Frank Raymond

Injective operations of the toral groups

THIS paper is a continuation of our earlier “Actions of Compact Lie Groups on Aspherical Manifolds” [3]. Our emphasis here is shifted away from aspherical manifolds. We find that a satisfactory hypothesis about the action (T’, X) is the requirement that at each x E X the map f” : T -+ X given byf”(t) = tx induce a monomorphism f* : q(Tk, 1) -+ OCR (X, x). We call such an action injectioe. If X happens to be a closed aspherical manifold then any effective action of a toral group will satisfy the hypothesis.

3-manifolds whose universal coverings are Lie groups

Abstract We classify those closed 3-manifolds whose universal covering space naturally admits the structure of a Lie group

Failure of Nielsen's theorem in higher dimensions

The following is known for closed orientable surfaces. If H: ii/l -+ &! is a map whose n-th power is homotopic to the identity, then H is homotopic to a homeomorphism K with Kqa = identity. The result is known as Nielsen’s Theorem on finite mapping classes. There are doubts (see [S]) as to the correctness of all parts of Nielsen’s arguments in [6]. Different (using complex analysis and the Smith theorems) and valid proofs have been given independently by Fenchel and Macbeath. Because we are dealing with surfaces, each self homotopy equivalence is homotopic to a diffeomorphism, and homotopic diffeomorphisms are diffeotopic. Consequently, Nielsen’s theorem may be equivalently stated as follows: If H: ii + ill is a diffeomorphism whose n-th power is homotopic to the identity then H is diffeotopic to a diffeomorphism K with Km = identity. The theorem is extremely useful in studying periodic maps on 3-manifolds, and there are obvious applications in surface theory. The question has been raised as to what extent Nielsen’s theorem holds for aspherical manifolds of dimensions greater than 2. An aspherical manifold is a closed manifold whose universal covering is contractible. Aspherical manifolds are therefore closed manifolds which are also K(n, 1)‘s. An important and interesting class of aspherical manifolds (generalizing the tori) are the nil-manifolds. A nil-manifold is simply the quotient of a connected contractible nilpotent Lie group by a uniform discrete subgroup. We shall show that in dimensions greater than 2 there exist closed nil-manifolds for which Nielsen’s theorem fails in a very strong sense. Specifically

Manifolds covered by Euclidean space

at infinity. However, a weak form of Theorem 3 would still have to be proved even for this method. Finally we call attention to 6.2 where we summarize the geometric content of the argument. This probably should be read next.

Local triviality for Hurewicz fiberings of manifolds

2. THE FUNDAMENTAL GROUP OF AN END Recall that if X is a connected, locally connected, locally compact space with a count- able basis we may define the ends of X as a maximal O-dimensional compactification of X. Specifically. if K is a compact set in X, then X -

Topological, affine and isometric actions on flat riemannian manifolds II

LET p: E -+ B be a fibering in the sense of Hurewicz. i.e. the map p has the path lifting property or what is the same the covering homotopy property holds for all spaces. As is well known, the fibering need not be locally trivial, i.e. 17: E -+ B need not be a fiber bundle. On the other hand, one might suspect that stringent conditions on the nature of E and B might force the map to be locally trivial. The following seems plausible: CONJECTURE. Let p: E + B he afiheriry in the sense of HurercYcz of a mcm{fold E (n.ith empty boundary) onto a rceakiy iocail,v contrnctible pnracompact base B. Then thejiberinq is locally trivial.

The index of manifolds with toral actions and geometric interpretations of the σ(∞, (S 1,M n)) invariant of Atiyah and Singer

Abstract Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π0(Homeo(M))→Out π1(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.

Rigidity of almost crystallographic groups

In the theory of transformation groups the smooth theory is certainly the most tractable. However, not all operations that one would like to perform remain in the smooth category. For example, analysis of the orbit space usually must go outside the smooth category. Furthermore, one would also like to study symmetries of interesting geometric spaces which often fail to be locally Euclidean such as spaces having manifolds as ramified coverings and analytic spaces. It has long been recognized that cohomology manifolds or generalized manifolds encompass all manifolds as well as many typical analytic and ramified spaces. The most important feature of generalized manifolds, from the point of view of transformation groups, is that one is often able to work with orbit spaces. In addition, one does have characteristic classes and with care one can often define workable invariant tubular neighborhoods of fixed point sets. In this paper we develop and exploit some of these ideas to prove results which even when specialized to the smooth category seem to be unobtainable from standard smooth methods. For example, we establish by cohomological methods alone formulae for the Atiyah-Singer invariant a ( ~ , (S 1, M)), Theorems 3, 4, 5, and 6. This enables us to conclude that its value is an integer and also to give explicit geometric interpretations of it in terms of the orbit space and the mapping cylinder of the orbit map. This makes it much easier to compute than by using smooth techniques alone, e.g. w 4. A very interesting geometric interpretation of ff(sl, m4k-1)=O is also given in terms of fibering M 4k-~ over a circle, w 4. There are several ingredients that enable us to obtain these results. First, we obtain by cohomological methods: