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1. Statements of results. Let Bm(Spin), Qm(SU), and Qm{e) denote the wth spinor, special unitary, and framed cobordism groups respectively (see [6]). In [4] Kervaire defined a homomorphism $ : Ö2w(e) —>Z2 for n odd and ttj^l, 3, or 7, and showed that $ = 0 for w = 5. Kervaire and Milnor state in [5] that $ = 0 for n = 9. One of the corollaries of our results is that $ = 0 for w = 4fe + l, ferlin [2] a homomorphism \F: 3>2n(Spin)—»Z2 was defined for In = 8&+2, k*zl, such that <I> = SEp, where p: 02n(^)—»Q2n(Spin) is the obvious map. \F induces a map from S22n(<S£/) into Z2 which we also denote by ^ . I t is easily verified that Qi(SU) = Oi(Spin) = Qi(e) =Z 2 . Let a be the generator. Let 6 be the secondary cohomology operation coming from the relation SqSq = 0 on an integer cohomology class [7]. If ƒ is a map, let 6/ denote the associated functional cohomology operation [7]. The main theorems of this announcement are the following.

The Kervaire invariant of (8𝑘+2)-manifolds