Singular Homology of non-Archimedean Analytic Spaces and Integration along Cycles

We constructed new theories on homology of Berkovich's analytic space and integration of an overconvergent differential form along a cycle in the sense of the homology. The new homology satisfies good properties such as what is called the axiom of homology theory, and has the canonical Galois action. The new integration is an extention of Shnirel'man integral, which is the classical integration of an analytic function on the non-Archimedean affine line, and it gives Fontain's p-adic periods for Tate curves.