1976 Dovermann, K.H.: Almost Isovariant Normal Maps. Preprint 1986 Dovermann, K.H., Masuda, M.: Preprint 1985 Dovermann, K.H., Masuda, M., Schultz, R.: Conjugation involutions on homotopy complex projective spaces. Jap. J. Math. 12, 1-35 (1986) Dovermann, K.H., Masuda, M., Suh, D.Y.: Rigid Versus Non-Rigid Cyclic Actions. In preparation Dovermann, K.H., Rothenberg, M.: The Generalized Whitehead Torsion of a G-Homo- topy Equivalence. Preprint 1986 Hattori, A.: Spin c Structures and S 1 Actions. Invent. Math. 48, 7-31 (1978) James, I.M., Segal, G.B.: On Equivariant Homotopy Type. Topology 17, 267-272 (1978) Masuda, M.: 7Z, 2 Surgery Theory and Smooth Involutions on Homotopy P(ll2n). Transfor- mation Groups, Poznan 1985, Springer Lecture Notes in Math., vol. 1217, pp, 258-89. Berlin Heidelberg New York : Springer 1986 Masuda, M.: The Kervaire Invariant of Some Fiber Homotopy Equivalences. Adv. Studies in Pure Math. 6. Amsterdam: Kinokuniya North Holland Masuda, M., Petrie, T.: Lectures on Transformation Groups and Smith Equivalence. AMS Contemporary Math. vol. 36, 1985 Masuda, M., Tsai, Y.D.: Tangential Representations of Cyclic Group Actions on Homo- topy Complex Projective Spaces. Osaka J. Math. 22, 907-919 (1985) Meyerhoff, A., Petrie, T.: Quasi-Equivalence of G-Modules. Topology 15, 69-75 (1976) Petrie, T.: Involutions on Homotopy Complex Projective Spaces and Related Topics. Proceedings of the Second Conference on Transformation Groups, Part 1, Lecture Notes in Math. vol. 298. Berlin Heidelberg New York: Springer 1972


Ep2]

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[Su] IT] [Wl] [W2]

Petrie, T.: Smooth S 1 Actions on Homotopy Complex Projective Spaces and Related Topics. Bull. AMS 78, 105-153 (1972) Petrie, T.: Pseudo-equivalences of G-Manifolds. Proc. Syrup. Pure Math. 32, 169 210(1978) Petrie, T., Randall, J.: Transformation Groups on Manifolds. Dekker Lecture Series vol. 48. New York Basel: Marcel Dekker Inc. 1984 Schultz, R.: Private communication to Y.D. Tsai, 1984 Sampaio, J.C.V.: Dissertation, Rutgers University 1987 Serre, J.P.: Linear Representations of Finite Groups, Grad. Texts in Math. vol. 42. Berlin Heidelberg New York: Springer 1982 Shaneson, J.: Wall's Surgery Obstruction Groups for G x Tl. Ann. Math. 90, 296-334 (1969) Sullivan, D.: Triangulating Homotopy Equivalences, Dissertation, Princeton University Tsai, Y.D. : Dissertation, Rutgers University 1985 Wall, C.T.C.: Surgery on Compact Manifolds. New York London: Academic Press 1970 Wall, C.T.C.: Norms of Units in Group Rings. Proc. Lond. Math. Soc. (3), 29, 593-632 (1974)

Received October 19, 1987

Finite group actions on homotopy complex projective spaces

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