Homology and cohomology of functional spaces

Abstract

Let {Xα} be an inverse system of compact spaces Xα and Y be an ANR. Consider a direct system {F(Xα,Y)} of topological spaces F(Xα,Y), where F(Xα,Y) is the space of all continuous maps f:Xα→Y, given the compact-open topology. In [5], S. Mardešić proved that for the singular homology there is an isomorphism(1)lim⟶H⁎s(F(Xα,Y))⟶∼H⁎s(F(X,Y)), where X=lim⟵Xα.In the present paper we prove that for the singular cohomology there is a finite exact sequence(2)0⟶lim⟵Hs1(2n−3)Fα⟶⋯⟶lim⟵Hsn−1(1)Fα⟶HsnF⟶⋯⟶lim⟵HsnFα⟶lim⟵Hsn−1(2)Fα⟶⋯⟶lim⟵Hs1(2n−2)Fα⟶0, where HsqFα=Hsq(F(Xα,Y),G), HsqF=Hsq(F(X,Y),G), X=lim⟵Xα, G is an abelian group.Let X be a compact space and S={Sm,σm} be the spherical spectrum. Consider the functional spectrum F(X,S)={Fm(X)}, where Fm(X)=F(X,Sm) is the space of all continuous maps f:X→Sm, given the compact-open topology. The functional spectrum F(X,S) induces the direct system {Cm−⁎(Fm(X))} of integral singular chains. The direct limit C⁎(X)=lim⟶mCm−⁎(Fm(X)) is a free cochain complex, where Cq(X)=lim⟶mCm−q(Fm(X)), q≥0. The Milnor homology H‾⁎(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‾⁎ we prove that there is an exact sequence(3)0⟶lim⟵Hsm−q−1(1)(Fm(X),G)⟶H‾q(X,G)⟶lim⟵Hsm−q(Fm(X),G)⟶0 and that the Milnor homology on the category Ac of compact pairs is a homology theory in the Berikashvili sense.