AN EXAMPLE FOR SO(3) BY PIERRE CONNER* AND DEANE MONTGOMERY SCHOOL OF MATHEMATICS, INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY Communicated September 7, 1962 1. Introduction.-The purpose of this note is to give an example to prove the following: THEOREM. There exists an action of SO(3) on Euclidean space EX, n >_ 12, which does not have a stationary point. In constructing the example to prove this theorem, we make use of earlier methods;1 see reference 2 for extensions by Kister. We also rely on a result of Floyd which is as yet unpublished. This is mentioned below and we wish to thank Floyd for informing us of this result. It is also necessary to use the fact3' 4that the product of a contractible manifold by a line is an open cell. 2. The map f.-we begin with the irreducible linear action of S0(3) in E6. This action leaves invariant the unit sphere S4. It has been proved by Floyd that there exists a mapf, f: S4 -> S4 which has degree zero and is equivariant; that is, fg = gf, g G G = SO(3). Floyd has used f to construct a four-dimensional continuum with trivial cohomology on which G = SO(3) acts without a stationary point. This map f is used as the basis of our example and a sketch of the construction of f is included for convenience. The group G = SO(3) contains a subgroup Z2 @ Z2. The set of fixed points of this subgroup, that is F(Z2 ED Z2; S4), is a simple closed curve. The orbits of G in S4 are of two types. Two of these orbits are two-dimensional and are projective planes. The remaining orbits are three-dimensional and all of the same type with isotropy group isomorphic to Z2 @ Z2. The simple closed curve F(Z2 (3 Z2; S4) intersects each three-dimensional orbit in 6 points and each two-dimensional orbit in 3 points. Let pq be an arc in F(Z2 Go Z2; S4) with p and q being in distinct two-dimensional orbits and all points in the interior of pq being in three-dimensional orbits. We first choose f as a map f: pq -F(Z2 E Z2; S4) in sueh a way that f(p) = p and f(q) is in the two-dimensional orbit G(p) and so that Gf(qv) = G, (G. is the isotropy group at q). The map f may be taken as strictly increasing. For any t in pq we define fg(t) = gf(t), g E G. This definesf for all of S4, Of: S4--.S4 and it can be seen that f has degree zero and is equivariant. 3. The mapping cylinder.-let X and Y be two copies of S4 each acted upon by G = SO(3) as above and let f be the equivariant map defined above f: X --Y. 1918
VOL. 48, 1962 MATHEMATICS: CONNER AND MONTcOMERY 1919 The join X o Y is defined as follows: X 0 Y = {xyt; x C X, Yy E Y. 0 < t < 1} with the usual identifications. Then G acts on X a Y by means of the following definition: g(x,yt) = (gx,gy,t), g G - SO(3). In X o Y there is the mapping cylinder K = {x,f(x),t; 0 < t < 1 and K is invariant under G because of the equivariance of f. A set U in X o Y is defined as follows: U = {x,y,t; d(y,f(x)) < e/2, 0. t < 1i, where e < 1/4. The set U is invariant. LEMMA 1. The set U can be deformed over itself to Y and this can be done in such a way that x X is carried tof(x) in Y. Let x be a point of X and let N(x) be defined as follows: N(x) = {x,y,t; d(y,f(x)) < e/2, 0 < t. 11, and let the points of N(x) where t = 1 be denoted by P(x), that is P(x) = {x,y,1; d(y,f(x)) < e/2}. The sets N(x) and P(x) are subsets of the join X o Y. The set Y was taken as the unit sphere in a Euclidean space and P(x) determines a set Q(x) in this Euclidean space as follows: Q(x) = { points on the intervals from the origin to P(x)}. Of course N(x) is homeomorphic to Q(x), and we may select a definite homeomorphism TX: N(x) -- Q(x) by requiring a point (x,y,t) of N(x) to be mapped to ty. The set Q(x) is a cone over P(x) with vertex at the origin. There is a deformation DX of Q(x) into itself which moves the origin along the radius joining it to f(x) and moves all of Q(x) to P(x). To define this, let z be any point of Q(x). Then z j on a ray parallel to the ray from the origin through f(x). The point z is to be deformed along this ray to P(x), in such a way that at time s, 0 < s < 1, the point has moved the sth part of the length from z to P(x) along the ray. We now define a deformation of N(x) over itself by the formula T-1DxTx; N(x) -- P(x). Or if the parameter s is exhibited for DX in Q(x) in the form Dr(s), then it is exhibited in N(x) by Fx(s) = Tx-'D(s)Tx: N(x) -N(x), T-I1(D (0)) TX = identity,
1920 MATHEMATPICS: CONNER AND MONTGOMERY PROC. N. A. S. Tj-I(Dx(1))Tx: N(x) -* P(x). Of course, points of P(x) remain fixed under this deformation. As x varies Fx(s) defines a deformation on all of U. That is, for u in U, there is an x in X and a y,d(y,f(x))..7/2 such that and then u = (x,y,t), Fx(s)u = T,-'Dx(s)Txu is defined; continuity can be verified. This completes the proof of the Lemma. 4. The neighborhood W.-It will be convenient to consider X and Y as being imbedded as unit spheres in two E5's. Let P and Q be solid balls in these two fivedimensional Euclidean spaces, each of radius 2. The group G = SO(3) acts on P and Q in such a way as to induce the actions on X and Y used above. Extend f radially, f. P-* Q, so f is equivariant and so the restriction is f: X - r. In the join P o Q, define a neighborhood WV of K in P o Q as followers: W= WI U W2 U Wi, where W1 = {x,yt; 1- E < x < 1 + E, d(yf(x)) < e/2}. (Note that a band around the sphere X o Y is the topological product of the sphere X o Y and a disk, actually a square an interval in X and another in Y. It has a product space metric and this metric is used above.) Continuing with the definitions, WH2 = lxyt; 1 - E < x K 1 + E, ye Q, 0 _ t <E}; W3 = {x,y,t; x E P, 1-3E/2 < 1Iy 11 < I + 3E/2, 1 - e < t. 1}. LEMMA 2. The set W is an open invariant set in P o Q and may be deformed in itself into the set U of Lemma 1. In order to see that W is an open set, we may proceed as follows. Note first that W2 and W3 are open sets in P o Q. The set W1 is not open but what we shall prove is that W1 is open at a point {x,y,t} with 0 < t < 1. This will suffice to prove that W is open since W2 and W3 are open. Hence, let (x,y,t), 0 < t < 1 be a point of W1. We must show that if (x',y',t') is a point of P o Q near to (x,y,t) then (x',y',t') is in W1. We know that d(y,f(x)) < E/2 so we have for some (', d(f,f(x)) < El < E2. (1) Given 8' > 0, we may choose a < 0 so that, if d(x,x') < 6, then d(f(x),f(x')) < V'. (2) We may assume 9 d (y,!/') <'. (3)
VOL. 48, 1962 MAf.1 T'HEMIA TIC'S: CON.NER AND MONTGOMER Y 1921 Then (1), (2), and (3) prove By choosing 6' properly, we then have d(y',f(x')) < E, + 26'. d(y',f(x')) < e/2. This implies, by definition, that (x',y',t') is in W, as was to be proved. We may deform W2 to t = 0 and WV3 to t = 1, where we leave fixed the set 1/3.. 2/3 and where each x,y,t moves (by varying t) along its own join interval. Then we deform x and y radially to X and Y and this carries W1 through itself to U. 5. The union of cells. The join P o Q is an 11-cell. We consider two of the constructions made above, denoting the first, by PI o Qi and the second by P2 0 Q2. In the second copy, we choose the left-hand band around the unit sphere to extend from 1 -.3E/'2 to 1 + &3E/2. For defining U and the W's, we use e/4 so that the right hand band runs from 1-7E7'4 to 1 + 7(,,/4. In successive copies, we make analogous adjustments so that we are always inside a band from 1-2E to 1 + 2E. In1P1o Q1, the set Bd P, o Qi is a 10-cell on the boundary and it is tamely imbedded. This 10-cell is given by OyIO; x E Bd P1, y E Qi, 0 < t _ 1$, and it is invariant under G. Inside this 10-cell there is a slightly smaller 10-cell Qi* as follows: Qi* =x,y,t; x C Bd P1, 11 y 2 l/2 _1 t 1. Then Qi* is also invariant under G. In P2 0 Q2 there is the 10-cell P2 o Bd Q2, and in it there is the slightly smaller 10-cell defined as follows: -2= t1',8,; x * 1 0 _ t _ 1/,, y E Bd Q2}. Next, we form the union P1' Q1 U P2 0 Q2 and identify Qi* and P2* equivariantly. The union (taken with this identification) is an 11-cell invariant under G. Continuing in this way with P3 0 Q3 and so on, we form Q2* and P3* analogous to the above and identify them to form the union P1 0 Q1 U P2o0 Q2 U P-3 OQQ. Step by step, we identify Qi* and P* and form the union U Xi a Qi with these identifications. In Pi o Qi there is a mapping cylinder Ki, and in the set above we take K2 U K3 U. This set can be seen to have an invariant contractible locally Euclidean neighborhood, which we shall denote by V. Note that we begin with K2 in order to provide
1922 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. a locally Euclidean neighborhood. We observe that V has vanishing homotopy groups and hence is contractible. See reference (1) for a similar argument. Now V is an open subset of an open cell, as can be seen from the construction of U P, o Qu; hence, V is a differentiable manifold. It is known (ref. 4) that and G has an action on V X El given by V X El = E12 g(v,r) = (g(v),r), g E G. This action has no stationary point, and this completes the proof of the theorem. 6. Concluding remarks.-corollary. The tetrahedral, icosahedral, and octahedral groups have an action on E'2 with no stationary point. This follows from the fact that the only isotropy groups occurring in the above action of G on E12are Z2 (3 Z2, Z2, e, N (N being the normalizer of a circle group T). By adding a point at infinity, we get an action of G = SO(3) on 512 with precisely one stationary point. This action can be seen not to be differentiable in a neighborhood of the stationary point. If it were, it would be equivalent to a linear action locally, and this linear action would be the sum of irreducible actions. However, the known isotropy groups of the irreducible actions do not fit with the isotropy groups listed above. * Alfred P. Sloan Fellow. 1 Conner and Floyd, "On the construction of periodic maps without fixed points," Proc. Am. Math. Soc., 10, 354-360 (1959). 2 Kister, "Examples of periodic maps on Euclidean spaces without fixed points," Bull. Am. Math. Soc., 67, 471-474 (1961). 3 McMillan and Zeeman, "On contractible open manifolds," Proc. Camb. Phil. Soc., 58, 221-229 (1962). 4 Stallings, "The piecewise linear structure of Euclidean space," Proc. Camb. Phil. Soc., 58, 481-488 (1962). A MEASURE DISTORTION MAPPING* BY G. T. WHYBURN UNIVERSITY OF VIRGINIA Communicated September 6, 1962 If a, b, a, 13, and 1 are real numbers with a < a < <3K b and 0 < 21 < 13 - a, by the central contraction c(x) of ab relative to a 1, and 1 is meant the homeomorphism of ab onto itself, linear on the intervals aa, a13, and 13b, which leaves a, b, and the midpoint 0 of ac3 fixed and maps a and 13 into points 1 units on the left and right of 0 respectively, i.e., if we take 0 as the origin so that a = -13, then c(x) is 1 1l+ a defined byy = x for -.< x <, y-a = (x- a) for a < x <-, b 1 y - b (x - b) for 13 <x < b. In this paper, these contractions are used to b -1