ON SL(3,R)-ACTION ON 4-SPHERESHINTARÔ KUROKIAbstract. We construct a natural continuous SL(3, R)-action on S 4 which is an extensionof the SO(3)-action ψ in [8]. The construction is based on the Kuiper’s theorem in [3] thatthe quotient space of CP (2) by complex conjugation is S 4 . We also give a new proof to theKuiper’s theorem.

1. Introduction
One of the central problems in transformation groups is to classify actions of a Lie groupon a smooth manifold (for example actions of a special orthogonal group SO(n) on a complex
projective space and actions of a special linear group SL(n, R) on a sphere). When the Liegroup is compact, there are a rich history and an abundant work. However, when the Liegroup is not compact, not much is known. We consider non-compact Lie group actions inthis paper.
The classification problem of group actions would be beyond our reach unless we put
some assumption on the actions. When the Lie group is SL(n, R) and the manifold is anm-dimensional sphere S m , there is a history in the classification problem. In 1974 C.R.
Schneider ([4]) succeeded in classifying real analytic SL(2, R)-actions on S 2 , more generally
on surfaces. F. Uchida classified real analytic SL(n, R)-actions on S n for n ≥ 3 in 1979 ([6])
and on S m for 5 ≤ n ≤ m ≤ 2n − 2 in 1981 ([7]). On the other hand, it is well known thatan effective continuous SL(n, R)-action on S n−1 is unique and any SL(n, R)-action on S m istrivial for m ≤ n − 2 ([2, Theorem 3.5]). From these result, the cases which remain unsolvedin the range m ≤ 2n − 2 are (n, m) = (3, 4), (4, 5) and (4, 6).
All the solved cases above have the property that the actions of SL(n, R) restricted to themaximal compact subgroup SO(n) are unique, and this property played an important rolein the classification. However the retricted actions are not unique in the remaining cases
listed above. In these cases, we need to see how many SO(n)-actions there are on S m (upto equivalence) and then check which of them extends to an SL(n, R)-action.
In this paper we will treat the case (n, m) = (3, 4). An irreducible SO(3)-representation isof odd dimension and unique in each odd dimension. Therefore there are exactly two SO(3)representations of dimension 5. One is the direct sum of the standard SO(3)-representation ofdimension 3 and the trivial representation of dimension 2. The other is the irreducible SO(3)representation of dimension 5. The unit spheres of these representations produce SO(3)actions on S 4 and it is known that any locally smooth SO(3)-action on S 4 is isomorphic toone of these two (see [1, Excercises 7 and 8 for Chapter IV]).
The former SO(3)-action on S 4 extends to a smooth SL(3, R)-action. In fact, the action(Ax,y), where (x, y) ∈ S 4 ⊂ R3 × R2 andSL(3, R) × S 4 → S 4 defined by (A, (x, y)) 7→ |(Ax,y)|
The author is supported by Fellowship of the Japan Society for the Promotion of Science.
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|(Ax, y)| denotes the length of (Ax, y), is a desired extension. However it is unknown whether
the latter SO(3)-action on S 4 called ψ in [8] extends to a smooth SL(3, R)-action. In 1985Uchida ([8]) constructed an SL(3, R)-action on S 4 extending the action ψ. His action iscontinuous but not smooth. So he gave the following problem (see [8] and [9, (P2)]):“Can the SO(3)-action ψ be extended to a smooth SL(3, R)-action on S 4 ?”We provide a continuous SL(3, R)-action on S 4 which is an extension of ψ but different
from Uchida’s. Our construction is based on the Kuiper’s theorem in [3] that the quotient
space of CP (2) by complex conjugation is S 4 . Ours is much more natural than Uchida’s,
but it turns out that our action is not smooth as well. So the Uchida’s problem above is still
open and the answer to the problem might be ‘No’.
This paper is organized as follows. In section 2 we exhibit two examples showing whether
an action of a compact Lie group extends to that of a non-compact Lie group. Section 3 gives
a new proof to the Kuiper’s theorem. In section 4 we construct a continuous SL(3, R)-action
on S 4 extending the action ψ and prove that the extended action is not smooth in the lastsection.
2. Examples
In this section we give two examples concerning the extension problem of a group action.
2.1. SO(4)-action on S 6 . Let us consider the SO(4)-action ρ4 ⊕ π on S 6 ⊂ R4 ⊕ R3 whereρ4 is a standard action on R4 and π is a linear action on R3 induced from a surjective
representation from SO(4) to SO(3). Then the following proposition holds.
Proposition 2.1 ([7]). The SO(4)-action ρ4 ⊕ π is not extendable to any continuous actionof SL(4, R) on S 6 .
This proposition was proved by F.Uchida in [7] and is one of the important facts to classify
SL(4, R)-actions on S 6 .
2.2. SO(2)-actions on R2 . Let us consider the SO(2)-action rp on R2 defined byrp : SO(2) × R2 3 (A, x) 7→ Ap x ∈ R2
where p is a natural number. Then the following proposition holds.
Proposition 2.2. The SO(2)-actions r1 and r2 are extendable to SL(2, R)-actions. However
rp for p ≥ 3 is not extendable to any SL(2, R)-action on R2 .
Proof. The SL(2, R)-action R1 : SL(2, R) × R2 → R2 defined by R1 (A, x) = Ax, whereA ∈ SL(2, R) and x ∈ R2 , is an extension of r1 . We identify R2 with the upper half space
H := {z ∈ C|Im(z) > 0}. Then the Möbius transformation of SL(2, R) on H provides anextension of r2 .
Assume rp extends to an SL(2, R)-action Rp on R2 for p ≥ 3. Then the kernel of Rp
includes the kernel of rp where the kernel of an action is the intersection of all isotropy
subgroups. In general the kernel of a G-action is a normal subgroup of G. In our case thekernel of rp is a cyclic group Zp of order p. But any proper normal subgroup of SL(2, R)
does not contain Zp (p ≥ 3). This is a contradiction. Hence rp is not extendable to any
SL(2, R)-action for p ≥ 3.
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The proposition above can also be derived from the classification result of SL(2, R)-actions
on R2 in [4]. But we gave a direct proof.
3. Preparation of construction
We describe the orbit structure of the SO(3)-action ψ on S 4 defined in the introduction.
Remember that the action ψ was obtained as the unit sphere of the irreducible SO(3)representation of dimension 5. This irreducible representation can be described expliclty as
the adjoint action of SO(3) on the vector spacesym(3) = {A ∈ M3 (R) : trace(A) = 0, A = At }where At denotes the transpose of a matrix A. We regard
S 4 = {A ∈ sym(3) : trace(At A) = 1}.
Then the determinant map π : S 4 → R identifies the orbit space of the action ψ with aninterval [− 3√1 6 , 3√1 6 ]. We can easily check that π −1 ( 3√1 6 ) and π −1 (− 3√1 6 ) are singular orbits
isomorphic to SO(3)/O(2) = RP (2) and other orbits are principal orbits isomorphic toSO(3)/K where K = Z2 ⊕ Z2 is   
−1 0 01 00−1 0 0
 1 0 0
 0 1 0  ,  0 −1 0  ,  0 −1 0  ,  0 1 0  .


0 0 1
00 10 0 −10 0 −1
We recall a lemma in [5].
Lemma 1 (Lemma 5.3.1 in [5]). Let G be a compact connected Lie group and K1 , K2 beclosed subgroup of G. Assume X1 = G ×K1 Dk1 and X2 = G ×K2 Dk2 where Dk1 and Dk2
are disks, Ki acts linearly on Dki (i = 1, 2) and dimX1 = dimX2 . Let f, f 0 : ∂X1 → ∂X2
be G-equivariant diffeomorphisms. Then M (f ) = X1 ∪f X2 is equivariantly diffeomorphic toM (f 0 ) = X1 ∪f 0 X2 as G-manifolds, if one of the following conditions is satisfied.
(1) f −1 f 0 is extendable to a G-equivariant diffeomorphism on X1 .
(2) f 0 f −1 is extendable to a G-equivariant diffeomorphism on X2 .
The following proposition gives a characterization of the action ψ. It also gives a criterionof when a smooth 4-manifold is diffeomorphic to S 4 .
Proposition 3.1. If a 4-dimensional compact connected smooth SO(3)-manifold has two
singular orbits RP (2) ' SO(3)/O(2) and the principal orbit type is SO(3)/Z2 ⊕ Z2 , thenthe manifold is equivarinatly diffeomorphic to S 4 with the SO(3)-action ψ.
Proof. It follows from [1] and [5] that the manifold in the proposition is of the form M (f ) =X1 ∪f X2 where Xi = SO(3) ×O(2) D2 (i = 1, 2) are tubular neighborhoods of two singular orbits and the attaching map f is an SO(3)-equivariant diffeomorphism from ∂X1 'SO(3)/Z2 ⊕ Z2 to ∂X2 ' SO(3)/Z2 ⊕ Z2 . Because f is an SO(3)-map on SO(3)/Z2 ⊕ Z2 ,we can think of f as an element of N (Z2 ⊕ Z2 ; SO(3))/Z2 ⊕ Z2 where N (H; G) is the normalizer of H in G. We can easily check that N (Z2 ⊕ Z2 ; SO(3))/Z2 ⊕ Z2 have the following
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representatives1I= 0000α=1

0 00 0 −10 1 01 0 ,A =  1 00  , A−1 =  0 0 −1  ,0 10 −1 0−1 0 0
0 1−1 0 00 −1 0

−1 0  , β =  0 0 1  , γ =  −1 00  .

0 00 1 000 −1

Hence the attaching map f can be thought of as one of the above matrices.
Consider the following commutative diagram where Z2 ⊕ Z2 = K.
'SO(3) ×H (H/K) −→ SO(3)/K↓ 1 × Rf
↓ Rf
'SO(3) ×H (H/K) −→ SO(3)/K.
Here Rf ([X]) = [Xf ] (X ∈ SO(3)/K), H is a subgroup of SO(3) which is isomorphic toO(2) and contains the element f and the group K, and horizontal arrows are defined by[g, hK] 7→ ghK. It is easy to see H/K ' S 1 and SO(3) ×H (H/K) ' SO(3) ×O(2) S 1 = ∂X1 .
If f = α, β or γ, then Rf is an antipodal involution of S 1 ; so Rf is extendable to adiffeomorphism on D2 . Hence 1 × Rf is extendable to an SO(3)-equivariant diffeomorphismon X1 = SO(3) ×O(2) D2 for f = α, β or γ. Therefore by Lemma 1 M (I) ' M (f ) if f = α,β or γ. Since αA = γ and αA−1 = β, M (I) ' M (α) ' M (A) ' M (A−1 ) by Lemma 1.
Therefore all M (f ) are SO(3)-equivariantly diffeomorphic to S 4 with the action ψ.
¤The following theorem was proved by Kuiper in 1974 ([3]), but we shall give a new proof
to it.
Theorem 1. Consider the Z2 -action on the complex projective space CP (2) = (C3 −{0})/C∗
by a complex conjugation. Then the orbit space is 4-dimensional sphere, that isCP (2)/Z2 ' S 4 .
Proof. The natural SO(3)-action φ on C3 − {0}/C∗ = CP (2) has two singular orbits; one is
the real projective plane RP (2) sitting naturally in CP (2) and the other is diffeomorphic to
S 2 . The principal orbits of φ are isomorphic to SO(3)/Z2 . Since the action φ commutes withthe complex conjugation, it induces an SO(3)-action φ̄ on CP (2)/Z2 . Because one singularorbit RP (2) of φ is the fixed point set of the complex conjugation on CP (2) and the otherone is S 2 , the induced SO(3)-action φ̄ on CP (2)/Z2 has two singular orbits both of which
are diffeomorphic to RP (2) and principal orbits of the induced action φ̄ are isomorphic toSO(3)/Z2 ⊕ Z2 . Therefore CP (2)/Z2 is diffeomorphic to S 4 (and moreover φ̄ agrees withthe action ψ) by Proposition 3.1.
¤4. Construction of the SL(3, R)-action on S 4Consider the natural SL(3, R)-action Φ on C3 − {0}/C∗ = CP (2). Since Φ commuteswith complex conjugation on CP (2), it induces an SL(3, R)-action Ψ on S 4 by Theorem 1.
The restricted action of Ψ to SO(3) agrees with the SO(3)-action ψ as observed in the proofof Theorem 1. Therefore our SL(3, R)-action Ψ is an extension of ψ.
4Remark . The action Ψ has one singular orbit RP (2) and one open orbit S 4 − RP (2).
Because the continuous SL(3, R)-action on S 4 in [8] has two open orbits and one singularorbit SO(3)/Z2 ⊕ Z2 , the action Ψ is different from the action in [8]. These two actions lookvery different but there might be some relation between them.
5. Non-differentiability of Ψ
We shall observe that our action Ψ is not smooth. Let p = [1 : 0 : 0] ∈ (C3 − {0})/(C∗ ×Z2 ) = S 4 . Take a small neighborhood Up of p and a local chart ϕ : Up → R4 defined byϕ([1 : x1 + y1 i : x2 + y2 i]) = (x1 , x2 , y12 − y22 , y1 y2 ).
The point p corresponds to the origin of R4 through ϕ. We consider a transformation Ψt
induced by the action Ψ of a matrix
1 0 0
 0 1 t .
0 0 1
By definition Ψt ([1 : z1 : z2 ]) = [1 : z1 + tz2 : z2 ] for [1 : z1 : z2 ] ∈ S 4 . For each t, there
is a sufficiently small neighborhood Up0 ⊂ Up such that Ψt (Up0 ) ⊂ Up . Therefore the map
Ψt around the point p can be described through the local chart ϕ. In fact, if we denoteϕ ◦ Ψt ◦ ϕ−1 by Ft , then we haveFt ((0, 0, y12 − y22 , y1 y2 )) = (0, 0, y12 − y22 + 2ty1 y2 + t2 y22 , y1 y2 + ty22 ).
p
If we set Y1 = y12 − y22 and Y2 = y1 y2 , then y22 = ( Y12 + 4Y22 − Y1 )/2; so we haveqq´³22222Ft ((0, 0, Y1 , Y2 )) = 0, 0, Y1 + 2tY2 + t ( Y1 + 4Y2 − Y1 )/2, Y2 + t( Y1 + 4Y2 − Y1 )/2 .
This shows that Ft with t 6= 0 is not differentiable at the origin, proving that Ψ is notsmooth.
The author is grateful to Professor Mikiya Masuda for his helpful comment.
References[1] G.E. Bredon: Introduction to compact transformation groups, Academic Press, 1972.
[2] G. Cairns, É.Ghys: The local linearization problem for smooth SL(n)-actions, Enseign. Math. (2) 43(1997), 133-171.
[3] N H. Kuiper: The quotient space of CP (2) by complex conjugation is 4-sphere, Math. Ann. 208 (1974),175-177.
[4] C. R. Schneider: SL(2, R) actions on surfaces, Amer. J. Math. 96 (1974), 511-528.
[5] F. Uchida: Classification of compact transformation groups on cohomology complex projective spaces withcodimension one orbits, Japan.J.Math. Vol 3, No.1(1977), 141-189.
[6] F. Uchida: Classification of real analytic SL(n, R) actions on n-sphere, Osaka J. Math. 16 (1979),561-579.
[7] F. Uchida: Real analytic SL(n, R) actions on sphere, Tohoku Math. J. 33 (1981), 145-175.
[8] F. Uchida: Construction of a continuous SL(3, R) action on 4-sphere, Publ. Res. Inst. Math. Sci. 21(1985), 425-431.
[9] F. Uchida, K. Mukoyama: Smooth actions of non-compact semi-simple Lie groups, A. Bak et al. (eds.),
Current Trends in Transformation Groups, Kluwer Acad. Publ., 201-215.
5Department of Mathematics, Osaka City University, Sumiyosi-Ku, Osaka 558-8585 Japan
E-mail address: d03sa004@ex.media.osaka-cu.ac.jp

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