Vol. 31 (1992) REPORTS ON MATHEMATICAL PHYSICS No. 2 Dedicated to Prof. Kazimierz Geba on the occassion of His 60th birthday BIFURCATION INVARIANTS FOR ACYCLIC MAPPINGS LECH GGRNIEWICZ Institute of Mathematics, N. Copernicus University, Torud, Poland WOJCIECH KRYSZEWSKI* Institute of Mathematics, Lodi University, tidi, Poland (Received January 31, 1992) The paper introduces a definition of the generalized topological degree for the class of acyclic set-valued mappings acting between spheres of different dimensions. The presented theory is applied in order to obtain a bifurcation index which constitutes a homotopy invariant yielding global bifurcation of solutions to equations (inclusions) involving acyclic maps.** Introduction In the paper we present an approach to introduce some topological invariants defined for acyclic set-valued maps cp zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED : U 4 R”, where U is an open subset of R”, m > n. These invariants provide a kind of algebraic indices for the existence of solutions to an abstract inclusion 0 E V(X)> x E u. First, we define the generalized topological degree of the above mentioned maps. To this end we use the auxiliary notion of a strongly acyclic map introduced in [15] and contrary to the case m = n, when homology methods are usually sufficient in order to define the topological degree (see [ll] and Section 4), we employ here homotopy theory technique. Our degree deg (or Deg) has all the natural properties and takes values in 7rm(S”) (or in rr,+i(P+‘)); therefore, in general, it is not integer-valued. The need for the generalized degree theory stems from the problems of the bifurcation theory of set-valued maps. Assume that * This work was done partially when the second author was visiting Munich, Germany, as the Alexander von Humboldt Research Fellow. ** The results of the paper without proofs were announced in Bolletino U.M.I. (7)6-B(1992). w71