On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory

Abstract

The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory. It suggests slight changes to Milnor’s and Thurston’s original definitions (giving differences for wild topological spaces only) which ensure that their homology theory is well-defined on all topological spaces. It further proves that Milnor-Thurston homology theory gives the same homology groups as the singular homology theory with real coefficients for all triangulable spaces. An example showing that the coincidence between these both homology theories does not hold for all topological spaces is also included.