proceedings of the american mathematical society Volume 58, July 1976 ON THE PRODUCT AND COMPOSITION OF UNIVERSAL MAPPINGS OF MANIFOLDS INTO CUBES W. HOLSZTYÑSKI Abstract. A map /: X -* Y is said to be universal iff for every g: X -* Y there exists x G X such that/(jt) = g(x). Let M,, t e T, and M" be orientable compact manifolds (in general with boundary). Let dim M" = n and let Q, be a cube with dim Q, = dim Mt. Let ft: M, -» Q,,f0: M" -* /" and/t: /"-*/" be universal mappings for f e Tand/fc = 1, 2,_Then (1.8) Theorem. The product map TXer-i ■Mt —> liier Qi 's universal. (2.1) Theorem. The composition fs » fs_x ° ■ • • » /, : M" —* I" is a universal map for s = 1, 2, .... (2.2) Theorem. The limit X of the inverse sequence in Ji_ jn Jx_ I" is an n-dimensional space with the fixed point property. Some "counterexamples" are furnished. Also the following variant of Proposition (1.5) from [3] is given: Theorem A (Proposition (1.5) of [3]). Let X be a compact space of (covering) dimension < n. Thenf: X —> /" is a universal mapping iff the element f* (e") of the nth tech cohomology group H"{X,f~x (S"~l ); Z) is different from Ofor a generator e" of H"{I", 51""1 ; Z) where (5""' = 3/"). 0. A mapping/: A —> Y is said to be universal iff for every g: X —> Y there exists x E X such that/(x) = g(x). We will use the following results. Theorem A (Proposition (1.5) of [3]). Let Xbea compact space of (covering) dimension < n. Then f: X —* I" is a universal mapping iff the element f* (e") of the nth Cech cohomology group H"(X,f~x(S"~x);Z) is different from 0 for a generator e" of H"(In,S"-x ;Z) (where Sn~x = dln). Proof. Since en = 8s"~x for some generator s"~x of H"~X(S"~X) hence f*(e") = 8f*(s"-x). By Proposition (1.5) of [3],/is universal iff f*(sn~x) $ lm[H"~x(X) -» H"-x(f-x(S"~x))] = Ker [8: H"-](rx(S"-x)) -» H"(X,f-x(S"-x))], i.e.,iff*(e") = 8f*(s"-x)*0. Theorem B (see [2]). Given a family of mappings of compact spaces, if the product of every finite subfamily is universal, then the product of all this family is a universal mapping. Received by the editors September 30, 1974. AM S (MOS) subjectclassifications (1970). Primary 54H25,55C20. £> American Mathematical Society 1976 311