Homology and cohomology of functional spaces

摘要

Let {X-alpha} be an inverse system of compact spaces X-alpha and Y be an ANR. Consider a direct system {F(X-alpha, Y)} of topological spaces F(X-alpha,Y), where F(X-alpha,Y) is the space of all continuous maps f : X-alpha - Y, given the compact-open topology. In [5], S. Mardesic proved that for the singular homology there is an isomorphismlim(-) H-*(s)(F(X-alpha,Y)) -(similar to) H-*(s)(F(X,Y)), (1)where X = lim(-) X-alpha.In the present paper we prove that for the singular cohomology there is a finite exact sequence0 - lim(-)((2n-3)) (HsF alpha)-F-1 - . . . - lim(-)((1)) H-s(n-1) F-alpha - H-s(n) F - . . .- lim(-) (HsF alpha)-F-n - lim(-)((2)) H-s(n-1) F-alpha - . . . - lim(-)((2n-2)) H-s(1) F-alpha - 0, (2)where H-s(q) F-alpha = H-s(q)(F(X-alpha, Y), G), H-s(q) F = H-s(q)(F(X, Y), G), X = lim(-) X-alpha, G is an abelian group.Let X be a compact space and S = {S-m, sigma(m)} be the spherical spectrum. Consider the functional spectrum F (X, S) = {F-m(X)}, where F-m(X) = F (X, S-m) is the space of all continuous maps f : X - S-m, given the compact-open topology. The functional spectrum F (X, S) induces the direct system {Cm-* (F-m(X))} of integral singular chains. The direct limit C* (X) = lim(- m) Cm-*(F-m(X)) is a free cochain complex, where C-q (X) = lim(- m) Cm-q(F-m(X)), q = 0. The Milnor homology (H) over bar (*) (X, G) of the compact space X over the coefficients group G is defined as the homology of the chain complex C-* (X) = Hom(C* (X), G) [8].In the present work for the compact space X and Milnor homology (H) over bar (*) we prove that there is an exact sequence0 - lim(-)((1)) H-s(m-q-1)(F-m(X), G) - (H) over bar (q)(X, G) - lim(-) H-s(m-q) (F-m(X), G) - 0and that the Milnor homology on the category A(c) of compact pairs is a homology theory in the Berikashvili sense. (C) 2019 Elsevier B.V. All rights reserved.