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Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly comnplex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps. Introduction. As the oriented analogue of the free equivariant unoriented bordism theory Rt*(X, A, r) of Stong [121, K. Komiya and C. M. Wu have respectively defined the free equivariant oriented bordism theories Q4+(X, A, r) and Q*(X, A, r) for involutions (X, A, r) (Komiya [9]), and Q*(X, A, r) for maps of odd prime period (X, A, r) (Wu [17]). The main object of the present paper is to apply Komiya's theories to the geometrical determination of the oriented bordism algebras Q* (Z2) of all orientationpreserving free differentiable involutions and Q(Z2) of all orientation-reversing free differentiable involutions. (Compare with the semi-geometric methods in Stong [11, Chapter VIII]). We also remark in this paper that the equivariant oriented and weakly complex theories of Wu, together with Mis'cenko's theorem [10, Appendix 1], give rise to a simple proof of the structure theorem for Q*(Z p) [2], U*(ZM, U*(Zm) ([3], [5k[6], [7]) and K'NLn(m)) [8]. These results are applied to the nonexistence problem for equivariant maps. In ?1, we define the bordism groups Q4(X, r) and Q(X, r), and then introduce the external product and the Pontrjagin product in these theories. In ??2 and 3, we give two kinds of direct sum decompositions of Q*(S , a) and Q(Sn, a) into isomorphic copies of Q(S, a). Deviating slightly from the Received by the editors July 28, 1971 and, in revised form, February 4, 1972. AMS (MOS) subject classifications (1970). Primary 57D75, 57D85, 57D90; Secondary 55B20, 55C35, 55G37.

Oriented and weakly complex bordism algebra of free periodic maps