Equilibrium Points in N - Person Games

* This note arose from consultations during the tenure of a John Simon Guggenheim Memorial Fellowship by MacLane. ' Whitehead, J. H. C., "Combinatorial Homotopy I and II," Bull. A.M.S., 55, 214-245 and 453-496 (1949). We refer to these papers as CH I and CH II, respectively. ' By a complex we shall mean a connected CW complex, as defined in §5 of CH I. We do not restrict ourselres to finite complexes. A fixed 0-cell e° e KO will be the base point for all the homotopy groups in K. 4 MacLane, S., "Cohomology Theory in Abstract Groups III," Ann. Math., 50, 736-761 (1949), referred to as CT III. ' An (unpublished) result like Theorem 1 for the homotopy type was obtained prior to these results by J. A. Zilber. CT III uses in place of. equation (2.4) the stronger hypothesis that XB contains the center of A, but all the relevant developments there apply under the weaker assumption (2.4). I Eilenberg, S., and MacLane, S., "Cohomology Theory in Abstract Groups II," Ann. Math., 48, 326-341 (1947). 8 Eilenberg, S., and MacLane, S., "Determination of the Second Homology ... by Means of Homotopy Invariants," these PROCEEDINGS, 32, 277-280 (1946). 9 Blakers, A. L., "Some Relations Between Homology and Homotopy Groups," Ann. Math., 49, 428-461 (1948), §12. 10 The hypothesis of Theorem C, requiring that v(1) not be cyclic, can be readily realized by suitable choice of the free group X, but this hypothesis is not needed here (cf. 6). 11 Eilenberg, S., and MacLane, S., "Homology of Spaces with Operators II," Trans. A.M.S., 65, 49-99 (1949); referred to as HSO II. 12C(k) here is the C(K) of CH II. Note that K exists and is a CW complex by (N) of p. 231 of CH I and that p-'K' K", where p is the projection p:K -K. 13 Whitehead, J. H. C., "Simple Homotopy Types." If W = 1, Theorem 5 follows from (17:3) on p. 155 of S. Lefschetz, Algebraic Topology, (New York, 1942) and arguments in §6 of J. H. C. Whitehead, "On Simply Connected 4-Dimensional Polyhedra" (Comm. Math. Helv., 22, 48-92 (1949)). However this proof cannot be generalized to the case W # 1.