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Consider a smooth effective action of a torus Tn on a connected C∞-manifold M. Assume that M is not a torus endowed with the natural action. Then we prove that there exists a complete vector field X on M such that the automorphism group of X equals Tn×R, where the factor R comes from the flow of X and Tn is regarded as a subgroup of Diff(M). Thus one may reconstruct the whole action of Tn from a single vector field. [ABSTRACT FROM AUTHOR]

We prove that the isomorphism type of a large class of groups is determined by the class of differential graded algebras over an integral domain with mild conditions, on which these groups act faithfully. [ABSTRACT FROM PUBLISHER]

We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper. [ABSTRACT FROM AUTHOR]

We consider classifying spaces of a family of p-groups and prove that mod p cohomology enriched with Bockstein spectral sequences determines their homotopy type among p-completed CW-complexes. [ABSTRACT FROM AUTHOR]

As it is well known, there is an element in the family of classical Lie algebras which presents more symmetries than the others. We are speaking, of course, about the algebra which, together with the usual two-fold symmetry which is common to the whole family, presents an order three symmetry which makes of it an “exceptional” classical Lie algebra. In this work we describe the fine group gradings induced by outer automorphisms of the Lie algebra . The qualitative approach used here emphasizes some algebraic aspects, as triality, and it is devoted to gradings involving the outer automorphisms of order three. The computation of the remaining fine group gradings, the ones induced by inner automorphisms and outer ones of order two, follows the same pattern as the classical case with n ≥ 5 and so it has been previously considered in the literature. [ABSTRACT FROM AUTHOR]