Abstract.When an arbitrary p-group G acts on a 7Lp-homology n-sphere X, it is proved here that the dimension function n: S(G)- ~7I(S(G) is the set of subgroups of G), defined by n(H)=dimX H, (dim here is cohomological dimension) is realised by a real representation of G, and that there is an equivariant map from X to the sphere of this representation. A converse is also established. O. IntroductionIt is well known that the fixed point set of a p-group G acting on a mod-p homology sphere is again a mod-p homology sphere and that for odd p, the codimension is even ([1; Chapter 4]). When G is elementary abelian (more generally, abelian) the action resembles a linear representation ([1; Chapter 13], [4,5]). When G is any p-group acting on a homotopy sphere with the stipulation that all fixed sets of subgroups of G are homotopy spheres of odd dimension, Petrie and tom Dieck [8] have shown that the action resembles the sphere of a complex representation. We prove here that when G is an arbitrary p-group acting on a mod-p homology sphere, the action resembles a real linear representation of G. It is shown in [63 that non-prime power order groups have actions which do not resemble linear representations.More specifically, suppose G acts cellularly on a finite CW-complex X satisfying Hi(X;7~p)=0 (i~:n, some n). In this context we prove the following result.

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Abstract. In this paper we prove that each compact flat Riemannian manifold is the boundary of a compact manifold. Our method of proof is to construct a smooth action of (TZ2)k on the flat manifold. We are independently preceded in this approach by Marc W. Gordon who proved the flat Riemannian manifolds, whose holonomy groups are of a certain class of groups, bound. By analyzing the fixed point data of this group action we get the complete result. As corollaries to the main theorem it follows that those compact flat Riemannian manifolds which are oriented bound oriented manifolds; and, if we have an involution on a "homotopy flat" manifold, then the manifold together with the involution bounds. We also give an example of a nonbounding manifold which is finitely covered by S 3 x S 3 x S 3.A complete flat Riemannian manifold is a smooth manifold whose universal cover is IR" where the covering transformations are rigid Euclidean motions. We shall be concerned only with compact manifolds. In this case, the translations among the covering transformations correspond to a finite regular covering by a torus. Thus the fundamental group n is a torsion-free extension of a finite group G, the holonomy group, by the fundamental group of the covering torus T". We then have the exact Bieberbach sequence 0 *Z" ,re P ,G ,1where the conjugation action of G on 7/" is a faithful representation. Our main result is the one stated in the title, namely that every compact flat manifold is the boundary of some compact manifold. Our method of proof is based on the construction of a group of affine involutions on the flat manifold we were independently preceded in this approach by Marc Gordon [4], who was able to show that very often there is no fixed point of the group Offprint requests to." G.C. Hamrick ~ 0011~I~'~ IEII~IglAI~ I~I~I QI~

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An oriented differentiable manifold M k admits an odd framing if the composition M k v ~BSO-e ,BSO(2 ~ is null homotopic, where v classifies the stable normal bundle and (~ is localization at 2. An odd framing of M is then a null homotopy of this composition. Under a suitable relation these give rise to the odd framed cobordism groups s r~2~. In analogy with framed cobordism, there is a Whitehead homomorphism (2) where S0c2 ~ denotes the localization of SO at 2.In this paper we will compute the cobordism groups in terms of the 2-primary part of stable homotopy and determine the image of J'. The motivation for this study lies in the fact [2] that all Z2-homology spheres admit odd framings. In [2] the structure of (2 frl2) modulo the image of J' is employed to analyze the groups of Z2-homology spheres and give applications to involutions. Section 1 gives some lemmas concerning localizations. The second and third sections introduce the cobordism theory and the homomorphism J' and establish the main result: Theorem. c)fr~z)~g2fr ~B k, where Bk=O for k~-3 rood 4, 7~(n) is the number of partitions of n, and Z(2 ) is the integers localized at 2.In the final section we show that the cobordism group is additively generated by odd-framed lens spaces and spheres together with framed manifolds.