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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.

In this paper we study differentiable orientation preserving involutions on S n § with non-trivial k dimensional fixed set. Define a cobordism relation between involutions T o and T~ by requiring that there exist an involution T on S n+k x I whose restriction to the ends yields T O and T~. The set of cobordism classes becomes a group d(n+k,k) by taking connected sum about a fixed point. We assume throughout that 5<_k<_n-3.By Smith theory the fixed set of T o, Fix(To), is a differentiable kmanifold with the Z2-homology of a sphere, and Fix(T) is a cobordism between Fix(T0) and Fix(Tt) such that H,(Fix(r), Fix(T/); Zz)=0 for i= 0, 1. We call such a manifold a ZE-CObordism and make the set of Z 2-homology k-spheres into a group 0(k 2~ under this relation via connected sum.Let O k denote the group of homotopy k-spheres studied by Kervaire and Milnor [10]. To calculate 0t, 2~ we study the natural map 0 ---, 0 (2)