, tome 62 (1985), p. 5-40

PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S.

TED PETRIE

JOHN RANDALLSpherical isotropy representations

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SPHERICAL ISOTROPY REPRESENTATIONSby TED PETRIE and JOHN RANDALL

i. History and discussion of ideas

An old question of P. A. Smith asks [Sm]: If a finite group acts smoothly on aclosed homotopy sphere S with fixed set S° consisting of two points p and q, are theisotropy representations Ty S and Tg2ofGat /» and q equal? Put another way: Describethe representations (V, W) of G which occur as (Tp S, T^ 2) for S a sphere with smoothaction of G and S0 == p u q. Under these conditions we say that V and W are Smithequivalent [P] and write V ^ W. Prior to the results of this paper all evidence suggesteda positive answer to Smith's question. We show here that the answer to Smith's questionis no. Specifically suppose G is an odd order abelian group with at least 4 non cyclicSylow subgroups. Then (Theorem A') there is a nontrivial subgroup (1.7) rI'(G) ofthe real representation ring RO(G) such that each z erI'(G) occurs as a differenceof two Smith equivalent representations. Even more generally there is an action of Gon a homotopy sphere S such that S° consists of an arbitrary number of points andthe pairwise differences of the isotropy representations realize given elements of rI'(G)(Theorem A). This paper presents proofs and elaboration of the results announcedin [PJ and [P^].

The question of Smith was undoubtedly motivated by the observation that ifS == S(V) is the unit sphere of a representation V, then T2 , the G tangent bundle

s0

of S restricted to S°, is S0 x V. Here V is V/V0. In particular this means thatthe isotropy representations T^ S for x in S6 are independent of x. One might thenwonder about the case where S is an arbitrary homotopy sphere with G action suchthat S° is a homotopy sphere. Smith's question deals with the case in which S° isthe o dimensional sphere. For interesting results for the positive dimensional case seeSchultz [ScJ and Ewing [E].

The problem of characterizing Smith equivalent representations has a rich historywhich we mention to motivate the material here. The first major breakthroughs onthis topic are due to Atiyah-Bott [AB] and Milnor [M] and rest on the Atiyah-Bott

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6 TED P E T R I E A N D J O H N R A N D A L L

fixed point theorem. The set ofisotropy groups {GJ A" e X} o faG space X is denotedby Iso(X).

Theorem 1.1. (Atiyah-Bott [AB]). — Let V be a representation of a finite group Gsuch that Iso(V — o) == {i}. If W is a representation of G with V - W, then V == W.

One important consequence of their proof is this:

Lemma i. a. — If G has odd order and V and W are Smith equivalent representations of G,then V-WeKer(RO(G)) -fl^ II RO(P)). {Here y(G) denotes the set of SylowPG^(G) l/ -'subgroups of G and a ij <z product homomorphism each of whose components is restriction to P.)

For even order groups there is a weaker result:

Theorem i. 3 (Bredon [BJ). — If G is cyclic of 2 power order and V - W, then W — Vis contained in 2^ RO(G) where f(V) is an explicit power of 2 which increases as dimensionof V increases.

Corollary 1.4. — Same hypothesis. If dim V is large compared with the order | G | of G,then V - W implies V == W.

This corollary indicates some of the complexity of Smith equivalence. Forexample V - W does not imply V ® S - W ® S. The next result when comparedwith the main results here shows the subtlety of the notion of Smith equivalence.

Theorem i. 5 (Sanchez [S]). — If | G | is odd and V and W are Smith equivalent represen-tations realized as Tp S and Ty S/or an action of G on a homotopy sphere S such that S11 is eitherconnected or p u q for all H C G, then V = W.

In order to state the main results here which deal with Smith equivalence wefix some notation. In this paper

1.6. — G is an odd order abelian group.

Let R(G) (resp. RO(G)) denote the complex (resp. real) representation ringof G. If H is a subgroup of G, resg : R(G) ->• R(H) is the restriction homomorphismand fixg: R(G) -> R(G/H) is the homomorphism defined by sending a represen-tation V to V11 its H fixed set. Realification defines a homomorphism r: R(G) -> RO(G).Let y denote the set of groups of prime power order.

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Define subgroups

(1.7) I(G) =Ker(R(G)^j[^R(P))

I'(G) = Ker(R(G) -^ ( n R(P) X n R(G/H)))PG^(G) G/HG^

where a is the product of resp for P e <^(G) and p is the product of resp for P e <9"(G)and fixn for G/H e^. The corresponding subgroups ofRO(G) are denoted by 10 (G)and 10'(G). One of the main results here about Smith equivalence is:

Theorem A'. — Let G be an odd order abelian group having at least 4 non cyclic Sylowsubgroups. Then every element ofrV{G) occurs as the difference of Smith equivalent representations.{Compare with 1.2.)

This theorem is a consequence of Theorem A and Lemma 2.6. In 2.5 we definea set St of complex representations of G.

Theorem A. — Lei K be any non empty finite set. Suppose for each u, v e K that R^e^Sand R^ — R,, e I(G). Then there is a closed homotopy sphere S with smooth action of G suchthat S°=K and {T^S | u e K} = {rRJ u e K}.

Theorem A' gives a sufficient condition to realize an element of RO(G) as adifference of two Smith equivalent representations. This condition involves the sub-group rP(G) of 10'(G). The latter is naturally related to necessary conditions (2.7and subsequent remarks). For a large family y of representations of G, Theorem A'together with Lemmas 2.7 and 2.8 lead to a necessary and sufficient condition,Theorem B', for representations in y to be stably Smith equivalent. (By definition,representations V and W are stably Smith equivalent if there is a representation S suchthat V © S and W®S are Smith equivalent.) Define y as follows: A represen-tation VofGisin^ if and only if dim V11 == o for each subgroup H for which G/H e 3P.

Theorem B'f. — A necessary and sufficient condition that two representations V, W e yhe stably Smith equivalent is that V — W e 10'(G).

Theorem B. — Let K be any non empty finite set. Suppose for each u e K, R^ is arepresentation in y. Then there exists a representation S of G and a smooth G action on a closedhomotopy sphere S such that 2° = K and {T^ S | u e K} = {R^® S | u e K} ifR^ - R, e IO'(G) for all u, v e K.

Using Theorems A' and B' it is easy to exhibit non isomorphic Smith equivalentrepresentations. Here is a representative example: Let L be the cyclic group of order pqwhere p and q are distinct odd primes. View L as the group of pq-th roots of unityin C and let f denote the complex one dimensional representation of L defined byasserting that g e L acts on v e f' by complex multiplication by g\ The represen-

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tations V = ^ (p+l ) (3+l )®^ and W == ^(P+ê^-^ of L where {a,pq) == i arenot stably Smith equivalent as representations of L; however, if G is a group satisfyingthe conditions of Theorem A' and 9 : G -> L is a surjective homomorphism, thenr<p* V and r<p* W are stably Smith equivalent representations of G. This follows imme-diately from Theorem A' because <p*(V - W) e I'(G) $ so r<p*(V - W) e IO'(G).

There is an interesting problem associated with Smith's questions which is nottreated here: namely, to relate the differential structure of S and the isotropy represen-tations T^ S for x e S° when S is a homotopy sphere. As noted by Schultz and othersthe results of this paper can be achieved on the standard sphere. See section 5.

In the long period since the discovery of Theorem A (see [Pa]) and the publicationof its proof together with the other results here, there have been a number of interestingpapers published on the topic of Smith equivalence. Treating the case of cyclic groupsof even order are papers of Gappell-Shaneson, Petrie, Dovermann and Siegel: see [CS],[Py], [Dov] and [Si], For noncyclic groups of even order, there are papers of Suhand Gho. See [Suh] and [C], A forthcoming paper of Dovermann-Petrie [DPg]treats the case of cyclic groups of odd order using many of the geometric methods ofthis paper as well as methods particular to cyclic groups of odd order. A good dealof [PR] is devoted to Smith equivalence.

There are three general ingredients to the proof of Theorem A. These are: theone fixed point actions of G on homotopy spheres of [P^, the Completion Theoremof Atiyah [AJ and the Induction Theorem in equivariant surgery given in section four.The main result of ^4] produces for each R e 3i a smooth action of G on a homotopysphere X with exactly one fixed point u such that T^ X = R and the equivarianttangent bundle TX is stably isomorphic to X x R. Lemma 1.2 ties together withanother result of Atiyah [AJ which asserts that I(G) is the kernel of the completionhomomorphism R(G) -> R(G) from R(G) to the completed representation ring R(G).This may be interpreted geometrically. Let E be an acyclic space on which G actsfreely. Then any representation R of G gives a G vector bundle E X R over E. Theabove result of Atiyah geometrically interpreted means that if R — R' eI(G), thenE X R and E X R' are stably isomorphic G vector bundles over E.

As an approximation to the G homotopy sphere S of Theorem A consider themanifold

w = u wûeK'

described as follows: Add a point o to K to give the set K/. Select any point z e Kand set T == R,. For u e K, W^ is a G manifold X^ with X^ == u, T^ X^ = Rând TXy = X^ X R^ as a stable G vector bundle. For u == o, W^ is a G manifoldwith W^ = 0, TW^ == W^ X T as a stable G vector bundle and the Euler charac-teristics of fixed sets W^ for H C G are arranged so that condition 4.18 (i) of the Induc-tion Theorem 4.19 is satisfied. Let R be the real line with trivial action of G and let

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Z = S(T C R) be the unit sphere of T ® R. There is an equivariant map F : W -> Zwhich collapses the complement of an invariant disk about z G W to a point. It hasdegree i. Since R,, — R^ e I(G) for u, v e K', Atiyah's result gives a stable G vectorbundle isomorphism

(B^ : E x TW -^> E x (W x T)

of G vector bundles over E X W. Then IT == (W, F, p^) is roughly what is neededto produce the sphere S in Theorem A. Observe that W° = K and Ty W == Ryfor u e K. If F were a homotopy equivalence, Theorem A would be established. Thegoal is to convert W to a G homotopy sphere S using H^ and without destroying theseside conditions.

The obvious tool for converting F to a homotopy equivalence is equivariantsurgery. The setting for this is a triple H^ == (W, F, (3) where F : W -> Z is anequivariant map of degree i and (3 : TW —^ F* ^ is a " bundle " isomorphism for someequivariant vector bundle ^ over Z. The triple /)^ is called a G normal map. Equi-variant surgery is a process designed to produce a G normal cobordism between H^and H ^ ' = (W, F', (B') where F' : W -> Z is a homotopy equivalence. This is notalways possible as there are obstructions. One powerful tool which guarantees successis an Induction Theorem which asserts that if for each hyperelementary subgroup Hof G resji^ is the boundary of some H normal map ^(H), then is G normallycobordant to H ^ ' == (W, F', (B') with (B' a homotopy equivalence.

The first such Induction Theorem, due to Dress, applies to free actions. Its chiefgeometric application was the construction of free actions on homotopy spheres [D],Induction Theorems for non free actions were established in [DPJ and [Pg] and wereemployed in the construction of one fixed point actions on homotopy spheres [Pg] [P^\.These induction theorems deal with G normal maps and depend therefore on thedefinition of " bundle " isomorphism occurring in the definition of G normal map. Inthe free case there is little ambiguity. " Bundle isomorphism " means a stable G vectorbundle isomorphism. In the case of non free actions the role of the " bundle " iso-morphism is far more substantial and the definition depends on the contemplatedapplication. (The idea of a flexible notion of " bundle " equivalence is dealt within [PR, pp. 91-95].) In the cases cited above TW is stably G isomorphic to F*(^) forsome G vector bundle ^ over Z. This definition must be altered to treat Theorem Aas the following result shows.

Lemma C. — Let S be given by Theorem A. Let z e K and F : S -^ S(R © R) = Z,(R = T^ S), be any G map. If there is a G vector bundle ^ over Z such that TS = F* ^ asa stable G vector bundle, then Ty S = T,, 2 for u,ve 2°.

Proof. — Since R e , dim R11 > o for each cyclic subgroup H of G by 2.3and 2.5. This means that Z11 is connected; so if^ is any G vector bundle over Z, then

/irt /•2252

io TED P E T R I E A N D J O H N R A N D A L L

the representation of H on Sa, (the fiber at x) is independent of x e Z11. If TS = F* S;,then for u,v e K = S0 res^T,, S - T, S) == res ' — ^) == o, where u' = F(^) andv ' = F(y). Since this holds for each cyclic subgroup of G, T^S = T,, S.

To apply these considerations then to prove Theorem A, we need a definitionof <( bundle isomorphism " for (3 : TW -> F* i; which is weaker than those previouslyused in equivariant surgery. One notion which works is called a Smith framing. Itincorporates the isomorphism (B^ constructed from Atiyah's Theorem and the P vectorbundle isomorphisms (3p : TW -> W X R for P e e^(G) arising from the isomorphismresp Ry = resp R. The definition and formal properties of Smith framings are treatedin section 3.

The proof of the Induction Theorem 4.19 separates into an algebraic part anda geometric part. The geometric part consists in establishing two fundamental lemmas(4.11 and 4.12) from equivariant surgery (see [PR, Ch. 3, § io]). They assert thatequivariant surgery is possible (with respect to the definition of G normal map usingSmith framing). These two results are useful in their own right as they are the keygeometric steps in producing actions on disks with isolated fixed points and distinctisotropy representations—the first such [Pg], The algebraic part of the proof of theInduction Theorem 3.19 and [DP^, 2.6] are identical; so we appeal to [DPi, 2.6]to complete the proof.

In section 2 we treat some algebraic preliminaries concerning the representationring and the Burnside ring. In section 3 we develop the notion of Smith framing. Insection 4 we treat the Induction Theorem. In section 5 we prove the main resultsby constructing a G normal map from the input from Theorem A. This normal mapsatisfies the requirements of the Induction Theorem which is applied to prove Theorem A.Theorems B and B' are also proved in section 5. Theorem A and Lemma 2.8 leadto Theorem B, while Theorem B' follows from Theorem B and Lemma 2.7.

2. The representation ring and the Bumside ring

If H is a subgroup of G, resg denotes restriction of G data to H data. E.g. if Xis a G space resg X means that X is viewed as an H space. The set of hyperelementary

^%/subgroups of G groups is denoted by 8^. Since G is abelian, these are the groups whichare a product of a cyclic group and a j&-group of relatively prime order. The Burnsidering of G, t2(G), is the ring of equivalence classes of smooth G manifolds with X equi-valent to Y if the Euler characteristics 5c(X11) and ^(Y11) are equal for all H C G. Theequivalence class of a manifold X is written [X], Additively Q(G) is the free abeliangroup generated by [G/H] as H runs over conjugacy classes of subgroups of G. WhenJf is a family of subgroups of G, Q(G, Jf) denotes the subgroup generated by{[G/H] | H e JT}.

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SPHERICAL ISOTROPY REPRESENTATIONS n

Let A(G) C ^(G) be the ideal defined in [0, p. 339]. By [0, Prop. 2]

A(G) == n^ Ker(resH : Q(G) -> Q(H))HE^

because G is abelian.If E eQ(G), its character ^ is the integral valued function defined on the set

of conjugacy classes of subgroups defined by

XE(H)=x(E11).

The definition of the equivalence relation in ti(G) means A(G) = { E | % E ( P ) = °for P C G and Pe^}. Set(2.1) J ^ = { H C G | G/H } and QQ = a(G,J^).

Lemma 2.9. — Suppose G is an abelian group with at least four non-cyclic Sylow subgroups.Then the unit i of t2(G) lies in the subgroup A(G) + Q.Q.

Proof. — For each Sylow subgroup S of G, let E = E(S) e 0(G/S) be a virtualfinite G set with ^g(P) = i for each P C G/S which is hyperelementary and/^(G/S) == o. Since G/S has at least 3 non-cyclic Sylow subgroups, the existence of Eis provided in [T^]. Since Q.(G) is contravariant in G, we may regard E(S) as beingin Q(G). Set

X==IlE(S)eQ(G).s

Then %x(G) = o and /x(P) :== I whenever P is a hyperelementary subgroup of Gi.e. P e^. Thus X = i - U with U eA(G).

Let X = Sa^G/H] and set Iso(X) = {H | =t= o}. Observe that if A and Bare G sets, the isotropy groups of A x B are intersections of isotropy groups of eachfactor. Viewing E(S) as an element of Q(G), Iso(E(S)) = { S X H(S) | H(S) is aproper subgroup of G/S}. This uses the fact that ^(G/S) == o. From the abovecomment, H eIso(X) implies H = fl (S X H(S)), where S runs through the set of

sall Sylow subgroups ofH. Then the index ofH(S) in G/S divides [H : G], the indexof H in G, for every Sylow subgroup S. Since there are at least four distinct Sylowsubgroups, [H : G] is divisible by at least 2 distinct primes so G/H ^ , i.e. H e .This shows that X e O.Q .

The following easy lemma is left to the reader.

Lemma 2.3. — Let G be as in Theorem A'. Then G ^JT and ^3^3^.

Let I'(G) be the subgroup defined in 1.7.

Lemma 2.4. — z e I'(G) if and only if there are representations V and W of G suchthat V11 = W11 = o whenever G/He^, V - W e I(G) and z == V - W.

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Proof. — Let % be an irreducible representation of G whose kernel K has index R(for some prime p. If < , V > denotes the multiplicity of 7 in V and V — W == ^ e I' (G),then <x, V> == <^ V^ = </, W^ === <^, W>. The middle inequality uses theassumption V1^ = W1^ in G/K. Removing all irreducible representations with kernelof prime power index from V and W gives V and W with V — W = z. Observethat V'^W'^o whenever G/He^.

If R is a complex (real) representation of G and ^ is a complex (real) irreduciblerepresentation of G, then ^(R) (w^(R)) is the multiplicity of / in R. Define a set Siof complex representations ofG by asserting a representation RofGis in^? if and only if

2.5. — (i) Iso(R — o) = Jf, dime R11 >_ 2 for He Jf, dime R33 3 P e ,

dime RK< - dime R11 whenever K contains H and H e f, and

(ii) dime R11 < w^(R) whenever H C G and ^ is a non trivial irreduciblerepresentation of H with ^(R) + o.

Lemma a. 6. — For each zeV(G) there are representations RQ and R^ in St withR, - R, = z.

Proof. — By 2.4 there exists V and W with V -- W == z and V11 = W11 == owhenever G/H e^. This means Iso(A — o) CJf for A = V, W$ so we can apply[^43 l ' 3 ] to see that there is a representation S of G such that VOS and W ® Ssatisfies 2.5 (i). It is an elementary exercise using Frobenius Reciprocity to see thatthat proof also shows that 2.5 (ii) is also satisfied (see [PR, § 9]); so that V®S andW © S are in ^. Their difference is again z.

It seems appropriate to motivate the appearance of the groups I'(G) and 10'(G)in the study of Smith equivalence of representations. The motivation comes from thefollowing necessary condition for Smith equivalence:

Lemma 2.7. — Necessary conditions that two representations V and W of G be Smithequivalence are: a) resp V ^ reSp W for all subgroups P of G in 8^. b) Whenever H is asubgroup of G such that G/H is cyclic and in ^, either (i) res^ V ^ resn W or(ii) V^W^o.

Remarks. — Note that the condition that V11 = o whenever G/H is cyclic andin 8^ is equivalent the condition that V e y (section i) i.e. V11 = o whenever G/His in y. This uses the fact that G is abelian. The real analog of 2.4 describes 10'(G)as the subgroup { V — W | V , W e ^ and reSp(V — W) = o whenever P e^}.This is the subgroup of RO(G) defined by 2.7 a) and b) (ii).

Proof'of'2.7. — The proof is based on a theorem ofAtiyah-Bott. This theorem [AB]asserts that a cyclic group G of odd prime power order cannot act smoothly on anoriented manifold M of positive dimension in such a way that M° is exactly one point.

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We apply this theorem to this situation: Let H be a subgroup of G such that G/H == Cis cyclic of odd prime power order. Suppose that G acts smoothly on a homotopysphere 2, S0 consists of two points x and y and T, S = V, Ty 2 == W. If resn Vis not isomorphic to reSg W, then x andj/ are not in the same component of211; so theconnected component of 211 which contains A: is a closed manifold M supporting an actionof the cyclic group G such that M0 is one point x. Thus dim M == o = dim V11 == oand similarly dim W11 = o or equivalently V11 == W11 = o. This verifies condition b).The necessity of condition a) has been noted in Lemma 1.2.

The next lemma enters into the proof (§ 5) of Theorems B and B'. For the proofof this lemma, we collect some facts and notation from representation theory. Let ^be a primitive k"th root of unity and Z[y the subring of C additively generated bypowers of ^. If g e G is an element of order k and x e R(G) is viewed as a complex-valued function, its value x{g) at g lies in ZR]. This leads to a ring homomorphismevg: R(G) -^Z[S] with eVg{x) = x{g). For each cyclic subgroup G of G, ^ denotesa primitive |C|-th root of unity and g{C) a generator of C. Set R(G) == riZ[Sc]where the product ranges over the cyclic subgroups of G. Then there is a homo-morphism ev : R(G) -> R(G) whose C-th coordinate is ev^ == ev^. A similarsituation applies for real representations. In this case Z[^c] is replaced by Z[^ ® Ic](where - denotes complex conjugate) and RO(G) is the product of these rings as Cvaries again over the cyclic subgroups of G. Comparing the real and complex settingleads to a commutative diagram:

R(G) -'-. R(G)

RO(G) -^> RO(G)

where r is realification and r{x) == x + x . Recall that I(G) is the ideal in R(G) definedby I(G) =={xeR{G) | reSpM =o for all Pe^} and IO(G) C RO(G) is similarlydefined.

Lemma 2.8. — The realification homomorphism r:I(G) ->IO(G) is surjective.

Proof. — First observe that coker r is annihilated by 2 because complexificationfollowed by realification is multiplication by 2. We assert that coker r is annihilatedby an odd integer too; so coker r = o. The assertion follows from a diagram chasein the above diagram and uses these facts:

(i) Ker r = {x — x | x e R(G)}. (Here ~" denotes the involution in R(G) induced bythe automorphism of G sending g to g~1.) Kerr == {x — x\ x e R(G)}. Here~~ denotes complex conjugation in each factor of R(G).)

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(ii) [G| is odd.(iii) If x e R(G) or RO(G), then resp(^) = o for all P e^ if and only if evc(^) == o

whenever G e^ is cyclic.(iv) |G|.R(G)Cev(R(G)).

The last statement has an easy proof due to R. Lyons. It suffices to exhibit foreach cyclic subgroup G an element f^ e R(G) such that ev(/c) has C coordinate [G|and all other coordinates o. View R(G) as a ring of complex functions on G. Notethat the function f^ which sends g e G to | G | if g generates G and to o otherwise isin R(G). To see that^ is in R(G) note that its inner product with an irreduciblecharacter is an algebraic integer (in Z[^c]) which is Galois invariant; so it is in Z. Thisends the proof of the lemma.

In order to apply the results of this paper, one needs to produce elements in I'(G)or 10'(G). Here are two relevant points about this (which lie behind the examplein section i): (i) If 9 : G -^ L is a surjective homomorphism, then 9*: F(L) -> F(G)where F is I' or 10'. An especially good choice for L is a cyclic group which is nota^-group. (ii) The Adams operations ^k can be used to construct elements in I'(G).For each j^-Sylow subgroup P of G choose an integer dp prime to the order of G andcongruent to i mod the order of P. Then for any representation R in SH,

n(p- i )Re l ' (G) .

Just note (^-i) ReSpR= o in R(P) and (â — i) R11 == [â — i) R]11 == oin R(G/H).

3. Framings and equivariant cell attachment

Background

We begin with a few words of motivation and notation. Recall that G is abelian.For any group F, E(r) is a contractible space with free F action and E = E(G). IfH C G and R is a representation of G, res^ R is its restriction to H and | H | is the orderof H. We denote by I(G) the ideal in R(G) consisting of those elements which arein the kernel of resp for each Sylow subgroup P of G. Suppose W is a sphere or diskwith a smooth action of G. Then Smith theory implies:

(i) W11 is a mod? sphere or disk if H C G, |H| == ^n, p prime. In particular7r,(W11) ® Zp == o if k < dim W11.

Smith theory or the Atiyah-Singer Index Theorem implies:(ii) If R,R' e^WjêW 0 }, then resp R ^ respR' for each Sylow subgroup P

of G. (If W is a sphere, p must be odd.)

The Atiyah Completion Theorem implies:(iii) IfR' and R are complex representations whose difference is in I(G), then E X R'

and E X R are stably isomorphic G vector bundles over E.

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Equivariant surgery is a process for reducing the homotopy groups of W11, H C G,where W is a smooth G manifold. In one setting it may be described like this: Givenis a representation R of G and some sort of bundle isomorphism (framing)& : T W - > W x R and an element a e image (^(8W11) -^^(W11)). There are twosteps to surgery using this data:

(iv) Using a and the framing 6, an equivariant handle is attached to W giving W 3 Wand a is killed in ^(W711).

(v) The framing is extended over TW.

If the definition of framing is too strong (e.g. if b is a stable G vector bundleisomorphism), then T^pW is isomorphic to R as a representation of G for all x e W°.If the definition is too weak, it may not be possible to achieve (iv) or (v).

It is natural to use surgery to convert a manifold W with framing into a sphereor disk by achieving (i) for all primes p. Since we do not want T^ W = R for allx e W°, we must avoid too strong of a definition of framing. Point (ii) givessome guidance. Since we must have isomorphisms gy: resp R' ^ resp R for each^-Sylow subgroup P of G, we might postulate stable P vector bundle isomorphismsby : resp TW -> res? W X R for each prime p as part of the definition of framing. Thisis sufficient to achieve (iv) but not (v). For that we introduce the notion of anR-P framing and insist that by be an R-P framing. This means b is a compromisebetween a stable G vector bundle isomorphism and a stable P vector bundle isomorphism.With the collection {by} we can achieve (iv) and (v) for all H =(= i. We treat thegroup H == i using a stable G vector bundle isomorphism Aoo : E X T) ^ E x W X Rmanufactured from (iii). The collection [by, by,} provides the needed data for a"framing". This is called a Smith framing and its precise definition occurs in 3.24.We should emphasize the role of the assumption that G be abelian. For these groupscomplex representations are represented by diagonal matrices; so ifR and R' are complexrepresentations of G whose restrictions to P C G are isomorphic, there is a P isomor-phism £p between them such that 6(ep) e Ap(R)—a torus. For a general group thiswould only be an element of Ap(R) which is not a torus. See the discussion after 3.33.

In this section we treat the part of surgery dealing with framings of equivariantbundles. This does not require restriction to manifolds or their tangent bundles. Themain results are 3.12 and 3.13. We use these in the next section to treat the manifoldcase which needs some added input concerning framings and embeddings.

Smith framing—desired formal properties

One of the basic constructions in homotopy theory is to attach a cell D^4'1 to atopological space W via a map i : S^ -> W representing an element a e TT^(W). Thisdata produces a new space 0 = W U^D^^ in which a is null homotopic. Surgerycan be viewed as an elaboration of this process in the smooth category. There the

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initial data includes also a vector bundle T] over W (which in practice is the tangentbundle TW) and a stable vector bundle isomorphism (framing)

(3.1) b : T] -> W x R where R == R^

This gives rise to a stable vector bundle isomorphism

(3.2) l(b): S^XR-^T) S' =1^s'

and if k < n a vector bundle isomorphism

(3.3) r (&) : S^XR-^T]whose stabilization is f(b). Using this we obtain a vector bundle F == I^T], <z, 6) over 0defined by

(3.4) r ^ u D ^ x R

and a framing b ' : F -^ 0 X R extending &. Here S^ X R is identified with i* Y)via r(A).

Now we discuss an equivariant analog of this construction. We have noted thereis ambiguity as to what the definition of framing should be; however, that is usuallydictated by application. We develop one notion of framing called Smith framing, whichreflects Smith theory and the Atiyah Completion Theorem mentioned earlier. First,let us specify what a Smith framing should provide. For this we need some notation:W is a G space, T) is a G vector bundle over W, R is a representation of G and a e (W11)for a given subgroup H of G. Let

(3.5) n + i = dim R11, S = S^ x D"-^ and D - D^1 x D"-^

View S and D as H spaces with trivial H action. Let

(3.6) t:S-^W11 be a map such that i represents a. Briefly we saythat i represents a.

Let X be an H space. Set ind^ X = G XH X. This is a G space. If X' is a G spaceand y:X->X' is an H map, ind^y: ind^ X -> X' is the G map induced by/i.e. indg/k, x] = gf{x) for [ , x] e G XH X. Set

(3.7) 0 = W u, indg D (/ == indg i : indg S W11).

Then 0 is a G space in which a is null homotopic. When R is a representation of G,we let R denote the G vector bundle over W whose total space is W X R. Since thebase space W of R does not explicitly appear in the notation, it must be determinedfrom the context. Let A == {p \ p is prime and p divides | G |} and P be the ^-Sylowsubgroup of G. Henceforth G space (resp. G subspace) means G c.w. complex (resp.G subcomplex).

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A Smith framing of T] is defined relative to a »S7m7A decomposition W of W—the basespace of T). By definition W == {Wp, z? | p £ A} where Wp is a G subspace of W suchthat the isotropy group G^ at x satisfies

(3.8) | GJ = p^ with n{x) > o for A: e Wp.

The inclusion of Wp in W is ip. If we delete the requirement that W be a subspaceof W, we call W a generalized Smith decomposition of W. We write W = {Wp \p eA}when each iy is an inclusion.

Here are two important examples of Smith decompositions:

(3.9) W is a smooth G manifold. Let N be an open G regular neighborhoodof { .yeW|G^^} defined with respect to a G G1 triangulationofW. See [IJ, [R], [ST]. For p e A, letW p = { ; c e W - N | |GJ ==p^\ n{x)>o}.

In this case ip is an inclusion. Then W == {Wp, ip \p eA}.

(3.10) W is a point XQ. Let ^p = E(G/P) and let ^:^p->A:o be theunique map. Then XQ = {xQp, iy \p eA} is a generalized Smithdecomposition of XQ.

Throughout this section H shall be a subgroup of G whose order is p^ for someprime p e A. We distinguish two cases referred to as Gase I and Gase II. Theseare respectively m> o and m == o (so H = i). In either case H is a subgroupof P—the j&-Sylow subgroup of G.

Now suppose W is a G space with Smith decomposition W and a e T^(W^) inGase I or a e TT^(W) in Gase II. In either case the G space 0 in (3.7) is defined.It has a natural Smith decomposition 0 defined in terms of W and a. The spaces 0^for q e A are in the two cases:

Case I: Op == Wp u^ ind^ D, 0, = W, q 4= A if/= indg i.Case II: Oy == W^ for all q.

Theorem 3.12 provides one of the important tools in the main geometric cons-truction in equivariant surgery. There it is applied with T) the tangent bundle of W.In 3.12 we use these notations and hypotheses:

(3.11) (i) a) is the H vector bundle over D defined by <o = D X R (3.5).(ii) W is a G space with this property: each component of W11 contains

a point whose isotropy group is H.(iii) W is a Smith decomposition of W.(iv) T] is a stable vector bundle over W (3.34).(v) a e (W^) in Gase I or a e T^(W) in Case II and k < dim R11.

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Theorem 3.12. — Assume (3.11). Let (B : T] -> R be a Smith framing ofr^ rel W (3.23).Then there is an H vector bundle isomorphism /"((3) : <*) -^ (^ | ) wAt'cA defines the G zw^r

s |s'A(W^ r == r(Y],fl,(B) ==T] uîndjo) ozw 0, (/== indgr((B)). TA^ y fl 5'wÂframing ^': F -> H r^ 0 extending p.

Smith decompositions and Smith framings restrict to subgroups and subspaces.Here is a discussion: First we show how to restrict a Smith decomposition W of aG space W to a subgroup H of G and to a G subspace X of W. For the latter thenotation is W . This is the Smith decomposition of X with X — X n Wp for each

xprime? which divides |G|. The H Smith decomposition res^W ofreSgW is definedas follows: resH W == {W; \p \ \ H |} where W; = {x e Wp | | G, n H | == p^, n{x) > o}.Now we discuss restriction for Smith framings. Let (B be a Smith framing rel W (3.23)where W is a G space. Let H be a subgroup of G and X a G subspace of W. Res-tricting G data to H data gives a Smith framing resg (B rel resji W and restricting G datato X gives a Smith framing (B rel W . Smith framings are related to the ideal I(G).

x x

Theorem 3.13. — Let R and R' be complex representations of G such that R' — R e I(G).Let W be any G space and W a generalized Smith decomposition of W. Then there is a Smithframing (B^(R', R) : R' -^RrelW which is natural with respect to restriction to subgroupsand subspaces.

Smith framings—definitions

We proceed with the definition of Smith framing and the proofs of 3.12 and 3.13by first providing some definitions and conventions. If M is a representation of Gand 7) is a G vector bundle over W, then s^(r^) == T] <9 M and s{f\) denotes s^{f\) forsome M. Similarly for the representation R, s^{R) == R <9 M and ^(R) = ^(R)for some M. If T]' is another G vector bundle over W and b: T) -> r\ is a G vectorbundle isomorphism, then s^{b) : b ® 15 : (73) -^s^^) is a G vector bundle iso-morphism. A stable G vector bundle isomorphism b : T] -> T]' is by definition aG vector bundle isomorphism b : Sy(r^) -> (7)') ôr some M. Sometimes we writeb: J(7)) -> s^) to mean that b is a stable G vector bundle isomorphism and if & is aG vector bundle isomorphism, s{b) : s{r^) -> s ^ ' ) means s{b) == s^{b) for some M.

There is one point where we need to use s^ where M is allowed to be a countabledirect sum of finite dimensional representations. This is to apply the Atiyah CompletionTheorem. In this case s^{r^) is only used when G acts freely on the base space W.Then G vector bundles over W are equivalent to vector bundles over W/G and a G vectorbundle isomorphism b : s^{r^)-> s-^(^) means an isomorphism S]£.W)IG-^ ^W)IGof UOQ vector bundles over W/G. Here Uoo is the infinite unitary group or infiniteorthogonal group depending on whether T] is a complex or real vector bundle.

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We associate certain G spaces to a representation RofG and a^-Sylow subgroup PofG:

U(R) — This is the space of n X n unitary matrices when R is a complex represen-tation of dimension n. When R is a real n dimensional representation, thisis the group 0^.

M(R) — This is the maximal torus in U(R).Ap(R) — This is the space of maps of G/P to M(R) which carry the identity coset

I e G/P to the identity matrix i eM(R). Denote by i eAp(R) the mapwhich carries G/P to i eM(R). We make Ap(R) a G space via

(gf)W=Ahg)Ag)~^Here g e G, g is its coset in G/P, h e G/P and fe Ap(R). Note that P C Gacts trivially on Ap(R). It is a torus.

Ap(P) — This is the space of maps of G/P to U(R) which carry I to i eU(R), soAp(R) is a subspace of Ap(R).

If T) is a G vector bundle over W and b : T] -> R is a P vector bundle isomorphism,define 6(6) : W -> Ap(R) by(3.i4) ^==Q{b)(x){g)gb^g-1

for g e G, x e W. Here b^: -> £3; is the map on the fiber over x. The proofof the next lemma is immediate from definitions. Compare [Bg, 11. i] in the case P = i.

Lemma 3.15. — If b : T] -^ R is a P vector bundle isomorphism and if

6(6) {x) e Ap(R) C Ap(R) for all x e W,

then 6(6) is a G map from W to Ap(R).

There is an obvious inclusion s : Ap(R) -> Ap(^R) and if b : T) -> R is a P vectorbundle isomorphism, sQ{b) == 6(j6).

By definition an R framing of T] is a stable G vector bundle isomorphismb : s(-^) -> J(R) and b is called the R framing. We usually write b : T) -> R deleting s.If b is only equivariant with respect to P C G, we say b is a res? R framing of •Y], Forsuch a resp R framing we define(3.16) 6(6): W^Ap(^R)

using (3.14). Of course Q{b) maps W to i if b is an R framing.The cone c(W) on W is a G space and the cone point XQ e c(W) is fixed by G.

The restriction of a map /: (W) ->W to WC c(W) is denoted by/o-

Definition 3.17. — An R-P framing b of T] is (i) a reSpR framing b such thatQ(A) : W -> Ap(A) together with (ii) a G map 6(6) : c(W) -> Ap(jR) such thatQ^b) == Q{b) and 6(6) {x^) == i. It is worth noting that Ap(R)° = Hom(G/P, M(R))

235

ao T E D P E T R I E A N D J O H N R A N D A L L

is discrete; so if b is an R-P framing of 73, Y}^ is isomorphic to R as a representation of Gfor all x e W°. This is the reason for excluding the G fixed points of W in Wp (3.8);so R-P framings of f\ impose no condition on T]^ for x e W0.

w?We observe that U(R) is a G space with

(3.18) gu R{g) uR{g)-1 for g eG and u e U(R).

Here R(^) is the matrix in U(R) given by the representation R. We abbreviate theright side of this equality by gug~1. When R and P are fixed, we abbreviate Ap(R)and U(R) by A and U. The next definition requires a G map 6 : W -> A.

Definition 3.19. — A G* map f: W ->V is a map which satisfies

(3.20) f{gx) = gf(x) g-^QW {g)-1 for x e W, g E G.

Here dot denotes multiplication in U. Sometimes to emphasize the role of 6 we saythat/is a G* map (wrt 6). Note that when 6 = i (the map which sends W to i e A),/is a G map with respect to the G action on U defined in (3.18).

The notion of a G* map is utilized as follows: Let b be an R framing of T] and let bbe an R-P framing of T], Then for each A: e W, we have b^ = Q(b, b) {x) b^ for someQ(b, b) {x) e U(A). This defines a map 6(^ b) : W -> U(^R) and

(3.21) y==6(M)^.

One easily verifies that 6(6, b) is a G* map {wrtQ{b)).

Definition 3.22. — An R framing b of T} compatible with the R-P framing b of T]is (i) an R framing H together with (ii) a G* map {wrtQ{b)) Q(b, b) : <:(W) -Û(^R)such that 6o(^ b) = Q{b, b) and 6(^ b) (ô) = i.

Recall that E = E(G). Let W be a G space and W = E X W. This is aG space. If /: W -» W' is a G map, then /: W -> W is the map IE X/.

Given a (generalized) Smith decomposition W of W and a G vector bundle T]over W, define

(3-23) ^-^ (rf- (3-9))-

Definition 3.24. — A Smith framing (B : 73 -^ R rel W is a collection {bp,b^\peA}==^where by is an R-P framing of f\y and b^ is an R framing of 73 such thatb^y == ^ b^ : Tjp -> R is compatible with by.

Remarks. — An implicit part of the data of a Smith framing is the collection ofmaps {9(ZL), 6(6 5 6 ) |^ eA} involved with the R-P framing by and the compatibilityof b and ? . Note that by === i^ X by is P equivariant! It is an R-P framing whereQ(by) is the composition <;(Wp) ->^(Wy) -Âp(jR). The last map is Q{bp).

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Proofs of 3.12 3.13

Since the techniques of the proof of 3.13 are used in the proof of 3.12, we treat3.13 first. In addition to the Atiyah Completion Theorem, it rests on these twoelementary lemmas.

Lemma 3.25. — If^isaG vector bundle over W, G/P acts freely on W/P, H^W, Z) = oand b is a resp framing off\ such that Q{b) has values in Ap(jR), then b defines an R-P framing°f 7]-

Lemma 3. a6. — Suppose W is a G space such that G acts freely on W and supposeK^W/G) ==o. If b is an R-P framing ofr^ and T ) i s an framing of'T], then T ) i s compatiblewith b.

What 3.25 asserts is the existence of Q{b) : c(W) -> Ap(^R) satisfying 3.17 and3.26 asserts the existence of Q(b, b) : (W) ->U(^R) satisfying 3.22. We begin pre-paration for 3.25 and 3.26.

The space of G* maps (3.19) from W to U = U(R) is denoted by (W, U)°*.This space is closely related to the space (W, W X U)° of G maps from W to W x Uwhere W x U is made into a G space via

g{x, u) = ( , gug-1 Q{x) C?)-1), x e W, g E G.

Lemma 3.27. — There are maps

^: (W, U)^-> (W, W x U)0 and X : (W, W x V^ -> (W, U)0*

such that ^X is the identity.

Proof. — Define ^(/) {x) = {x,f{x)) and define X(A) = nh where n : W X U -> Uis projection.

There is a G* homotopy extension theorem which goes like this. Let W be aG space, I the unit interval with trivial action, S ' = = W x I , S = A x I u W x owhere A is a closed G invariant subspace of W and 6 : S' -> A == Ap(R).

Lemma 3.28. — Given any G* map f: S -> U, there is a G* extension F : S' -> U.

Proof. — Let H : S' -> S' X U be a G map which extends ^(/). This existsby the G homotopy extension theorem. Define F == X(H). Then F is a G* map byLemma 3.27 which extends f because X^ = i.

Decompose the cone cW as W' u W" where W' = {{x, t) \ x e W, o < t < 1/2}and W' n W" = W X 1/2. Gall a G map 6 : c\N -> A special if 6 maps W" to i.Given any G map 9 of<:(W) to A, such that 6(^0) == i, there is a special map 6' suchthat 6' = 6 on W = W X o. Just take any G map T : cW -> cW such that T is

wthe identity and TW" = XQ. Then 6 o T = 6' is special. Let [W, W']0 denote the

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set of G homotopy classes of maps from W to W. Let U == U(R) be the G spacewith action defined by (3.18). Write [W, U]0 == o if this set contains only the mapof W to i e U.

Theorem 3.29. — If [W, U]° = o and 6 : c(W) -> A is special, then any G* map{wrt 6 ) /: W ->• U extends to a G* map F : cW -> U such that F(^) == i e U.

w

Proof. — By the G* homotopy extension lemma 3.28 with A = 0, thereis a G* extension F' of / to W. Since 6 is special, 6(W X 1/2) = i; sof •==. F e [W, U]°. (See 3.19.) Since this set is zero by hypotheses, /' extends

Wxl/2to a G map F" : W" -> U such that F"(A:o) == i. Then F' u F" = F is the desiredG* extension of f.

Let C°° denote the countable direct sum of copies of C with trivial action of G.Then the inclusion of U(C°°) into U(C°°®R) which sends u to i^@u is a G mapwhich is a homotopy equivalence for any complex representation R of G; so[W, l^C00)]0 -> [W, U(C°° + R)]° is bijective if G acts freely on W.

Corollary 3.30.—If G acts freely on W, K^W/G) == [W, U^R)? for any R.

Proof. — The left side is [W/G, U(C°°)] = [W, l^C00):^ by definition. Theright side is [W, U(C°° + R)]° for sR == C°° + R.

Theorem 3.31 [AJ. — If^' is a contractible space with free G action^ K^E'/G) = o.

Lemma 3.32. — IfWisaG space such that H^W, Z) = o and G/P acts freely on W/P,then any G map 6 : W -> A extends to a (special) G map Q: c(W) -> A with Q(xo) == i.

Proof. — As P acts trivially on A, 6 factors through W/P. Now, G/P acts freelyon W/P and A is a torus; so TT,(A) = o for i > i, and G/P homotopy classes of mapsof W/P are in i — i correspondence with I-P(W/G, ^(A)) == o. To see this, notethat G maps of W to A are in i — i correspondence with sections of the fibrationA -» W XQ A -> W/G [BJ., Since the fiber A is a torus, the statement follows. Thus 6is G homotopic to the map which takes W to i. A homotopy between them produces 6.By the discussion preceeding 3.31, we may suppose that 6 is special.

Proof of 3.25. — Lemma 3.32 provides an extension 6 : c(W) ->-A of Q(b). SetQ(b) == Q. Then b and 6(6) define an R-P framing of T].

Proof of 3.26. — We have [W, U(jR)]° == K^W/G) = o (3.30). Apply 3.29to 6 = Q{b) : c(W) -^ Ap(jR) and /= Q(b, b) : W -> U(^-R). This produces an exten-sion 6(6,6) : (W) -> U(jR) such that 6(6, 6) (^o) = i; so b is compatible with b.(To use 3.29 we have implicitly assumed 6(6) to be special. As noted, this does notrestrict generality.)

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The next lemma needs the notion of a map f: W -> W between generalizedSmith decompositions of W and W. By definition this means

{={f,J^\peA}

where fy: Wp -> W; and /„ : W -> W are G maps such that /„ ip == i^ for j& e A.Lemma 3.33 shows that XQ (3.10) is universal in the sense that any generalized Smithdecomposition maps to it.

Lemma 3.33. — IfW is any G space and W is a generalized Smith decomposition o/W,then there is a map f: W -^Xo.

Proof. — Let f^ : W -> XQ be the unique map. Since G/P acts freely on W /P,there is a G/P map fp : Wp/P -> x^ which classifies this free action. Let fy be thecomposition of this with projection of Wp on Wp/P. Then f = {fp,f^ \p eA}.

Let R and R' be two complex representations of G of the same dimension. SinceG is abelian, we may suppose the matrices R{g) and R'(^) for g e G representing Gvia R and R' are diagonal. This means R(^) and R'(^) lie in M(R) = M(R'). Thenormalizer ofM(R) in U(R) is the group of matrices permuting the complex coordinatesof R. This means that if R and R' are isomorphic as representations of P, there is aP isomorphism £p : R' -> R such that e? normalizes M(R). Then for all g e G

W {g) - R'te) ep-1 RQ?)-1 e M(R)

by 3.14; so

6(e,)eAp(R).

If R' — R eI(G), then the P isomorphisms £p are defined for all p eA.

Proof of 3. i3. — First we treat the case in which W = XQ is a point and W isXp (3.10) and produce a Smith framing p^: S/ ->Rrelxo. Of course in this caseR' and R are just R' and R viewed as G vector bundles over Xy. To fit previousnotation, set T] = R'. Let by == i; s? : -> £. Here ^ = i; 73. This is a G vectorbundle over E(G/P) which is contractible. In addition Q{bp) {x) = 6(ep) e Ap(^R)for all A:eE(G/P); so 3.25 implies that by defines an R-P framing of T]?. SinceR' -- R eI(G), the Atiyah Completion Theorem implies that there is an R framing^ : 7 j = E x R ' - > E x R = = R . Note that 7]p is a G vector bundle over E x E(G/P)on which G acts freely and K^E X E(G/P)/G) == o by 3.31. Then 3.26 impliesthat b^=i*pb^ is compatible with by. Taken together these facts imply thatPUR'? R) = = { ^ 5 ^oo \P eA} is a Smith framing. This completes the case whenW = XQ. Now let W be a G space and W a generalized Smith decomposition. Thenthere is a map f : W - ^ X o (3.33). Set (Bw(R', R) = f* (BJR', R).

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Now we begin the treatment of 3.12. Let W be a G space and T) be a stableG vector bundle over W. This means

(3-34) For each x eW and each irreducible representation % of G^ eitherthe multiplicity m^{^) of ^ in the Gj^ representation ^ is o ordim 7)^ <^ d^m^{^). Here d^ is i, 2 or 4 (5.5) depending on ^.See [PR,~Gh. 3, § 9] or ^3, 3.6].

As before H C P. We shall use one of the framings(3.35) (i) b is an R-P framing of T] or

(ii) b is an R framing of T],

and an element a e (W") with k < dim R11 to define an H vector bundle isomorphism

(3.36) - •r(A) : co ), <o = D x R == R

and the G vector bundle F = F(7], a, i) over 0

(3.37) r=7]U^indg(o, /=ind^'(&)

extending T], As before z : S -> W11 represents a and S' = i'S. We also view i as amap of S to W with image S' as follows: Let z e F be any point. Then S' is identifiedwith - ? x S ' C E x W = = W and i is the composition S -> S' C W. Then TJ

and in either case in 3.35 we have that i*[b\ ) : i* s{^ ) ->• z*jR == ^(co ).|s' s' s

The definition of/"(6) depends on the choice of b in 3.35 which in turn dependson whether H falls in Case I (| H | == p^ m > o) or Case II (H = i). The conventionis to take 3.35 (i) in Case I or 3.35 (ii) in Case II. In either case define

(3.38) \-i •f{b)==^b : ^co t J7)

This is a stable H vector bundle isomorphism. Since Y] is stable and k < dim R11 andW satisfies 3.11 (ii), there is an H vector bundle isomorphism

(3.39) r{b) : <o •^

such that s £ ' ( b ) =={{b) up to regular H homotopy. See [PR, Ch. 3, § 9] or [?3, 3.8].By definition two H vector bundle isomorphisms are regularly H homotopic if there

is a homotopy of H vector bundle isomorphisms connecting them.

Lemma 3.40. — Case I: Any R-Pframing ofr\ extends to an R-P framing b'ofY. Case H:Any R framing of TJ extends to a R framing V of Y.

Proof. — Since s t ' ( b } ==t{b) up to regular H homotopy, F' == JY] u^ induce,{f== ind^/'(&)), is G isomorphic to jF; so we must produce a P vector bundle iso-morphism b ' : r" -> sSL together with 6(&') : c{0) -> Ap(jR) in Case I and in Case IIa G vector bundle isomorphism b ' : f" -> sR.

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First case. — By the G homotopy extension theorem, there is a G map6':c(0) -»Ap(^R) which extends 6(&). Define V on ^ to be b. On (indgjco)^define b^ == %(x) {g) for x e D and g e G. If y e (<o )^ observe that in indg

sis identified with gb-^v) by definition of f{b), md^(b) and F'. Then

b(gb-\v)) = Q{b) {x){g)gv

by definition of Q(b). Since this is b\gv), b' is well-defined. It is a P vector bundleisomorphism. Define 6(^) to be 6'. Then 6', 6(A') defines an R-P framing V of Fextending b.

Second case. — Note that {{b) = ?(6[ )-1 = i up to regular H homotopy. This

follows from the fact that E is contractible and {b \ ) = {b ) . This means that-, ^ . ^ Is' s' s'F =JTJ U^md^jco, (/==ind^), up to G vector bundle isomorphism; so we needan extensum b ' : f -> j£ of b. Set 6' = 6 on s^. Let &' be the identity onmd§ ^co = jR. This is a well-defined G vector bundle isomorphism which extends b.

Lemma 3.41. — Let Y] be a stable G vector bundle over W. Suppose b is an R-P framingofr^andb is an R framing of^ compatible with b. Let a e 7^(W11) where | H | == m, w > o.^^<? W ri 3 . 1 1 (ii) W ^<dimR11. Then there is an R-P framing b' ofr = r(^ a, A) extending b and an R framing of f extending H1 which is compatible with b'.

Proof. — The first statement is Case I of 3.40; so we treat the second claim. Use theG" homotopy extension lemma (3.28) to produce a G* map (wrtQ(b')) 6': c(0) -> U(jR)which extends 9(^ b). Define^' to be ^b' on F. Then T ) ' is an R framing whichextends "b, is compatible with V and Q(b\ b ' ) == 6'.

Proof of Theorem 3 .12 . — Case I (|H| =^m m> o). Let t : S — Wp11 be amap which represents a, and set S' = ?S. Note that T] = (T]J by definition

s' s'of7)p as ?; 73. Let p == {^, 6^ | y e A} be the given Smith framing of T]. Define /"((B) by

^'(P)-^)^ ^^] ) (cf. 3.39).s s'

This is the H vector bundle isomorphism which defines r(7], a, (3) == F as a G vectorbundle over 0 extending T]. By definition Fp = F . This is F(7] , a, b ) by ins-

°^pection; moreover, Fy == for y + ^ . By Lemma 3.41 there is an R-P framingbp : Ty -^ R extending ^ ^and a G framing b^ : fp -> £ extending ^ = y"; which is compatible with by. Since ^ == 7] n and F = T] u Fy, we can defineb^ : f -> R by ^ -= on ^ and ^^ == b^ on Fp. Finally set by == b^ for q ^ p.Then j3' == [b^ b'^ \ q e A} is a Smith framing of F which extends p.

^74


Case II (H == i). — Define/'(P)=/'(^):(. - *(.)| ) (cf. 3.39).

s |s'

This is an H == i vector bundle isomorphism. Again r(7), a, (B) is a G vector bundleover 0. The Smith decomposition 0 =={0p\p eA} has Op = Wy for all j& eA$so •y]p == Fp for all p e A; hence to define (B' we need only define b^ : P -> R exten-ding b^. This is provided by 3.40.

4. An Induction Theorem

In this section we prove the Induction Theorem 4.19. It is analogous in statementto the Induction Theorem 2.6 of [DPJ. There is an essential geometric differencewhich appears in the definition of a normal map. See 4.9 and [DPi, 2.4]. Theproof of the Induction Theorem [DP^, 2.6] is algebraic but appeals to two geometricresults about normal maps pDP^, 3.12 and 3.13] which give condition for doing surgeryto kill a homotopy class. Their analogs here 4.11 and 4.12 are much deeper withthe definition 4.9. Once they are established, the proofs of the two induction theoremsare identical. The main bulk of this section is devoted to treating 4.11 and 4.12.These are the main geometric steps in any equivariant surgery procedure. See e.g.[W], [DP^L [PR].

A manifold triad (W; Wo, Wi) is a triple of manifolds such that BW == Wo u W^and Wo 0 Wi == 8Wo == ^W^. A G manifold triad is a manifold triad such that G actson W and respects the triad structure. A G map of triads is a G map which respectsthe triad structure. Let W be a G manifold triad. Set X = Wo and assume thefollowing gap hypotheses:(4.1) For each subgroup K of G in 8ft and each subgroup L > K, every

component of W1' has dimension less than one half the dimension ofthe corresponding component of W1^.

If X is not empty, then 4. i holds for X in place of W and this implies each componentof X1^ has a point whose isotropy group is K provided XK 4= 0 and K e 8ft.

In the preceding section we treated Smith framings rel W where W was a quitegeneral Smith decomposition. We shall need more structure for its role in the processof equivariant surgery on a G manifold W. In this case W = W^) is defined interms of a set y {Smith set) of submanifolds of W which are untouched in the surgery process.

Let W be a G manifold with boundary X and y == «9^(W) be a set of invariantsubmanifolds, called a Smith set for W, with these properties:(4.2) (i) If Mey and BM + 0 then 8M = M n X and if M C W"

where H is a non trivial p group for some prime j&, thenjr- iArH _ — - i -r— ~\ r ^ 'v ^dim M < - dim W11 and dim M n X < - dim X11.

2 2

242


(ii) If L^, W^^.(iii) W has an equivariant G1 triangulation ([IJ) such that each

M e y is a subcomplex.

The Smith set <^(W) for W, determines the Smith sets

^(W)[ ={MnX|Me<^(W)}

and resH ^(W) == {resn M | M e <^(W)}

for X = aW and resn W.Let N(^) be an open G regular neighborhood [St] with respect to the C1 trian-

gulation in (iii) of the union Wy of the A's in e$^(W). For example take interior ofthe second derived neighborhood ofWy in W [St, p. 44]. Then N(^) n X == N(<$^[ )

|xis an open regular neighborhood in X of X^ where X^ is the union of the manifoldsin y . With respect to this data define the Smith decomposition of W:

xW(^)={W^||G|}

where Wp == {x e W - N(^) | | GJ == p^ and n{x) > o}.

The Smith decomposition W(^) is determined by y alone by the uniqueness of Gregular neighborhoods [R], Since we shall work with a fixed G regular neighbor-hood N(<9") of W^, the full strength of this is not needed.

Whenever a Smith set V === c^(W) is given explicitly for W, we shall use theSmith decomposition W(<^) ofW implicitly and abbreviate the phrase Smith framing rel W(<^7)by Smith framing. This abbreviation works smoothly with respect to resg and restrictionto boundary because of these two identities:

resH W(^) = W(resn ^) and W(^) == 8W(^ ).8W 9W

Let S stand for either W or X from above and let H be a subgroup of G. SetS^^eS-N^ ) |G,=H}.

s

Lemma 4.3. — If H e^, W satisfies 4. i and k < - dim S11, ^% ^yy/ map of Sfc

ZTZ^O S11 is homotopic to one into S^.

Proof. — Let Q^be the union of the M in y . By 4.2 (i) and general position,s

any map of S^ into S11 can be assumed to miss Q11. Since there exist G regular neigh-borhoods of Q in S in any open neighborhood of Q^ and since G regular neighborhoodsare G isotopy invariant, we may suppose ^ misses N(<97 )11. Similarly using 4.1

swe may suppose S^ misses S1' whenever L > H. Then S^ lies in S^.

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Now let Z be a G manifold triad, F : W -> Z be a G map of triads and letX = Wo, Y == ZQ and /==F :X->Y. In what follows we suppose that

y == <^(W) is a Smith set (4.2) for W, W == W(^) and X = W .x

Let H be a subgroup of G. If Cyn is the mapping cylinder of/11, then by defi-nition ^k+^f^) == Trfe+i^/H, X11). An element ^e^^/11) ls represented by ahomotopy class of diagrams

Sk X11

(4.4) ^ : u /H

jy+i K > ^ Y11

Set a = B(JI e 7^(X11). Let TZ = dim X" and leti X11

^

yH

D ^D^1 xD"-^S == S^ x D"-^(4.5) (X :

^ == ^pt' represents a in the sense of 3.6D

be a diagram which gives (JL by restriction to (D^"^1, S^) C (D, S). We remark thatby property 4.2, we may suppose iS C X" C X^; so in particular pi is actually repre-sented by a class in ^+i{f^)9 Set D' == ind^D where H acts trivially on D. Asin 3.7 define(4.6) 0 == W u^, D' where ind == indg

and extend F to F" : 0 -> Z by setting F" = indg h.D'

Fix this notation:

(4.7) P : TW -> R is a Smith framing rel W; H e^; n = dim R11 == dim W11;Q is the H representation R/R11; A < dim R11, (JL e Tr^^/11);D = D^ X D^ and S = S^ X D^^ viewed as H manifolds withtrivial H action; D = ind^ D(R) and Do = indj S X D(t2) C D.

Definition 4.8. — An extension ^ of (A e^+i(/11) is a commutative diagramof G maps

XDO

+D

^:

Y

244


such that the restriction to (D^ +1, S^) C (D, Do) is pi. If i in 4.8 is an embedding, then

W = W u,D

is a smooth G manifold. There is a G map F': W -> Z extending F withF' == K. Let y == ^(W) = <^(W). Then y satisfies 4.2 for W and we take

DN(^(W')) =N(^(W)). Then W'(^) = = { W ; | ^ e A } is defined. Observe that 0is a G deformation retract ofW. In fact there is a G retraction r which carries each Wpto Op (defined in § 3) for each p e A.

The aim is to give conditions for which there is an extension % with & an embedding.This requires some added structure which we now discuss. A normal mapifr = (W, F, p) consists of

(4.9) (i) A G map F: (W, BW) -> (Z, 8Z) of degree i between smoothG manifolds of the same dimension. It is required that W satisfiesthe gap hypotheses 4.1, Iso(A) contains for A == Z, W, andIso(BA) contains ^ if BA 4= 0. In addition TW must be a stableG vector bundle.

(ii) A Smith framing p : TW -> R rel W (3.24) for some represen-tation R of G. (Here R == W X R and W = W(^) for agiven Smith set y for W (4.2).)

Remarks 4.9'. — When we need to keep track of the group acting, we call thisa G normal map. This definition of a normal map is a variation of those in [W], [DPJ[DPJ and [Pa]. For theorems 4.11 and 4.12 the degree i assumption is irrelevant(it is needed in 4.21). The Smith set ^(W) for W and the G regular neighborhood N(<$^)of Wy are an implicit part of the structure of the normal map . These are used indefining W(^) and the Smith framing p.

A normal map of triads is a normal map which is also a map of G manifold triads.Observe that if R is a representation of G, W is a smooth G manifold with boundary Xand (B : TW -> R © R is a Smith framing, then (B : TX ©ft^S.©& is a Smith

xframing. Because of the definition of Smith framing, we may express this asP : TX -> R. This means that a normal map of triads H^ == (W, F, p) defines two

xadditional normal maps ^ = (W^, F^, (y for i == o, i as follows:

^(W,) = <^(W) == {M n WJ M e e$^(W)},w,

(3 ,==P and F, = F :W,^Z,.w, w,

Suppose i^ == (W, F, (B) and ^/ == (W, F', p') are normal maps of triadswith F : W -> Z and F' : W' -> Z. We say that IT9 extends IT if W C W, F'

245


extends F and |B': TW -> R extends (3 : TW -> R. Denote ^o = (^ Po)by (X,y, (3o) and let H be a j& subgroup of G for some p e A.

Definition 4.10. — We say that ' arises from H^ by surgery on (x e^ +i(/11) if'^/ extends and there is an extension ^ of (JL (4.5) such that i is an embedding,W == W u, D and F' === K. (In particular (B' : TW -» K is a Smith framing rel W.Here R == W x R.) D

If there exists an H ^ ' which arises from i^ surgery on (JL, we say that surgery on \Lis possible.

The input for 4.11 and 4.12 consists of a normal map H^ == (W, F, p) as abovewith i^Q == (X,/, (Bo) and (A e Tr^^/11). Choose any map i: S == S^ X D"-^ -^ X11

representing 9[L (3.6 and 4.4). Then^ and p define a stable vector bundle isomorphism/'(B^rTS-^TX" (4.14).

Two equivariant bundle isomorphisms are called (stably) regularly homotopic ifthere is an equivariant homotopy of (stable) bundle isomorphisms connecting them.See [PR, pp. 92-93] for elaboration if needed.

We state 4.11 now and prove it later. Compare [DPi, 3.12] and [PR, 10.2,p. 141].

Theorem 4 .11 . — Suppose IT is a normal map of triads with 9iF =^ u^ and^ == (X,/, po). Let H be a subgroup ofG ofp power order, p e A, pi e T^.^/11), k < w/2,n == dim X11^ 5. Then surgery on (A is possible if there is an embedding i' : S -> X11

representing S[L such that ind§ i' (i' = i == ^pi) is an embedding and the differential di's*

{defined below) is stably regularly homotopic to /"(B)11.

Corollary 4. i2. — Surgery on [L e n^ + i(y11) is possible if k< - n and n = dim X11 >: 5.

Proof. — This is an immediate consequence of 4.11 and 4.16 below. Takei' == L in 4. i6.

sLet S and X be two manifolds of the same dimension and let t : S -> X be an

immersion. Its differential d^: TS -> TX induces an isomorphism between TS andi* TX also denoted by

A : TS -> i* TX

and also called the differential of t. Suppose S' and X' are two manifolds of the samedimension having S resp. X as submanifolds. Suppose L : (S^ S) —> (X', X) is animmersion of pairs. The composition

v(S, S') — TS' - > TX' -> v(X, X')s x

246

SPHERICAL ISOTROPY REPRESENTATIONS 3»

induces an isomorphism

^(Os: v(S,S')^i v(X,X')s*

called the normal differential of i at S.Here is a mild generalization of a lemma ofHirsch [H] which produces embeddings

(immersions) from vector bundle isomorphisms. It is proved in [DPJ, [Pg], [PR,10.4, p. 142].

Lemma 4.13. — Suppose G acts freely on the smooth manifold X of dimension ^^5.Let S == S^ X D""^ k<_nf2 and let i: S -> X be a map. Any stable vector bundleisomorphism I : TS -> i* TX determines a G immersion of ind^ S in X, G homotopic to ind^ 1,whose differential is stably regularly G homotopic to ind^ I. If k < ^/2 ^ immersion maybe taken to be an embedding.

Theorem 4.11 is the smooth analog of 3.12 and is partially reduced to thattheorem. This we now explain. Let IT == (W, F, (3) be a normal map with Smithframing (3 : TW -> K rel W. The H vector bundle <o over D defined in 3.11 (i) is TD

by inspection (see 4.7). Equally clear are the equalities (i) TD| ==TD(J ®R

and (ii) TW =TX^ I M '"

) R. Here R denotes the space of real numbers and

S' == iS has trivial action. Then the H vector bundle isomorphism

r((B): co i*TW

in 3.12 becomes an isomorphism

r((3) : TD| ^t*TW

Let ^ be an H vector bundle over an H space S. We suppose E; is given anH invariant inner product on fibers. Define an H vector bundle ^g over S11 via

s"=^ H <^^g, i.e. $H is the orthogonal complement of ^H in ^ . If b : S ->

is an H vector bundle isomorphism, then Schur's Lemma implies b =.b^@b^s°

where b^:^-^^ and ^^H-^SH- Apply this to TD| , TW and A = = r ( ( B )

noting (i) and (ii) and S11 == S. This gives H vector bundle isomorphisms:

(4.14) /'(P)11: TS®R -^(TX11®]^)s'

r((B)H: v(S,Do)^^•*v(XHX)s'

because (TD(J )11 == TS and (TD(J )n == v(S, Do).

247


Lemma 4.15. — Let % be an extension of [L with i an embedding. If

rf(i )Cin: TSCR-^TX^R {i = i )s s

zj regularly homotopic to ^((B)11 W I/ n(t)g =r((B)H, then F = F(TW, a, (3) == TW(Jyi?r 0 is defined by ^—the restriction of % to (D, S). See 4.5-4.7.)

Proof. — Both TW and r are obtained from TW by gluing on ind^(TD| ).

The gluing isomorphism in the first case is ind^(fl?(i) © i^) and in the second it iss

ind^'(p). Since rf(i) © IR = d{\. )@ I R ® ^ ( O S ^^'(P) u? to regular H homotopy,s s

these G vector bundles are isomorphic.

Lemma 4. i6. — Let i^ === (W, F, (3) be a normal map of triads^ H a p-sub group of Gand y-^'^k+i{f11) with k <^ w/2, ^^5 (^ + i = dim W11) be given. Then there isan extension W of(A such that i is an immersion {embedding tfind^ L' (i' == i ) is an embedding^

^in particular if k < yz/2). The differential of L is stably regularly homotopic to ^'((B)11 and

sthe normal differential n(i)g of i: (S, Do) -> (X11, X) is f^)^

Proof. — Let ^ be the diagram 4.5 which gives (JL by restriction to (D^S S^.We may suppose that iS C X? = {^ e X^ | G^ == H} by 4. i and 4.3. Since G/Hacts freely on X", Lemma 4.13 applied to i and the group G/H gives a G immersionof ind^E S into X;" whose differential is stably regularly G homotopic to ^'((B)11. Note

that ind^8 is the same as ind^ S as a G manifold. (If k < - dim X11, the immersion

may be taken to be a G embedding. Note 4.9 (i).) Thus we may suppose that ind§ iis an immersion. The G Tubular Neighborhood Theorem [BJ provides a G immersion iof Do == ind^ S X D(t2) into X extending ind^ i such that the normal differential %(t)gof i : (Do, S) -^ (X, X11) is/"((B)H. Since (D, Do) retracts equivariantly into ind^(D, S),there is an extension of h: D -> Y to K : D -> Y giving the diagram % as in 4.8.

Proof of 4.11. — The existence of the extension ^ of ^ is a consequence of thepreceding lemma. The embedding i given there satisfies the hypothesis of 4.15$ so

By 3.12 there is a Smith framing (3" : rF(TW, (A, (B) == TW' -> R. Let r : W -> 0

be a G retraction, i.e. r == {r^,r^\p e A} and r^ and /p are G retractions for p eA.Since Smith framings are functorial with respect to such maps, there is a Smith framingr^ P" *o> r -> R. Since ^ is a deformation retraction, TW' == r^(TW ) = r^ F.

oThis identification gives the required Smith framing p': TW' -> R. Moreover,F = F" : W' -> Z extends F (see 4.6) and -5T' = (W, F', p') is the effect of surgeryon (A.

^5


A normal cobordism between two normal maps = (W^, F^, (3,), i = o, i, with^ : W, -^ Y consists of a normal map of triads IT == (W, F, j3), F : W -> Z withthese properties: Z is the triad (Y x I, Y x o, Y x i) with I == [o, i]; ^ ==^\There is a G embedding j : (No X I, No X o, No X i) -> (W, Wo, W^) whose imageis (N, No, N1) where N == N(^), N, == N(^), ^ = ^(W) and e^ = ^(W,).Finally require y == {j\A x I) | A e <$^} and note that the condition ^ =^ thenimplies that <9^ and <$^ are canonically isomorphic because <9^ = <97 emphasizethat surgeries of the kind in 4.11 produce normal cobordisms. w.

Remark 4.17. — If ^i = (W,, F^, (3J, z == o, i, are G normally cobordant andG ^^, then W^ and WJ8 are canonically isomorphic; so we write W^ = W^. Inaddition T^Wo==T^Wi as representations of G for xeW^=W^. For ifIT == (W, F, (B) is a normal cobordism between and '5T1 then V^ ^ Vf^ X I forany K ^ , in particular for K = G. For by 4.2 (ii) W1^ € ^(W) and Wo1^ e ^(Wo);so W^ =J(Woc X I) where j is the G embedding given in the definition of normalcobordism above.

Let W be a G manifold with boundary X, R a representation of G and(3 : TW -> £ C R be a Smith framing rel W. If IT == (W, F, (3) is a normal mapwith F : W — Z, then 8iT == (X, F , (3 ) is a new normal map with Smith

x xset <$^(X) == ^(W) , F : X -> Y = BZ and Smith framing (B : TX ® R -. K. © R.

X X X

I f^ i saG normal map and H is a subgroup of G, then resg^ is the H normal map(reSnW, resn F, res^).

Before treating the Induction Theorem we need to discuss some related pointsfrom [DPJ. There we used a variant of the notion of a normal map called a prenormalmap. This is a triple H^ = (W, F, (B) like the normal map in 4.9. The essentialdifference appears in the vector bundle isomorphisms occurring in these definitions.This is 4.9 (ii) here and 2.4 (iv) there (where ^ == R for some representation R of G).Briefly 2.4 (iv) requires a stable G vector bundle isomorphism b : sTW -> jR togetherwith a collection c == {c(H.) : TWg -> Rjj | H C G} of G vector bundle isomorphisms.Let us denote (&, c) by (3, write (B : TW -> R and call this a bundle isomorphism. Thisis stronger than a Smith framing ofTW; so the condition that TW be a stable G vectorbundle (3.34) is not required for a prenormal map.

Let H^ = (W, F, (3), F : W -^ Z be a normal map (3.9) which satisfies theseconditions:

(4.18) (i) [Z] - [W] e A(G) + 2Q(G, ),(ii) 3 Z = a W = 0 ,

(iii) dim Z is even, Z11 is simply connected, dim Z11 == dim W11 andexceeds 5 for H effl.

2495


Induction Theorem 4.i9« — Let IT be a normal map which satisfies 4.18. If for eachH e^ there is an H normal map 1T(H) such that 8iT(H) = resn^, is normallycobordant to == (W, F', p') wA^ F': W -^ Z ij a homotopy equivalence.

Proof. — Since Z11 is connected for H e and since each component of W11

contains a point whose isotropy group is H (4. i), we may alter IT by a normal cobordismwithout destroying 4.18 and achieve also that W11 is connected. This is a simple standardconstruction. See e.g. [DPg, 9.1]. We therefore add to the hypothesis 4.18 (iii)the condition that W11 is connected for all H e 8P. With this change the conditionsof this theorem imply those of [DP^, 2.6] with K there taken to be { K C G J K ^ ^ } .There is one essential difference namely the induction theorem there involved a pre-normal map and here deals with a normal map. (In the presence of the assumptions 4.18,the only difference is 4.9 (ii) vs. [DP^, 2.4 (iv)].) The proof of the InductionTheorem 2.6 of [DPJ is entirely algebraic but appeals to two geometric results [DP^,3.12 and 3.13] about surgery on a prenormal map. We have established their analogshere (4.11 and 4.12) for surgery on a normal map. There is one minor differencebetween [DPi, 3.12 and 3.13] and 4.11 and 4.12. The former deals with arbitrarysubgroups H while the latter with subgroups H e^. The only use of [DP^, 3.12and 3.13] for H^ occurs in [DP^ 3.15] to achieve [Z] — [W] eA(G). Theassumption that [Z] — [W] e A(G) + 20(G, ) (4.18 (i)) means this can be achievedvia [DP^ 3.15] using 4.11 and 4.12. This means the proof of the Induction Theoremof [DPJ for prenormal maps applies to prove the Induction Theorem here for normalmaps.

Remark 4.20« — We should emphasize one aspect of the equality <W(H) = res^^in 4.19 which might otherwise go unnoticed. Let iT(H) = (U, Fy, (3(U)) be anH normal map and let IT = (W, F^, (B(W)) be a G normal map. Then from thedefinitions involved, resg^ == c^(H) means in particular that resji ^(W) ==- <^(U) .

wFor later application we give examples of Smith sets <$^(W) = ^(W, G) and^(U) = ^(U, H, G) defined when reSnW = 8V for which resn <^(W) == ^(U)'By definition

<^(W, G) = {W1 -1 L C G, L }and (U, H, G) = (W, G) u { U1-1 L C H, L }.

To verify 4.2 (i)-(iii), note that 4.1 implies 4.2 (i) and 4.2 (ii) is clear. Here aresome remarks about 4.2 (iii). The G equivariant triangulation theorem of Illman [IJgives a triangulation ofW for which 4.2 (iii) is satisfied for <$^(W, G). Take an IllmanH triangulation of the closure U' of the complement of an H collar neighborhood CofWin U. Take an Illman H triangulation ofG which agrees on C n W and C n U'with the given triangulations there. One uses [1 , Theorem 4.3] for the last triangu-lation. Together, these triangulate U and (iii) is satisfied for ^(U, H, G).

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5. Proof of theorems A, B and B'

The proof of Theorem A combines 3.13, the Induction Theorem 4.19 and theone fixed point manifolds constructed in [P^].

Suppose 7) is a G vector bundle over W and R and R' are complex representationsof G whose difference lies in I(G). Then by 3.13 there is a Smith framing

(5.1) Pw(R ' ,R)^ R ' -^RrelW

for any Smith decomposition W of W. If b: .?(•/]) ->• s^) is any stable Gvector bundle isomorphism (R' framing) then there is an obvious Smith framing(3 = Pw(^'? R) o ^ : T] 5. obtained by composing b and the Smith framing Pw(^'? R)«This is used in the proof of Theorem A.

We begin with preparation for the proof of Theorem A by recalling materialfrom [P4], Remember R is the one dimensional trivial real representation of G.Henceforth G is as in Theorem A' (section i).

Results from [P^]: For each representation R e SS, there is a G manifold X == X(R)with these properties:

(5.2) (i) X° is one point u.(ii) There is a stable G vector bundle isomorphism

b(X) : ^(TXCR) ->J(RCR).

(iii) 2 — [X] lies in the subgroup F == A(G) + 20(G, ) of Q(G).(iv) For each H e , there is an H manifold U(H, R) whose boundary

is reSiiX(R) and there is a stable H vector bundle isomorphism&(U(H, R)) : J(TU(H, R)) -> resH s(fi C R) such that

resH^(X(R))=&(U(H,R)) .X(R)

(v) dim X is even; dim X11 == R11 and exceeds 5 (resp. 3) for H e^(resp. HeJ^)$ X satisfies 4.1 and HeIso(X).

(vi) TX is a stable G vector bundle and TU(H, R) is a stable H vectorbundle for H e^.

Remarks. — Condition (vi) is not explicit in [PJ, but is verified at the end of thesection. Actually the main result of [PJ is that X = X(R) may be taken to be ahomotopy sphere with the properties in 5.2. We do not require the additional depthof that condition but note that it implies 5.2 (iii) because then 2 — [X] GA(G). Tosee this, note from section 2 that that condition is equivalent to the condition that theEuler characteristic ^c(X11) is 2 whenever H e 8ft. To verify this use the fact that H,being in ^, has a subgroup P of prime power order for some prime p such that H/P

251


is cyclic. Note that X1' is an even dimensional mod? homology sphere because |P|is odd and dim S is even. The odd order cyclic group H/P acts on X1' with X11 asfixed set. Apply the Lefschetz Fixed Point Theorem to this situation to see that /(X11)is 2. Finally we remark that the existence of b(X) in 5.2 (ii) implies T^X(R) is iso-morphic to R.

We emphasize that the main conceptual conditions involved in the proof ofTheorem A below are 5.2 (i)-(iv). The others in 5.2 are necessary but technical.

The next lemma and its corollary are needed to construct a G manifold W whichamong other properties satisfies the condition that 2 — [W] eA(G). See section 2for definition of A(G).

Lemma 5.3. — For e == o, i, any R e S9 and any H e Jf, there is an R © R framedG manifoldV^ whose boundary Z^ has no fixed points and [Zy = (2 — 6e) [G/H] in t2(G, JT).

Proof. — For each subgroup H in Jf, we construct H manifolds Z^H) and V^H),s = o, i, with these properties: (i) V^H) has a res^ROR) framing; (ii) Z^H) isthe boundary of V^H); (iii) the class [Z^H)] in Q(H) is 2 (resp. -4) for s == o(resp. i). We shall then define V^GXnV^H), Z^ == G X n Z^H). Beginwith an oriented surface S with trivial H action and ^(S) == — 2. Since dim R11 : 2by (2.5 (i)), resH R == A © R2 for some representation A of H. Set Z°(H) == S(R © R),Z^H) == S x S(A © R), V°(H) == D(R © R) and V^H) == S x D(A © R), whereD(- ) denotes unit disk. Condition (iii) is a consequence of the fact that^(Z^H)1^) == 2 (resp. — 4) for all K. C H for e == o (resp. i). Conditions (i) and (ii)are nearly obvious. Their verification is left to the reader. From properties (i)-(iii),one may verify: (i') VH = G XH V^H) has an R © R framing; (ii') Z^ == G XH Z^H)is the G boundary ofV^; (iii') the class [Z^] in i2(G, Jf) is (2 - 6s) [G/H]; (iv') theG fixed set of Z^ is empty (since G ^J^).

Remark. — 2.Q(G,Jf) is freely generated by {2. [G/H] | H eJT}.

Corollary 5.4. — Let K be a finite set and {R, R^ [ u e K} be a set of representationsin 3i. Then there is a G manifold V with these properties:(i) There is a stable G vector bundle isomorphism 6(V) : ^(TV) -^(R©R).(ii) The boundary of V has no fixed points and 2 - [X(K)] - [BV] e A(G) for

X(K)= U X(RJ.u G •"•

proof. — Assertion: 2 — [X(K)] e F. For this, note from 2 .2 that

i eA(G) +^(G,^).

This implies 2 e F. In addition 2 — [X(RJ] e F for each u eK by (5.2 (iii));so [X(RJ] (u e K) and 2 are in F. This implies the assertion. It means that2 — [X(K)] == X + 20) for some X e A(G) and 2<o e 2t2(G, Jf7). By Lemma 5.3,

25^


and the subsequent remark, there is some choice of nonnegative integers flg, e = o, i,H ejf; so that [VJ = 2<o for Vo = UK.ZH. Let V = U^.V^. Then Vo = 3Vand conditions (i) and (ii) of this corollary hold.

Proof of Theorem A (§ i). — Let K be the finite set given in the statement ofTheorem A. For each u e K, we are given a representation R^ e . Fix a point zin K and set T == Rg. Add an additional point o to K to give a new set K' and setRO = T; so R^ is defined for all u e K'. Let Z = S(T®R) and D = D(TCR)be the unit sphere resp. unit disk in TOR. These are G manifolds. We shallconstruct a G normal map IT == (W, F, (B(W)), F : W -> Z and for each H e^ anH normal map ^T(H) = (U(H), F(H), (3(U(H))), F (H) :U(H)^D such thatres^ = a^(H).

First we construct W and U(H) for H e^. For u e K, let W^ = X(RJ (5.2)and for each H e^, let UJH) == U(H, RJ (5.2 (iv)). Corollary 5.4 providesa G manifold V such that (BV)0 = 0, 2 — [X(K)] — [BV] e A(G) and V has aT©R framing 6(V) of its tangent bundle. Set Wo = BV, Uo(H) = V andW = = U W,; so W^K and T ,W=R, for u e K. Set U(H) = U UJH)

M G K ' ^ ^K'whenever H e .

Then by 5.2 and the construction of V and Z we find

(i) reSnW = BU(H) for each H e^,(ii) 2 - [W] eA(G).

The degree one G map F : W -> Z is obtained by collapsing the complementof an invariant open disk centered at z e W (and identified with T) to (o, i) eZ0

and maps the disk homeomorphically onto Z — (o, i). Let F(H) : U(H) -^ D beany H map which extends res^F; so it too has degree i.

Next we construct Smith framings

(B(W) : TW © R -> T ® R and (B(U(H)) : TU(H) -^ T © R;

so that reSnp(W) == (B(U(H)) . Recall that the definition of a normal map (4.9)w

requires that these Smith framings are defined rel W and U(H) where W == W(<$^)and U(H) = U(H)(^') for Smith sets y and y for W and U(H). The equationresH (B(W) = (B(U(H)) requires resn ^ = ^ by 4.20. We take y = ^(W, G)

w wand y = (U(H), H, G); so by 4.20 the preceding equality is satisfied. Havingsaid this, set

P(W)= U ^(R,®R,R©R)o&(WJM£K'

(3(U(H)) = U pu«(H)(resHR«®R,reSHR®R)o&(UJH)).u £ K'

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See 5.1, 5.2 and 5.4 for notation here. Then(iii) resH (3(W) = (B(U(H)) for each H e^f.

wThis follows from the naturality of (3 ( • , •) with respect to restriction to subgroups andto subspaces (3.13). By construction(iv) resHF=F(H)( .

w

We assert that H^ and each ^(H) for H e^ are normal maps (4.9). Theconceptual elements of a normal map—an equivariant degree i map and Smith framingof tangent bundle—have been defined for^ and^(H). The added technical require-ments of 4.9 which must be verified for H^ and (H) follow from conditions 5.2 andare left to the reader. In order to apply 4.19,^ must satisfy 4.18 and resg == S^(H)/^/for each H e8^. Concerning 4.18, only 4.18 (i) is not obvious and that has beenachieved in (ii) above. The equations res^^ = <W^(H) for He^ have beenverified in (i), (iii) and (iv) above. As a consequence of 4.19, is normally cobordantto ^ == (W, F', p') where F': W -> Z is a homotopy equivalence. Then W isa homotopy sphere with W70 == W° = K and T^W == T^W == R^ for u e K byRemark 4.17. This completes the proof of Theorem A.

As promised earlier we verify 5.2 (vi) which requires that TW is a stable G vectorbundle and TU(H) is a stable H vector bundle for H e 8P. These conditions arerequired as part (4.9 (i)) of the definition of that H^ and ^(H) be normal maps. Itsuffices to show that TW^ and TUJH) for u e K' and H e are stable G resp.H vector bundles. By 5.2 (ii) and 5.2 (iv), TWy and R^ are stably isomorphic G vectorbundles and TUy(H) and R^ ® R are stably isomorphic H vector bundles for H e .In particular the G^ representations Ty^ Wy and R,, are isomorphic for all x e Wy andsimilarly for T^ UJH) with A:eUJH). From this stability (3.34) both for TW^and TUy(H) follows from:(i) dim R^ < d^ w^(RJ whenever H is a subgroup of G and ^ is an irreducible

representation of H such that ^(^) =^ ° (p^vided ^ 4= R).(ii) ^(R,CR)=^(RJ if x + R .

We must justify (i). By definition

(5-5) ^==dimaD^

where D^ is the division algebra of real linear H equivariant endomorphisms of %. Inour case G and hence H is odd order abelian; so d^ = 2 whenever % is a non-trivialirreducible representation. By hypothesis R^ eS9\ so if / is a non-trivial irreduciblecomplex representation of H it remains irreducible as a real representation and 2.5 (ii)gives

dim R^ === 2 dime R^ < 2^(RJ <_ 2w^(RJ

which is (i) because d^ is 2.

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Proof of Theorem B (§ i). — Let {Ry | u e K) be the representations in y givenin the statement of Theorem B. Since each R^ lies in «$ , R^ == o whenever G/H e 8ft.This implies that Iso(Ry — 0) C J?7; so by [PR, 9.163 p. 139], there is a representation SofG such that each R,, @ S is the realification of a complex representation R^e SS (2.5).

Now modify the proof of Theorem A. Use the representations { R ^ j ^ e K }to replace the representations {R^ | u eK} there. We do not know (and in generalit is false) that R^ — R^ e I'(G) for each u, v! e K; however, r : I(G) -> IO(G)is surjective and r(R^ — RJ = R^ — R^ e IO'(G) C IO(G). At the point in theproof of Theorem A where Theorem 3.13 is applied, use the fact that r(Ry — R^) == r{z)for some z e I(G). With this modification the proof of Theorem A yields the proofof Theorem B.

Proof of Theorem B\ — This is an immediate consequence of Theorem B andLemma 2.7.

We end this paper with the referee's comments showing that the homotopy sphere Sin Theorem A can be taken to be the standard sphere. The key point to note is themanifold W = II Wy constructed in the proof of Theorem A is a framed boundaryand the modification of W to W' = S using 4.19 is by framed surgeries; so S is aframed boundary. Since dim S is even, the famous work of Kervaire-Milnor implies Sis the standard sphere. To see that W is a framed boundary note Wo is by construction.The manifolds Wy for u eK come from [Pg] (see especially 1.7 and 2.10 there) andreferring to this one notes the Wy are framed boundaries for u e K. This means W isa framed boundary too.

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Department of Mathematics,Rutgers University,Newark, N.J. 07102

Manuscrit refu Ie g mai 1983.

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, tome 62 (1985), p. 5-40

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