Properties of Group Manifolds

MA THEMA TICS: P. A. SMITH

674

PN PROC. N. A. S.

PROPERTIES OF GROUP MANIFOLDS By P. A. SMITH DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY

Communicated November 3, 1931

Let M be an n-dimensional topological manifold, compact or not and suppose that there can be defined over M a correspondence T which associates in a continuous manner a single point Z to every ordered pair X, Y, - in symbols, XY = Z. Suppose further that for one point E, XE = EX = X for every point X. The Fundamental Group r of M Is Abelian.-This theorem was proved by Schreier' for the special case in which T defines a continuous group. A simple proof is the following: let a and P be two elements of r; they are oriented closed curves defined by the equations X = a(s), X = ,(t), where s, t are points ranging over the oriented circles o, T, respectively. We may suppose that a(so) = ,B(to) = E. Now the torus a X r is mapped on M by the continuous single-valued correspondence t defined as follows: to the arbitrary point s X t of a X T, there is made to correspond the point a(s) jP(t) of M. Under t, the images of the oriented curves a = so X r and b = a X to are a and P, respectively. On o- X T the curve aba1 bV' is deformable to a point. Under t this deformation is imaged by a deformation of a,a- 1 to a point. Hence a and g are permutable,

and r is abelian. We shall now assume that (XY)Z = X(YZ) for arbitrary points X,

Y, Z of M. Then there exists a uniquely determined n-dimensional submanifold M1 of M over which T defines an n-parameter continuous group G. M is the group (or parameter) manifold of G. We shall outline the proof. Let U be a neighborhood of E which is an n-cell. The set of points XU, determines, for a fixed X, a singular image of U on M, and if X lies in a suitably restricted neighborhood V of E, the points of XU will be uniformly near the corresponding points of U. By simple continuity considerations, it follows that X U covers E whenever X is in V. Hence for X in V, there is at least one X' in U such that XX' = E-that is, each X in V has a "right-itverse' in U. We can in the same way choose a neighborhood W of E such4hat each point of W has a right-inverse in V. Let X be in W, X' a right-inverse of X in V, and X' a right inverse of X'. From (XX')X' = X(X'X'), we have EX' = XE or X' = X, so that X'X = E. Hence if XY = E, we have Y = X'. Hence each point of W has a true and unique inverse. It now follows readily that the set W` consisting of the inverses of the points of W, contains E and is homeomorphic with W. Thus the set W, = W + W` is a connected open set containing E, and such that W,-' = W1. The same is true of

VOL. 17, 1931

MA THEMA TICS: P. A. SMITH

675

the set W1 W1... W1, hence of the set M1 = W1 + WWI +... and since M1 Ml = M1, it follows that T defines a group over M1. The proof that M1 is unique offers no difficulty. A simple example shows that M1 need not be identical with M. A necessary and sufficient condition that M1 be identical with M is that if Xn - X and Xn Y. = E, then the sequence Y. has at least one converging subsequence. This condition is automatically satisfied when M is compact. Suppose now that M is the group manifold of a group G. From the preceding results we deduce immediately that if N is a k-dimensional sub-manifold of M containing E and with the property NN = N, then G defines a group over N-namely, a k-parameter subgroup of G. In particular if N is a simple closed curve j, then j is the group manifold of a one-parameter subgroup g of G. Such subgroups are of frequent occurrence. We shall say that a number of them gi, g2, . . ., gp are independent if their group manifolds ji, j2, .. ., jp, when oriented and regarded = 1. Conas elements of r, satisfy no relation of the type jl j22... cerning the structure of abelian groups, we have the theorem that an n-parameter abelian group admits at most n independent one-parameter subgroups. The proof consists in showing that if j1, j2,.... jn+l are manifolds of independent one-parameter subgroups, then the product manifold it X j2 X ... jn+1 is.mapped on M in a continuous non-singular manner, at least.in the neighborhood of E; this contradicts the invariance of dimensionality. We -remark finally that a group manifold M is orientable. For suppose that an oriented n-cell H can be isotopically deformed on M into its negative. Let Ht (O < t < 1, H = Ho = -H1) represent the deformation, and let Xt represent the successive positions of a definite point X of H. For each value of t the set Kg = X-'H1 is a homeomorph of H, and hence an oriented cell containing E. Since Xi = Xo, it follows that Ko = -K1. It is clear that the deformation H, can be modified if necessary so that Kg is always within a given n-cell L, neighborhood of E. Thus K1 defines an isotopic deformation within L of an oriented n-cell into its negative, which is impossible. I 0. Schreier, Hamb. Abh., 4, 15-32 (1926).