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In this article, we investigate the topological structure of large scaleinteracting systems on infinite graphs, by constructing a suitable cohomologywhich we call the uniform cohomology. This provides a new perspective for theidentification of macroscopic observables from the microscopic system. As astraightforward application of our theory when the underlying graph has a freeaction of a group, we prove a certain decomposition theorem for shift-invariantclosed uniform forms. This result is a uniform version in a very generalsetting of the decomposition result for shift-invariant closed $L^2$-formsoriginally proposed by Varadhan, which has repeatedly played a key role in theproof of the hydrodynamic limits of nongradient large scale interactingsystems. In a subsequent article, we use this result as a key to proveVaradhan's decomposition theorem for a general class of large scale interactingsystems.

Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms