Singular (Lipschitz) homology and homology of integral currents

We compare the homology groups $H_n ^{IC}(X)$ of the chain complex of integral currents with compact support of a metric space $X$ with the singular Lipschitz homology $H^L_n (X)$ and with ordinary singular homology. If $X$ satisfies certain cone inequalities all these homology theories coincide. On the other hand, for the Hawaiian Earring the homology of integral currents differs from the singular Lipschitz homology and it differs also from the classical singular homology $H_n(X)$.

💡 Research Summary

The paper investigates three homology theories defined on a metric space (X): the homology of integral currents with compact support (H_n^{IC}(X)), the singular Lipschitz homology (H_n^{L}(X)) built from Lipschitz simplices, and the classical singular homology (H_n(X)) built from arbitrary continuous simplices. The authors first set up the necessary background. Integral currents, following the Ambrosio‑Kirchheim framework, are metric‑space analogues of Federer‑Fleming currents: they are multilinear functionals on Lipschitz forms with finite mass and finite mass of the boundary. The chain complex (I_n(X)) consists of (n)-dimensional integral currents with compact support, and its homology is denoted (H_n^{IC}(X)). Singular Lipschitz chains form a free abelian group generated by Lipschitz maps (\sigma:\Delta^n\to X); the associated boundary operator is the same as in the classical singular complex, yielding (H_n^{L}(X)). Finally, the ordinary singular chain complex (S_n(X)) uses all continuous maps, giving the usual homology (H_n(X)). Natural inclusion maps (I_n(X)\hookrightarrow S_n^{L}(X)\hookrightarrow S_n(X)) are defined, and the paper asks when these induce isomorphisms on homology.

The central technical condition introduced is a family of “cone inequalities”. For a fixed dimension (n), the space (X) satisfies an (n)-cone inequality if there exists a constant (C>0) such that for every ((n-1))-dimensional integral current (S) with compact support there is an (n)-dimensional integral current (T) with (\partial T=S) and (\mathbf{M}(T)\le C,\mathbf{M}(S)), where (\mathbf{M}) denotes the mass. Intuitively, any boundary can be filled by a current whose mass is controlled linearly by the mass of the boundary. Under this hypothesis the authors prove a “homology equivalence theorem”: if (X) satisfies the cone inequality for all dimensions up to (n), then the three homology groups coincide, \