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Let the pair ( G , M ) R denote a group G acting on an R -module M , R a unitary ring. We ask for the existence of an R -local space X such that ( E ( X ) , π k ( X ) ) R is equivalent, in a natural way, to ( G , M ) R , for some k ≥ 2 , where E ( X ) denotes the group of homotopy classes of self-homotopy equivalences of X . If such an X exists, we say that X realizes the group action ( G , M ) R . We prove that if G is finite and acts faithfully on a finitely generated Q -module M , there exist infinitely many rational spaces realizing ( G , M ) Q . Our proof relies on providing a positive answer to Kahn's realizability problem for a large class of orthogonal groups that strictly contains finite ones. As a matter of fact, we enlarge the class of groups that is known to be realizable in the classical Kahn's sense. [ABSTRACT FROM AUTHOR]

We show that the vertex coloring problem for finite simplicial complexes can be translated into the algebraic homotopy problem of ellipticity for rational spaces. We follow the ideas of Lechuga-Murillo for the classical vertex coloring of graphs. [ABSTRACT FROM AUTHOR]