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

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