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Let $S$ be a minimal smooth projective surface of general type withirregularity $q=2$. We show that, if $S$ has a nontrivial holomorphicautomorphism acting trivially on the cohomology with rational coefficients,then it is a surface isogenous to a product. As a consequence of this geometriccharacterization, one infers that no nontrivial automorphism of surfaces ofgeneral type with $q=2$ (which are not necessarily minimal) can be homotopic tothe identity. In particular, such surfaces are rigidified in the sense ofFabrizio Catanese.

Surfaces of general type with q=2 are rigidified