Measure homology
S. K. Hansen · MATHEMATICA SCANDINAVICA · 1998
Let X be a topological space, Sin k X the space of singular k-simplices with the compact-open topology, and let c k Xbe the real vector space of all compactly supported signed Borel Measures of bounded total variation on Sin k X.There are linear operators d X c k X 3 c kÀ1 X, so that c à X Y d f gis a chain complex.The homology H " à X is the measure homology of X of Thurston and Gromov.The main results in this paper are that H " à À satis¢es the Eilenberg-Steenrod axioms for a wide class of topological spaces including all metric spaces, and is ordinary homology with real coe¤cients for CW-complexes. Introduction.Measure homology was introduced by Gromov and Thurston in [T] ½6 in connection with Gromov's theorem that the Gromov norm of a closed oriented hyperbolic n-manifold M equals the volume of M divided by the supremum of the volumes of the geodesic n-simplices in the hyperbolic nspace.For a measurable space X Y }, let vX Y } be the vector space of all signed measures of bounded total varation.The total variation of a signed measure " on X Y } is k"k " X À " À X where " " À " À is the Jordan decomposition of " into its positive and negative variation.A measure " on X Y } has support in A P }, Supp" A, if "A B "B for all B P }.We write bX for the Borel '-algebra on the space X , and de¢ne a linear subspace of vX Y bX byLet Sin k X be the set of continuous maps from the standard k-simplex Á k to the space X with the compact-open topology, and set