Weight Systems for SO(3)-Actions

Annals of Mathematics

Weight Systems for SO(3)-ActionsAuthor(s): Robert OliverSource: Annals of Mathematics, Second Series, Vol. 110, No. 2 (Sep., 1979), pp. 227-241Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1971260 .

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Annals of Mathematics, 10 (1979), 227-241

Weight systems for SO(3)-actions

By ROBERT OLIVER*

Let G be an arbitrary compact connected Lie group, acting smoothly on a manifold M. Let H ( G be any (closed) subgroup and x e M any point fixed by H; then the action of G on M induces a linear representation of H on the tangent plane T,(M). If x and y lie in the same connected component of MH, then the H-representations of T,(M) and Ty(M) are isomorphic (the tangent bundle being equivariantly locally trivial). These tangential representations, together with the orbit types in M, provide the main "local information" about the action.

If M_ Di or R', and T is the maximal torus in G, then the fixed point set MT always has the cohomology of a point (see [1], Chapter III, ? 10), and in particular is non-empty and connected. One thus gets a unique tangential representation of T, not depending on the choice of point fixed by T, which is called the weight system for M (see [2]). Since any representation of T extends to at most one representation of G, the weight system contains as much information as any tangential G-representation can, if G has a fixed point.

If there is a fixed point x E MG, the weight system of M is isomorphic to the tangential T-representation T,(M), and is thus the restriction of a G-representation. Conversely, if a representation V of T is the restriction of an (orthogonal) G-representation V, then V is the weight system for the action of G on V or its unit disk D( V). So in the case of action of G with fixed points on either Euclidean spaces or discs, a representation of T can occur as the weight system if and only if it extends to a G-representation. Since the G-representations are known (in principle, anyhow), the interesting case left is that of which representations can occur as weight systems for actions with fixed points.

The main goal of this paper is to answer the question for fixed point free actions of SO(3) on disks. It turns out that this gives information not

0003-486X/79/0110-2/0227/015 $ 00.75/1 (? 1979 by Princeton University Mathematics Department

For copying information, see inside back cover. * Work supported in part by NSF grant MCS76-0146-AO1.



228 ROBERT OLIVER

only about what dimensions such actions can occur in, but also about what isotopy subgroups can or must occur. Thus, it provides a basis for classify- ing these actions based on local structure.

The subgroups of S0(3) are well known; the following notation will be used:

1 the trivial subgroup Zn the cyclic subgroup of order n (n ? 2) D? the dihedral subgroup of order 2n (n > 2) T the tetrahedral subgroup 0 the octahedral subgroup I the icosahedral subgroup SO(2) the maximal torus 0(2) the normalizer of S0(2).

As for the irreducible representations of the maximal torus S0(2), S will denote the trivial representation on R', and tV (n > 1) the representation on R2 defined by

(A, v)-> Av (A E SO(2), v e R2).

The following description of the irreducible representations of S0(3) follows easily from [7] (Section 20). It gives an explicit description of which SO(2)-representations can occur as weight systems for actions on disks with fixed points:

THEOREM. Any irreducible (real) representation of S0(3) is odd dimensional. For any n > 0, there is a unique irreducible S0(3)-representation on R2n ', whose restriction to S0(2) is s + t' + * + tn.

COROLLARY. The representation a0s + alt' + *-- + ajt? of S0(2) can occur as the weight system of an S0(3)-action with fixed points on a disk if and only if

ao > aa > .*- >an

In contrast to this, the following will be proved as the main result of this paper:

THEOREM 1. Let V = a0s + alt' + - * - + ants be an arbitrary SO(2)-repre-

sentation. For s > 1, set

ks = a8 + a28 + a38 + *

Then V occurs as the weight system of some fixed point free SO(3)-action on a disk if and only if



WEIGHT SYSTEMS FOR SO(3)-ACTIONS 229

(iii) O < k -2k2< a0, (iv) k2?+ k3+ K14> k-a0, (v) k3> k38 and k4> kc8 for all s > 2.

The motivation for these restrictions will be made clearer in Section 1. As consequences of this result (and the constructions used in proving it),

we get 1. If S0(3) acts smoothly on DI without fixed points, then n > 8 and

Iso(D-) 2 {1, Z2, D2, D3, D4, O 0(2)}. 2. There is a fixed point free action of S0(3) on D8 with only the above

seven subgroups as isotropy subgroups. 3. The only possible weight systems for actions on D8 are t1 + tk+ t3+ t'

with k +2 (mod 12). 4. There exist smooth actions of S0(3) on S7, all of whose orbits are

3-dimensional (all isotropy subgroups are finite). These results thus give counterexamples to a theorem of Montgomery and Samelson ([3], Theorem 4), that any smooth action of S0(3) on S7 has orbits of dimension less than three. The error in their proof occurs in Case 3 (p. 656), in the description of the space T- F(H).

Section 1

The classification of weight systems for fixed point free S0(3)-actions on disks will be based on comparing the tangential representations at fixed points of 0(2) and 0. We must first see that fixed points for these subgroups always do occur.

LEMMA 1. Assume that SO(3) acts smoothly on a disk D. Then D0'2) and D' are both non-empty.

Proof. By Smith theory (see, e.g., Chapter III of [1]), DSO(2) has the integral cohomology of a point, and so

DO (2) = (DsO (2))Z2 (Z2 = 0(2)/SO(2))

has the Z2-cohomology of a point. In particular, it is non-empty. The octahedral group 0 has normal subgroups D2 < T < 0, where D2

and O/T are 2-groups, and T/D2 _ Z3 is cyclic. We thus have 0 e 9 (in the notation of [4]), and any smooth action of 0 on a disk has a fixed point by Proposition 2 of [4]. D

Assume now that S0(3) has a given smooth action on a disk D. For any g e S0(3) and any x e D fixed by g, we let xx(g) denote the character of g for its representation on TX(D). The following lemma will make it possible to compare the tangential representations at fixed point sets of different sub-



230 ROBERT OLIVER

groups.

LEMMA 2. Assume that g1, g2 e SO(3) are conjugate elements of prime power order and let x1, x2 e D be points fixed by g1, and g2, respectively. Then xX1(g1) = X2(92)-

Proof. Choose h e SO(3) such that g2 = hglh-'; then hx, is fixed by g2. Since the subgroup generated by g2 has prime power order, its fixed point set is connected (it has the Zp-cohomology of a point if g2 has p-power order). Thus, the representations of <g2> on T.2(D) and Th,,(D) are isomorphic, and XX2(g2) =hll(92)

Furthermore, the action of h on T(D) sends T,1(D) isomorphically to Th,1(D), and sends the action of g, on the first space to the action of g9 = hglh- on the second. So X.,(g9) = Xh.,(g2)- D-1

Any real representation V of SO(2) is invariant under the action of the Weyl group 0(2)/SO(2), and its character Xv thus extends to a unique character X of SO(3). If X 10 (X 10121) is the character of an actual 0-(0(2)-) representation, then this will be referred to as the associated 0-(0(2)-) representation to V. We can now list restrictions necessary for an SO(2)-representation to be a weight system:

PROPOSITION 1. Assume that SO(3) acts smoothly and without fixed points on a disk D, and with weight system V. Then

( 1 ) V has an associated 0-representation W, (2) V has an associated 0(2)-representation, (3) V has tV as a direct summand, (4) dim (WD4) > dim (WO), (5) dim (Vz3) > dim (VZ38) and dim (Vz4) > dim (VZ48) for all s > 2.

Proof. Let X denote the character of SO(3) extending X, Fix a point x e DO (Lemma 1). All elements of 0 have prime power order; thus for any g e 0 and any g' e SO(2) conjugate to g,

X%(g) = Xv(g') = X(g)

by Lemma 2. It follows that the character of the 0-representation on T,(D) is X 1, and so T,(D) is the associated 0-representation to V. Similarly, for any y e D0'2', T,,(D) is the associated 0(2)-representation to V (any element of 0(2) not in SO(2) has order 2).

To see that (3) holds, fix a point y e D0o2'. Since the action is assumed fixed point free, y must lie in an orbit of type SO(3)/0(2) -RP2, and the tangential representation of SO(2) on the orbit must be a direct summand of V_ T,,(D). The action of SO(3) on RP2 is that induced by the standard




action on S2, and so the tangential representation at the point fixed by SO(2) is easily seen to be tV.

To prove (4), fix subgroups 0(2) D, C: 0 in SO(3). Then

DD4 _ DI U D0'2' and Do n D0'2) = DG = 0

(G being the smallest closed subgroup containing 0 and 0(2)). Furthermore, DD4 is connected (D4 being a 2-group), D0O2) # 0 by Lemma 1, and so

dim (DD4) > dim (DO) . On the other hand, since W is the tangential representation at any point fixed by 0, we have

dim (WD4) = dim (DD4) and dim (WO) = dim (DO),

and so dim (WD4) > dim (WO). Similarly, to prove (5), fix a subgroup SO(2) C SO(3), and let Zn, for all

n > 2, denote the order n subgroup of SO(2). If 0 denotes an octahedral subgroup such that o f SO(2) = Z3, then for all s > 2,

DZ3 _ DZ3s U DO and DZ38 f D0 - DG = 0.

DZ3 is connected, and so dim (DZ3) > dim (DZ38). Again, dim (VZ3) = dim (DZ3) and VZ38 has the same dimension as at least one connected component of DZ38,

so the first part of (5) holds. The second part follows similarly by choosing o such that On SO(2) = Z4. El1

We must now make these conditions more explicit, in terms of the coefficients of s and tV in V. This clearly involves studying the representations of 0. Since 0 is isomorphic to S4, we will use permutations of {1, 2, 3, 4} to denote elements of 0, when necessary.

The five irreducible representations of S4 will be denoted W1, WY1, W2 T W3, and W3': W. denotes the orientation preserving representation on RI, and W' the unorientable representation on R%. Thus, W, is the trivial representation, W,' the representation fixing A4 C S4, and W2 the standard representation of S4/D2- S3 on R2. Furthermore, W3 is the restriction of the SO(3)-representation on R3, and W3 T3? W,' can be obtained by taking the permutation representation of S4 on R4 and removing a trivial summand.

We now calculate the coefficients of the associated virtual representation to an SO(2)-representation.

LEMMA 3. Let V = a0s + alt' + *-- + ant, be an SO(2)-representation. For all s > 1, set

ks = a. + a28 + a38 + * Let W denote the associated virtual 0-representation to V; namely, the



232 ROBERT OLIVER

representation whose character is obtained by extending Xv to an S0(3)- character and then restricting to 0. Then

W = (ao - ki + k2 + k3 + k4) W1 + (k3-k4) W1 + (k2- k3) W2 + (k - 2k2 + k4)W3 + (k2- k4) W3.

Proof. Since any g e 0 has order at most 4, it is conjugate (in S0(3)) to any element of S0(2) of the same order. The character of W can thus be directly calculated from X, giving the following:

Xw(J) = ao + 2k, X %w((123)) = ao - ki + 3k3, Xw((12)) = Xw((12)(34)) = ao - 2k1 + 4k2, Xw((1234)) = ao - 2k2 + 4k4.

Upon comparison with the character table for 0 _ S4 (see, e.g., ? 5.8 of [6]):

g W1 W' W2T W3 W3

1 1 1 2 3 3 (12) 1 -1 0 -1 1 (123) 1 1 - 1 0 0 (1234) 1 -1 0 1 -1 (12)(34) 1 1 2 -1 -1

the result follows by direct calculation. D The conditions listed in Proposition 1 can now be rewritten as follows:

PROPOSITION 2. Let V = a0s + a *t' + + ant" be an SO(2)-representation, and for all s > 1 set

ks = a. + a2. + a3. + *

Then VfulJills conditions (1)-(5) of Proposition 1 if and only if

(i) a, > O, (ii) k2> k3> k4, (iii) O < k -2k2 <a0, (iv) k2+ k3 + k4?> k1-ao, (v) k3> k3.y k4> k4s for all s > 2.

Proof. Condition (1), that V have an associated 0-representation W, is clearly equivalent to requiring that all of the coefficients given by Lemma 3 are non-negative; i.e., that

k2+ k3 + k4?> ki-ao, k2> k3> k4, and k1 + k4> 2k2- Note that the last is implied by (iii) above.

We now show that (iii) is equivalent to condition (2), the existence of an associated 0(2)-representation to V. Let g denote the element of order 2 in SO(2), and h any element of 0(2) not in SO(2). Then g has characters

x&g) = 1 %tn(g) = 2 - (-1) y




and so Xv(g) = a, - 2k, + 4k2. For any n > 1, t" extends to a unique representation th of 0(2) on R2, such that Xtn(h) = 0. Let s+ and s- denote the trivial and non-trivial 0(2)-representations on RI, respectively; thus any 0(2)- representation V extending V has the form

V= a+s+ + as- + at' + a2t2 a with xv(h) = a+- a. So V has an associated 0(2)-representation if and

only if one can write ao = a+ + a- (a- , a- > 0) such that

ao-2k + 4k2 = a+-a-; namely k1-2k2 = a

and this can be done if and only if 0 < k - 2k2 < ao. One easily checks that W1 and W2' are the only 0-representations on

which D4 has positive dimensional fixed point set. Thus dim ( WD4) > dim ( W0) if and only if the coefficient (k2 - k3) of W2' is positive. In other words, (4) holds if and only if k2> k3.

Clearly, (3) is equivalent to (i) and (5) to (v). Combining the above observations now gives that conditions (1)-(5) are equivalent to (i)-(v). D-1

The following additional observation about the associated 0-representations will be needed for the constructions in the next section.

PROPOSITION 3. Let V be an SO(2)-representation fulfilling conditions (i)-(v) above, and let W be its associated 0-representation. Then W contains (W2' D W ') as a direct summand.

Proof. Referring back to Lemma 3 for the coefficients of these representations, one checks:

W':k2-k3 by (ii), W3: k, - 2k2 + k4 > 0 by (iii) and (v), (k4> 0 by (v)) W3 : k2-k4 > 0 by (ii). L

Section 2

Actions of S0(3) will now be constructed to prove the converse of Proposition 1. The original motivation for the constructions comes from Theorem 4 of [5] and Theorem 6 of [4]. It is based on the use of G-complexes: complexes built up by attaching cells of the form G/H x DI to lower dimensional skeletons.

Theorem 4 of [5] (and its proof) describe a procedure for constructing a finite SO(3)-complex, such that S0(3) and I act without fixed points, and all other subgroups have contractible fixed point set. The procedure is simply to construct the complex explicitly step by step, at each stage finding a



234 ROBERT OLIVER

maximal subgroup H whose fixed point set is not contractible (and should be), and adding cells G/H x DI to make it so. Carrying this out, one obtains a contractible SO(3)-complex with orbit space D2, and with orbit structure described by the following diagram:

SO(3)/D4

SO(3)/0(2) O3Z2 SO(3)I0

SO(3)/D3

In other words, the complex contains 5 orbits of cells: SO(3)/0(2), SO(3)/O, SO(3)/D4 x D', SO(3)/D3 x D', and SO(3)/Z2 x D2.

One then wants to "thicken up" the above complex replacing cells by handles to obtain a smooth compact G-manifold of the same homotopy type (which can be shown to be a disk using the h-cobordism theorem). Theorem 1 below can be proved in this way, but it turns out to be easier to use the following alternative procedure. We start with equivariant disk bundles M and M' over SO(3)/0(2) and SO(3)/O, respectively, which have the correct tangential representations. A complex X is defined, consisting of two orbits SO(3)/D3 and SO(3)/D4 and a 1-cell SO(3)/Z2 x D'. Embeddings of X in a M and AM' are constructed, and shown to have equivariantly diffeomorphic regular neighborhoods N and N'. Then M U NM' is a compact manifold with smooth SO(3)-action; it has the equivariant homotopy type of the contractible SO(3)-complex described above, and is shown to be a disk.

First let G be an arbitrary compact Lie group with subgroups Ho, H1 C G and Hc Ho n H1. By the simple G-complex of type (HO, H1; H) will be meant the complex

(G/H x I)/I (I = [0, 1]) where the relation is defined by setting (gH, t) - (g'H, t) whenever

(a) t = 0 and g-lg'e Ho or (b) t = 1 and g-lg'e H1 . The complex consists of two 0-cells G/HO and G/H1 and a 1-cell G/H x D'.

Let X be the simple G-complex of type (HO, H1; H), and assume that Ho and H, are finite. By a smooth embedding of X in a G-manifold M is meant an equivariant embedding f: X -> M such that the composite

G/HxI >X f M

is a smooth immersion. Embedding X smoothly is clearly equivalent to choosing an arc f: I--> M transverse to orbits, still an embedding in MIG,




and such that f(O) e MH0, f(l) e MH1, and A() C MH (MK denoting the set of elements with isotropy subgroup K, for any K _ G). As described above, we need some procedure for showing that two smooth embeddings of X have equivariantly G-diffeomorphic regular neighborhoods.

For any x e M with isotropy subgroup K, the slice representation at x is as usual the normal K-representation v,(Gx, M) to the orbit of x. Note that two points x, x' e MK have isomorphic slice representations if and only if they have isomorphic tangential representations.

Now fix Ho, H, _ G and H( H, f H1. Let V0 be an HO-representation and V, an H1-representation, such that V0 and V1 are isomorphic as H-representations. Define an action of the group

AutH0 ( V0) x AutH1 ( V1) on S( VO)H X S( VJ)H X ISOH ( V0, V1) I

by setting: (p, *)(a,, al, f) = (9(a0), W(a,), *,SWlq). This induces an action of

ro(AUtHO( VO)) x ro(AutH,( V1)) on r0(S( VO)H) X w0(S( VJ)H) X w0(ISoH ( V0, V1)) Let 2(Ho, VO; H1, V1; H) denote the set of orbits of this action.

Let X again be the simple G-complex of type (HO, H1; H), and let f: X-> M be a smooth embedding such that x0 = f(e, 0) and xl = f(e, 1) have slice representations isomorphic to VO and V1, respectively. Let f be the arc from x0 to xl defined by f(t) = f(e, t).

For i = 0, 1, fix an identification of Vi with the slice representation at xi, and embed it via an exponential map into M. By reparametrizing X if necessary, we may assume that f lies in the images of these embeddings near its endpoints. Elements

ao(f) E wO(S(VO)H) and al(f) E r0(S(V)H) can now be defined as those components of (Vi - O)H in which f((O, 3)) and f(( - 3, 1)) lie for 3 sufficiently small.

Furthermore, the slice representations at points of f form an H-vector- bundle over the arc, with VO and V, identified with the fibers over x0 and x1, respectively. This determines a connected component a2(f) in ISOH(V1, V2). Changing the identifications of VO and V, with the slice representations corresponds to the action of some automorphism of VO x V1 on (ao(f), a,(f), a2(f)). So upon letting a(f) be the orbit of this triple, we get a well defined invariant in 2(HO, VO; H1, V1; H).

PROPOSITION 4. Assume Ho and H, are finite subgroups of G, He' Ho n H1, and let X be the simple G-complex of type (HO, H1; H). Let f and f' be smooth embeddings of Xinto M and M', respectively, such that a(f) = a(f'). Then f(X) and f'(X) have G-diffeomorphic regular neighborhoods.



236 ROBERT OLIVER

Proof. Consider first the embedding f, and let f be the arc f(t) = f(e, t), with endpoints x, = f(O) and xl = f(1). Let

9o:G X HOD(VO) - M, 9l:G X HD(Vl)> M

be G-diffeomorphisms onto tubular neighborhoods of Gx0 and Gx, (with 9i(e, 0) = xi), and identify D(Vi) with the slice 9i(e x D(Vi)) at xi. By choosing the neighborhoods small enough, we may assume that f intersects their boundaries transversely and uniquely. As before, X can be repara- metrized if necessary so that f(I) n Im (pi) lies in D(Vi); let ai e S(Vi) be the point of intersection of f with S(Vi).

A regular neighborhood N of f(X) can now be constructed by attaching to G XH0D(VO) and G xH1D(VJ) a 1-handle of the form (G xHD(V)) x Di, where V is the H-representation obtained by restricting VO (or V1) to H and subtracting a trivial summand. Such a construction is determined by the points ao and a, (where (e, 0) x DI is attached), and an H-linear isomorphism ,8: Vo -> V1 such that f8(ao) = - a. Then a(f) is the connected component of the triple (ao, al, f8) in

S(VO)H X S(VI)H X ISOH (VO V1)-

Define corresponding elements a' e S(Vi)H and A' G ISOH(VO0 V1) for f', and let N' be the corresponding regular neighborhood of f'(X). Since a(f) = a(f'), the identifications of Vi with the slice representations at xi and x' (=f'(e, i)) can be chosen such that (ao, al, ,8) and (a', al, ,f') lie in the same connected component of

S(VO)H X S(Vl)H X ISOH (O V1)-

Using Theorem VI. 3.1 of [1], we can construct G-diffeomorphisms of D(VO) and D(Vl) (isotopic to the identity) sending a, to a' and a, to a'. Thus, N' is G-diffeomorphic to some regular neighborhood N" of X obtained by attaching the 1-handle at ao and a, via some f8" G ISOH ( VT, V1) in the same component as fi'. But f8" then lies in the same component as f, and so N" is G-diffeomorphic to N (just reparametrizing the 1-handle). D

We can now prove the converse of Proposition 1:

THEOREM 1. Let V = a0s + alt' + * + ant" be an SO(2)-representation,

and set k8 = a8 + a28 + a38 +

for all s > 1. Then V is the weight system of some smooth fixed point free action of SO(3) on a disk if and only if

(i) a, > O, (ii) k2> k3> k4,




(iii) Ok1-2ck2?ao, (iv) k2+k3+?k4>ki-ao, (v) k3> k38 and k4> k48 for all s > 2.

Proof. It has been seen (Propositions 1 and 2) that the above conditions are necessary for V to be the weight system of such an action. Conversely, assume that V fulfills (i) through (v) above, and let V and W be its associated 0(2)-and 0-representations, respectively (existing by Proposition 2). Note that

VID3- WID3 and VID4 - WID4

by the construction of V and W. Let W' denote the orthogonal complement of some W3 C W (applying

Proposition 3). Let V' denote the complement of some T' V (T' again denotes the extension of t' to an 0(2)-representation, and is found as a sub- space by (i)). Define manifolds

M= SO(3) x0(2,D(V') and M' = SO(3) xOD(W').

Before continuing the construction, we must fix some specific subgroups of SO(3). First set

ab 0 ab 0 SO(2) = d 0 e SO(3) and 0(2) c d 0 e S(3)

For any n > 2, Zn, will denote the order n subgroup of SO(2), and Dn will denote the subgroup generated by Zn, and

Z2 it I -1 0 08.

Now fix 0 as the subgroup of all matrices with one non-zero entry in every row and column. We have 0 containing D4, but not D3. To fix this,

0 1 0 define 1D3 to be the subgroup generated by Z2 and 001.

1 0 0 Let X and X denote the simple SO(3)-complexes of type (D4, D3; Z2) and

(D4, D3; Z2), respectively. To see that X X, it suffices to find an element g in the identity component of N(Z2) such that gD3g-1 =D3. The identity component consists of all rotations around the axis {x, + X2 = x3 =0}; take g to be the rotation which sends the axis {x, = X2 = 0} to {X1 = X2 = X3}.

By (v) above, D3 and D4 occur as isotropy subgroups in S(V'), and an arc f can be defined on S(V'), transverse to the orbits and embedded in the orbit space, such that f(O) and f(l) have isotropy subgroups D4 and D3,



238 ROBERT OLIVER

respectively, and f(IJ) C S( V'),2. Then f extends to a smooth embedding f of X in AM.

In order to embed X in AM', first fix subspaces W2, W3' C W' (using Proposition 3). Choose elements xs e S( W ) and xl e S( W') with isotropy subgroups D, and D3, respectively, and let f' denote the geodesic on S( W,' W') connecting them. Then f'(I) C S( W')z2, and so f ' induces a smooth embedding of X X into AM'.

Now, f(O) and f'(O) have isomorphic tangential representations (namely, VID, and Wj ,, respectively), and similarly for f(1) and f'(1). It follows that the slice representations for Do and D3 are also isomorphic for f(X) and f'(X); let V0 and V, denote these slice representations (in AM or aM'). We have

VO - W'I Do -es and V1 _ W'ID3 (e denoting a trivial representation on R'). In order to show that f(X) c AM and f'(X) C AM' have equivalently diffeomorphic regular neighborhoods, it will suffice to show that

12(D,, VO;D3, V1;Z)| = 1.- We have

VO_ W2'LD, ? W ,I - R1 EIR2ED Y2 and

VlWD3? W+ D3-6$e2 V1 _ W2 I3+W3 |D 2 f e

Here, 772 and $2 denote the (unique) irreducible representations on R2 of Do and D3, respectively. R1 and R2 denote non-trivial representations of Do on R'; R2 fixes D2 (=Z2 x Z'), and R1 fixes the other Klein 4-subgroup of Do.

Since $2 e $2 ' V1, Vf3 has codimension at least 2 in Vi", and so S(Vi)z; is connected. In (R D R2 72) the fixed point set of D2 is contained in that of Z' with codimension 1; thus S( VO)z; has either 1 or 2 connected components.

Let Vi+ and Vi- denote the + 1 and -1 eigenspaces, respectively, of the Z2-action on Vi. Then

Autz;(Vo) _O(n+) x O(n-),

where n+ (n-) is the dimension of Vo+ (V,-). Iso,2(Vo, V1) is clearly homeo- morphic to this, and thus has four connected components, depending on the orientations of the maps from VO+ to V + and VO- to V-. Similarly, AutH% (Vi) is a product of groups of the form 0(n), U(n), and Sp(n), one for each isomorphism class of irreducible summands in Vi; and so wro(AutHi (Vi)) is a product of Z2's: one for each irreducible summand which cannot be given a complex structure.




The following table describes the action of irO(AutHO ( VO) x AutH1 (Vi)) on

rO(S(V0)z2) x r0(S(V,)Z) x iro(IsoZ,(Vo, V1)) (SO or pt.) x So x So.

In each row the action is that of the automorphism of VO or V1 which negates the summand listed on the left and fixes its complement. A "+" means that the appropriate So-factor is fixed by the action, and a "-" that the two elements are switched. It is clear from the table that the action is transitive, and thus that | (D4, VO; D3, V,: Z') = 1.

Summand S(Vo)zj Iso(VO+, V,+) Iso(V0-, Vi-) R1 - +

R2 + _+

$2 + _ _

Proposition 4 now applies to show that f(X) and f'(X) have SO(3)-diffeomorphic regular neighborhoods N A dM and N' _ AM'. Let (p: N -+ N' be a diffeomorphism, and set D = M U ,M'. SO(3) acts smoothly on D without fixed points, and with weight system V. Since dim(D) > 8, the h-cobordism theorem applies to show that D is a disk if H*(D) = 0, and if D and aD are both simply connected.

Note first that the inclusions

X_-- N N', SO(3)/0(2) C M, and SO(3)/0 M'

are all homotopy equivalences. fi*(D) can thus be computed from the Mayer- Vietoris sequences

*. -* H%(SO(3)/Z2) - fti(SO(3)/D3)) eFi%(SO(3)/D4)-* Hi(X) - * .

*. -* ,H)(X) - Fi%(SO(3)/O(2)) fti(so(3o/0) -* H(D) -* * *

where maps between orbits are induced by inclusion of subgroups. One now simply computes:

SO(3)/Z' SO(3)/D3 SO(3)/D4 X SO(3)/0(2) SO(3)/0 D H, Z Z5 Z2 E Z2 Z2 0D Z2 Z2 Z2 ? H3 Z Z Z Z 0 Z 0 Ai (i 1,3) 0 0 0 0 0 0 0

To compute 7r1(D), apply the van Kampen theorem to the triple (M, M', N). For any finite subgroup HC SO(3), 7r1(SO(3)/H) H*, where H* is the "double cover" of H in the universal covering group Sp (1) of SO(3). Since 0* is generated by D* and D*, we have a surjection

Ths(N) _r1M(X) o r1(MD ) a Dr(Sc(3)/0) n

Thus, 7rl(M) Z2 surjects onto 7r,(D), and D must be simply connected



240 ROBERT OLIVER

(H1(D) = 0). To compute 7r1(aD), first define AM = AM - N and AM' = AM' - N'.

Then AD is the union of AM and AM' intersecting at AN. Since N X, M:D SO(3)/0(2), and M' Q SO(3)/0 are all regular neighborhoods, the inclusions

aNd-N-X, aMzdM-X, aMzdM'-X, aMzM- (S0(3)/0(2)), and aM'M'- (SO(3)/0)

are all deformation retracts (this is, in fact, clear from the constructions). Furthermore, all of the subcomplexes which are subtracted off have codimension at least 3 (dim (X) = 4 and dim (D) > 8), and so the inclusions

AN _M N. A M C AM C M and AM' AM' M'

all induce isomorphisms of 7r,. By the van Kampen theorem,

ir1(aD) _ r1(D) = 0. D By Proposition 2, Theorem 1 can also be stated in the alternate form:

THEOREM 1'. Let V be any (real) SO(2)-representation. Then V is the weight system of a smooth fixed point free action of SO(3) on a disk if and only if

( 1 ) V has an associated 0-representation W, (2) V has an associated 0(2)-representation, (3) tC Vt (4) dim (WD4) > dim (WI), (5) dim(Vz3)> dim(VZ38) and dim(Vz4)> dim(VZ48) for all s>2. D Among the corollaries of this result are:

COROLLARY 1. (a) If SO(3) acts on D" without fixed points, then n > 8 and Iso (DI) D

{1, Z2, D2, D3, D4, 0, 0(2)}. (b) There is an action of SO(3) on D8 with only the above seven groups

as isotropy subgroups.

Proof. (a) Let W be the tangential representation for 0; by Proposi- tions 1, 2, and 3, W contains We W3D W3. So n dim(W) ? 8. Fur- thermore, any orbit SO(3)/O in Dn must contain

SO(3) X 0(W2 W3) in a tubular neighborhood, and 1, Z2, D2, D3, D4, and 0 all occur as isotropy subgroups in that space. We have already seen that 0(2) must be an isotropy subgroup.




(b) Consider V = t' + t2 + t3 + t'; it clearly meets conditions (i) through (v) in Theorem 1. Thus, SO(3) has a fixed point free action on D8 with V as weight system, and the action can be constructed so that D8 is the union of

SO(3) X 0(2)D(t te t) and SO(3) x . D( W2ti WD.

One easily checks that the above seven subgroups are the only ones which occur as isotropy subgroups in these spaces. D

COROLLARY 2. There is a smooth action of SO(3) on S7 consisting of 3-dimensional orbits.

Proof. Consider any SO(3)-action on D8 with weight system t' + t2 + t3 + tV. Then (D8)s0(2) is zero-dimensional, is thus contained in the interior, and all isotropy subgroups in S7 are finite. El

In the case of actions on D8, one can simplify the description of which weight systems are possible.

COROLLARY 3. The 8-dimensional SO(2)-representations which can be weight systems for fixed point free SO(3)-actions on D8 are

t D tk ' t3 D t for any k _ +2 (mod 12).

Proof. Any such representation clearly satisfies conditions (i) through (v) of Theorem 1. Conversely, if V satisfies these conditions, and is 8-dimensional, then

2k2??k, 4 and k2> k3>k4> O

imply that k, = 4, k2 = 2, and k3 = k4 = 1. By (i) and (v), tl ' t3 t4 must be contained in V, and the fourth term tk must be such that 21 k, 3 4' k, 4 t k; namely, k _ ?2 (mod 12). El

STANFORD UNIVERSITY, CALIFORNIA

REFERENCES

[1] G. BREDON, Introduction to Compact Transformation Groups, Academic Press, 1972. [2] W.-C. and W.-Y. Hsiang, Differentiable actions of compact connected classical groups,

II, Ann. of Math. 92 (1970), 189-223. [31 D. MONTGOMERY and H. SAMELSON, On the action of S0(3) on Sn, Pacific J. Math. 12

(1962), 649-659. [41 R. OLIVER, Fixed point sets of group actions on finite acyclic complexes, Comment. Math.

Helv. 50 (1975), 155-177. [51 , Smooth compact Lie group action on disks, Math. Z. 149 (1976), 79-96. [61 J.-P. SERRE, Representations Line'aires des Groupes Finis, Hermann, 1971. [7] B. L. VAN DER WAERDEN, Group Theory and Quantum Mechanics, Springer-Verlag, 1974.

(Received October 31, 1977)



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Weight Systems for SO(3)-Actions