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Cohomology of deformations

In this article we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.