Pacific Journal of Mathematics SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS KARL HEINZ DOVERMANN AND DONG YOUP SUH Vol. 152, No. 1 January 1992 PACIFIC JOURNAL OF MATHEMATICS Vol. 152, No. 1, 1992 SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS KARL HEINZ DOVERMANN AND DONG YOUP SUH Dedicated to Professor Y. Y. Oh on his 60th birthday For certain even order cyclic and some non-cyclic abelian groups G we construct smooth actions on homotopy spheres Σ with exactly two fixed points, ΣG — {p, q}, such that the tangential representations TΣ and TΣ are not isomorphic. P q 1. Introduction. Two real representations V and W of a finite group G are called Smith equivalent if there is a closed smooth mani- fold Σ which is homotopy equivalent to a sphere and G acts smoothly and effectively on Σ with exactly two fixed points, ΣG = {p, q}, such that the tangent spaces T Σ and TΣ at p and q are isomorphic P q to V and W as representations of G. Several authors have stud- ied the question of which groups do and which groups do not have non-isomorphic Smith representations. We shall recall their results below. In this paper we want to contribute two other classes of groups which have non-isomorphic Smith equivalent representations. The first one is as follows. Let H be a cyclic group of odd order, and G = H x Zk, k > 0. Below we shall give a list of conditions (see 2 Condition 2.2) for a pair (A, B) of representations of H. There are cyclic groups H of odd order and non-isomorphic representations A and B of H such that these conditions are satisfied. In Theorem 2.3 and Theorem 2.4 we quote two results from [DP2] and [DW] which provide examples. Theorem B of [DP2] shows that such groups H have non-isomorphic Smith equivalent representations. In §4 we will prove our first principal result, which extends the just quoted theorem. THEOREM A. Suppose H is a cyclic group of odd order which has non-isomorphic representations satisfying the conditions in 2.2. Assume that the order of H is divisible by at least three distinct primes. Then G — H x Zk has non-isomorphic Smith equivalent representations. 2 In particular, if A and B are two representations of H which satisfy Condition 2.2, then there exists an action of G on a homotopy sphere Σ with exactly two fixed points x and y, and T Σ - TΣ = I (A - B) X y 41 42 KARL HEINZ DOVERMANN AND DONG YOUP SUH for some I Φ 0. Here A and B are considered as representations of G with trivial action of Z*. // AφB then T Σ φ TΣ. 2 X y We give the details of the proof only in the case k = 1. For k > 1 more notation is needed but the argument is basically the same. In our second result we consider abelian groups G with at least three non-cyclic Sylow subgroups. Let A and B be real representation of G such that Condition. 1.1. (I) Aκ = Bκ = 0 whenever G/K is of prime power order. (2) Resp A = Resp B whenever P c G is of prime power order. (3) dim,4* = dim Bκ for zλ\ K c G. We shall show in Lemma 5.1 how to construct non-isomorphic rep- resentations A and B for any such group G. Our second principal result generalizes one of T. Petrie [PI] which we recall below. THEOREM B. Suppose G, A and B as in Condition 1.1. There exists an action of G on a homotopy sphere Σ with exactly two fixed points x and y such that T Σ- TΣ is a non-zero multiple of A-B. X y In particular, if AφB then T Σ φ T Σ. X X This is not only an improvement of Petrie and Randall's result, but we also give a proof which shows that Σ may be chosen to be equiv- ariantly cobordant to a product of surfaces as they are constructed in §3 of [DP2]. (See also §3 of this paper.) This will be used in [DKS]. In the proof of Theorem B we will use information about a surgery obstruction group for which we thank A. Bak. The study of Smith equivalent representations is motivated by a question of P. A. Smith [Sm] who asked whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [AB] and Milnor [M] established an affirmative answer to the question for semi-free actions and for actions of cyclic groups of odd prime power order. By definition, a semi-free action has the whole group and the trivial group as its only isotropy groups. Sanchez [Sz] showed that the an- swer is also affirmative for cyclic groups of order pq, where p and q are odd primes. Additional elementary considerations show that the answer is affirmative for any group whose order is a product of two primes. Bredon [B] showed for 2-groups that Smith equivalent repre- sentations are isomorphic if their dimension is large in comparison to SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS 43 the order of the group. Two Smith equivalent representations are said to be s-Smith equivalent if the action is semi-linear, i.e., we require in addition that the fixed point sets Σκ are homotopy spheres for every subgroup K c G. Sanchez's result also implies that s-Smith equiva- lent representations of any cyclic group of odd order are isomorphic. Petrie announced the first negative answer to Smith's question [PI], see [PR1] for the details of the proof, THEOREM. Suppose G is an odd order abelian group with at least four non-cyclic Sylow subgroups. There are non-isomorphic Smith equivalent representations of G. In this reference Petrie also posed the problem of finding all groups which have non-isomorphic Smith equivalent representations. Our Theorem A is a contribution to the solution of this problem. Since Petrie's announcement several authors provided classes of groups which have non-isomorphic s-Smith equivalent representations. One such class are cyclic groups of order Am, where m > 1. See the work of Cappell-Shaneson [CS1], Petrie [P2], Siegel [Si], and Dover- mann [D]. Non-isomorphic s-Smith equivalent representations were also constructed by Suh [Sul] for some non-cyclic abelian groups and by Cho [Cl] and [C2] for certain quaternion and dihedral groups. Non-isomorphic Smith equivalent representations of odd order cyclic groups were constructed by Dovermann-Petrie [DP2]. The groups were of rather large orders. Dovermann-Washington [DW] showed that such non-isomorphic Smith equivalent representations also exist for cyclic groups of small orders. The topic of Smith equivalent rep- resentations was surveyed in [DPS], [MP], [CS2], and Petrie-Randall [PR2] wrote a book about it. This review of the history shows that, basically, Petrie's question has been answered for cyclic groups, except for those groups whose order is of the form 2m where m is odd, and this is the class of groups we are treating in this paper. In Theorem A this is the case when k = 1. In case k > 1 the result is interesting for the discussion in [DKS], because we get some additional conclusion based on the specific construction. There we conclude that the actions described in Theorems A and B can be chosen to be real algebraic. In the construction of s-Smith equivalent representations for groups Z , with m > 1, the papers mentioned above use in an essential way 4m that the subgroup Z2 occurs as isotropy group. This implies that the m s-Smith equivalent representations restrict to the same representation 44 KARL HEINZ DOVERMANN AND DONG YOUP SUH of Ίj2 . This is not the case in [Sul], [Cl], and this paper. Here m one supposes non-isomorphic Smith equivalent representations of the group H, and then one uses them to construct such representations for the group G in which H is an index 2 subgroup. In fact, if V and W are Smith equivalent representations of a cyclic group G, and H is an index 2 subgroup which is an isotropy group of either V or W, then V and W are isomorphic (see [Su2]). Thus the non-isomorphic representations of G in Theorem A must also be non-isomorphic as H representations. Based on the different constructive approaches it happened that the technique of proof implied if one constructed non-isomorphic (s-) Smith equivalent representations for the group H, then one could also construct such examples for the group G in which H is a subgroup. These groups G had to be again of the same form as those groups one started out with. In [PR1] one would assume that H and G are abelian of odd order with at least four non-cyclic Sylow subgroups. In [DP2] and [DW] one would assume that H and G are cyclic of odd order and that H has non-isomorphic representations which satisfy 2.2. More generally we like to conjecture: Conjecture. Let H be a subgroup of G. If H has non-isomorphic Smith equivalent representations, then so does G. 2. Preliminary material. We shall formulate Condition 2.2 which is the essential assumption in our Theorem A, and we shall describe how to satisfy it. Let us fix some notation. For any group G we denote by 3°{G) the set of all subgroups of G of prime power order. Also, let 3? denote the set of all groups of prime power order. We use the follow- ing standard notation for the complex 1-dimensional representations of a cyclic group Z of order n. Consider Z as being identified with w w the ft-th roots of unity, so ZcC. The underlying vector space of n the representation tk is C, and under the action (g,v) is mapped to gkυ. For any cyclic group G of order n the complex represen- tation ring R(G) is isomorphic to Z[t]/(tn - 1). Thus any complex representation can be written as a linear combination of the elements in {tk\k = 0... , n- 1}. 9 Let G be cyclic of order n, and let V = Y^a^tk be a complex representation of G. For g e G such that the fixed point set Vg = {0} Atiyah and Bott [AB] defined a complex number SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS 45 Note that v carries sums to products; so we can define for any two representations V and W of G for which v is defined. Suppose V and W are Smith equivalent representations of Z sup- n ported by an even-dimensional homotopy sphere Σ, i.e., ΣG = {p, q) and TpΣ = V, TΣ = W. The Atiyah-Singer G Signature Theorem, q the Lefschetz Fixed Point Theorem, and Smith Theory imply Condition. 2.1. (1) Sign(G, Σp) = 0 for Pe^(G). In particular, v(Vp - Wp)(g) = ±1 if K<p>s> = W<p>s) = 0. Here (P, g) denotes the group generated by P and g GG. (2) The Euler characteristic /(Σ*) = 2 for all subgroups K cG. (3) Res/> F = Resp PF for each P e &>(G) of odd prime power order. An easy computation shows that v(V)(g) = ±v{V'){g) whenever V and V are isomorphic as real representations. Let U and U' be real representations of G such that Ug = £/'£ = 0. We write v(U)(g) = ±v(U')(g) or v(U - U')(g) = ±1 if C/ and t/' are realifications of complex representations V and V such that v{V){g) = ±ι/(F/)(g). This explains our notation in 2.1(1). In order to find non-isomorphic Smith equivalent representations we have to start out with two non-isomorphic representations which satisfy 2.1(1) and 2.1(3). Actually we will make some additional as- sumptions in 2.2 which will allow us to carry out the construction of the homotopy sphere which supports the Smith equivalent represen- tations. From now on, unless specifically stated otherwise, G denotes a cyclic group of order 2n where n is odd and H is the index two subgroup of G. Sometimes H just denotes an odd order cyclic group. Consider pairs (A, B) of real representations of H satisfying Condition. 2.2. (1) Ah = Bh = 0 for each h eH which generates a subgroup of prime power index in H. (2) dim Aκ = dimBκ whenever \H/K\ is divisible by at most 3 distinct primes. (3) Resp A = Res/> B whenever P e &>{H). (4) v{Ap - Bp)(h) = ±1 when ever P e & and heH generate a subgroup of prime power index in H. 46 KARL HEINZ DOVERMANN AND DONG YOUP SUH Observe that 2.2 (3) and some of the conditions in 2.2 (4) are nec- essary if we have Smith equivalent representations V and W of G which restrict to the representations A and B of H. For this com- pare the reference to [Su2] in the introduction. There are two ref- erences which guarantee the existence of groups H which have non- isomorphic representations satisfying all conditions in 2.2. THEOREM 23 ([DP2, Corollary C]). There are odd order cyclic groups which have non-isomorphic real representations satisfying Condition 2.2. If Z has non-isomorphic representations which satisfy Condi- m tion 2.2 and m divides an odd integer m!, then Z > has also non- m isomorphic representations which satisfy Condition 2.2. The groups in this theorem are rather large, and the representations cannot be given explicitly. The next reference improves on this result. In [DW] the reader can also find examples of non-isomorphic repre- sentations which satisfy the conditions in 2.2 for groups as in the next theorem. THEOREM 2.4 ([DW, Theorem A]). Let m=p\, ...Pk be a square- 9 free odd integer such that p\ is congruent to 5 modulo 8, the Legendre symbols [j^] are 1 for j > 2, and k > 4. Then the group Z has m non-isomorphic representations which satisfy 2.2. 3. One fixed point actions on manifolds. Our starting point is the construction of cyclic actions on surfaces. We shall use the following conventions. A real representation U of L is L oriented if Uκ is an oriented vector space for all K c L. A smooth L manifold X is L oriented if for all K c L each component of Xκ is oriented. Since a complex vector space understood as a real vector space has a canonical orientation, the realification of a complex representation of L is canonically L oriented. If X is an L oriented manifold, then T X is an L oriented representation. Here L = {g e L \ gx = x} X x x is the isotropy group at x. Let U be a representation. A product bundle X x U over X is denoted by U_ when the base space X is understood from context. Let L = Z be a cyclic group of order m, and tk the complex m representation of Z from the previous section, (fc, m) — 1. Let A+ m and A- be finite Z sets of the same cardinality, \A+\ = \A-\. m SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS 47 PROPOSITION 3.1. (See [DP2, 3.15].) There exists an oriented closed surface S with smooth orientation preserving action of Z such that m (1) Sκ = A% U A* for all proper subgroups K of L. (2) The tangent bundle TS is stably isomorphic to the product bun- dle Sxtk. (3) T S = Res t±k if x e Aξ and K c L. (Observe that T S X κ X and Res t±k are oriented representations, and the isomorphism is as- sumed to preserve orientations.) The problem in the application of Proposition 3.1 is the choice of the sets A and A- . To indicate our choice we need some more + notation. Let tk = ψ be an irreducible representation of G = Z . 2n We suppose that 2n/(k, 2ή) is divisible by at least two odd primes. We assign to it the group L(ψ) which acts effectively on ψ. It is obtained as follows. Let ker(^) be the kernel of the homomorphism ψo'. G —• 17(1) associated with ψ. This kernel is also the isotropy group of any non-zero vector in ψ. Then L(ψ) = G/kcv(ψ). Let m(ψ) be the order of L(ψ). Then m(ψ) = 2n/(k, 2ή). Observe that (m(ψ), k) = 1. Now A+ and A- are chosen as L(ψ) — % ( ) m ψ sets. The choice will depend on m(ψ) only. Choice 3.2. (1) If m(ψ) is odd we decompose m(ψ) as a product m(ψ) = r(ψ)s(ψ) such that (r(ψ), s(ψ)) — \. If m(ψ) is divisible by four primes we suppose that r{ψ) and s(ψ) are divisible by at least two primes. Choose a(ψ) and b(ψ) as natural numbers such that a(ψ)r(ψ) + b(ψ)s{ψ) + 1=0 (mod m(ψ)). (2) If m(ψ) is even, we set mr(ψ) = m(ψ)/'2. As in (1), we decom- pose mf(ψ) as r(ψ)s(ψ) such that (r(ψ), s(ψ)) = 1. Then we choose natural numbers a(ψ) and b(ψ) such that a(ψ)r(ψ) + b(ψ)s(ψ) + l = 0 (mod m(ψ)). In case (I) we set A+(ψ) = b(ψ) - [Z /Z ]ua(ψ) [Z /Z ] u [Z /Z ] m{ψ) r{ψ) m{ψ) s(ψ) m{ψ) m{ψ) In case (2) we set A+{ψ) = b(ψ) - [Z IZ ] υa(ψ) - [Z /Z ] u [Z /Z ]. m{ψ) lr{ψ) m{ψ) 2siψ) m{ψ) m{ψ) In either case \A+(ψ)\ = 0 (mod m(ψ)) such that we can choose A-(ψ) as free Z ( ) set with the same cardinality as A+(ψ). m ψ In the second step we assign to each irreducible complex represen- tation ψ = tk , for which m(ψ) is divisible by at least two primes, a 48 KARL HEINZ DOVERMANN AND DONG YOUP SUH surface X{ψ) with G action. First we use Lemma 3.1 to define an L(ψ) action on a surface which we call X'(ψ). Reduction modulo m(ψ) defines a homomorphism G—> L(ψ) and this induces a G ac- tion on X'{ψ). The surface with this induced G action is denoted by X{ψ). We describe the properties of these surfaces X(ψ), which are al- most identical with those listed in [DP2, Corollary 3.5]. As before we suppose that m(ψ) is divisible by at least two odd primes. LEMMA 3.3. Let X{ψ) be as above, and G{ψ) = ker(^). (1) Res ( )X(ψ) (with its trivial G{ψ) action) is a G(ψ) oriented G ψ boundary. (2) There is a representation A of G such that TX(ψ)®A = ψ_®A. For all subgroups K of G such that \G/K\ = 1, 2, an odd prime, or twice an odd prime (3) X{ψ)κ is a finite set (4) \X{ψ)G\ = 1, and \X(ψ)κ\ = 1 whenever \G/G(ψ)\ = \L(ψ)\ is divisible by at least four distinct odd primes. (5) \X(ψ')κ\ = \X{ψ")κ\ whenever G{ψf) = G{ψ"). (6) If G(ψ) = 1 or G(ψ) D Z2 then every isotropy group of X{ψ) is 1, or it contains Z. Thus G acts freely on X(ψ) - X{ψ)zz. 2 Proof Only (6) does not occur in [DP2]. It is an immediate conse- quence of our choice of A+ and A- . D Let U be a complex representation of a cyclic group G such that UG = 0 and Uκ = 0 whenever \G/K\ = 1, an odd prime, or twice an odd prime. Then U is a direct sum of non-trivial irreducible representations, U = Σaw(U)ψ. For each irreducible representation ψ for which a (U) Φ 0 the assumption on m(ψ) is satisfied and ψ X(ψ) is defined. We now define a G oriented manifold (3.4) The exponent a indicates an α^-fold cartesian product of X(ψ) ψ with itself. Next we study the properties of this manifold. They are derived from Lemma 3.3. These properties are exactly those in [DP2, 3.6-3.11], and the proof is unchanged as well. COROLLARY 3.5. Let U be a complex representation of G satisfying JJK = 0 if and only if \G/K\ = 1, 2, an odd prime, or twice an odd prime. Let X{U) be the G oriented manifold in 3.4. Then SMITH EQUIVALENCE FOR FINITE ABELIAN GROUPS 49 (1) There is a representation C of G such that TX(U) Θ C and U_®C_ are isomorphic G vector bundles. (2) άimX(U)κ = 0 if and only if \G/K\ = 1, 2, an oddprime or f twice an odd prime. (3) If \G/K\ Φ 1, 2, not an odd prime, and not twice an odd prime Resx X(U) bounds as an oriented K manifold. (4) χ(X(U)G) = 1 and χ{X(U)κ) is even whenever \G/K\ is not 1, 2, an odd prime, and not twice an odd prime. (5) Suppose \G/K\ = 1, 2, an odd prime, or twice an odd prime. Then X(U)K is a finite set and if y e X(U)K, then TX(U) = Res* U y as K oriented real representation. The cardinality of X{U)K depends only on {{K, dim Uκ) \ K c G and \G/K\ is divisible by at most three distinct odd primes }. To obtain Theorem A we will start with a collection S? of complex representations of G — H x Z2 where H is an odd order cyclic group. If U is a representation of G we denote its Z fixed point set UZ2 by 2 U2 which we also consider as representation of H. The complement is denoted by Uf, so U = U2 Θ Uf. The representations in ^ are assumed to satisfy Condition. 3.6. (1) If U e <9> and # c G, then I/* = 0 if and only if \G/K\ = 1, 2, an odd prime, or twice an odd prime. (2) If K e lso(U) then K = 1 or K D Z . 2 Each pair (Z), is) of representations in 5? satisfies (3) dimDκ = dimJS^ if \G/K\ is divisible by at most three distinct odd primes. Furthermore dim/) = dim is and dimZ) = dim is . 2 2 (4) v(Dξ-Eξ)(g) = 1 whenever Pe^(H) and g e H generates a subgroup of prime power index in H. (5) Df = Ef as representations of G. Note on Condition 3.6 (2). This condition expresses that if ψ is an irreducible summand of U then ψ is a summand of U2 or G acts freely on ψ - {0}. Hence G acts freely on U - U2. We list the essential properties of the manifolds X{U) obtained from a collection of representations 5? as in 3.6 when £/ e *5*\ It should be compared with [DP2, Theorem D on page 289]. We denote X(U)Z2 by X2(U) and consider it as an oriented H manifold. THEOREM 3.7. Let G = Z (n odd) and S? a collection of rep- 2w resentations of G which satisfy 3.6 (l)-(4). There is a collection of