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Mathematics  1999 

On the cobordism classification of manifolds with Z/p-action

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Abstract:

We refer to an action of the group Z/p (p here is an odd prime) on a stably complex manifold as simple if all its fixed submanifolds have the trivial normal bundle. The important particular case of a simple action is an action with only isolated fixed points. The problem of cobordism classification of manifolds with simple action of Z/p was posed by V.M.Buchstaber and S.P.Novikov in 1971. The analogous question of cobordism classification with stricter conditions on Z/p-action was answered by Conner and Floyd. Namely, Conner and Floyd solved the problem in the case of simple actions with identical sets of weights (eigenvalues of the differential of the map corresponding to the generator of Z/p) for all fixed submanifolds of same dimension. However, the general setting of the problem remained unsolved and is the subject of our present paper. We have obtained the description of the set of cobordism classes of stably complex manifolds with simple Z/p-action both in terms of the coefficients of universal formal group law and in terms of the characteristic numbers, which gives the complete solution to the above problem. In particular, this gives a purely cohomological obstruction to the existence of a simple Z/p-action (or an action with isolated fixed points) on a manifold. We also review connections with the Conner-Floyd results and with the well-known Stong-Hattori theorem.

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