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1. Introduction. The classical Lef schetz fixed point theorem states that with each compact self-mapping ƒ of a metrizable ANR one can associate a number L(f), the Lefschetz number of/, such that if L(J) is nonzero, ƒ has a fixed point. If the ANR in question happens to be a smooth Banach manifold and ƒ a smooth map with isolated fixed points, we can attach to each fixed point an index equal to ± 1 or zero, and the sum of these indices is equal to L(f). In other words, L(J) is the algebraic number of fixed points of/. See (l, §11). Here we announce a coincidence point theorem for certain classes of Banach manifolds and certain classes of maps, which extends the Lefschetz-Fuller coincidence theorem (3) and which specializes to give the Lefschetz fixed point theorem for Banach manifolds. The theorem is formulated in terms of an ( oo —p) -dimensional cohomology theory for certain Banach manifolds. These results form part of the author's Cornell doctoral disserta­ tion, written under the direction of J. Eells. Full details will appear elsewhere. We use notation and terminology of (l). 2. (oo —£)-dimensional cohomology. In what follows, E will denote a C°°-smooth, separable Banach space. All Banach manifolds in con­ sideration will be C^-smooth, paracompact and such that their tangent

Coincidence theory for infinite dimensional manifolds