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We raise the question of the realizability of permutation modules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we consider a collection { M i } i = 1 n of finitely generated Z G -modules that admit a submodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each M i as its i-th homology, G as its group of self-homotopy equivalences E (X) , and the action of G on each M i as the action of E (X) on H i (X ; Z) . [ABSTRACT FROM AUTHOR]

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category C and for arbitrary groups H ≤ G 1 × G 2 , is there an object ϕ : A 1 → A 2 in Arr (C) such that Aut Arr (C) (ϕ) = H , Aut C (A 1) = G 1 and Aut C (A 2) = G 2 ? We are interested in solving this problem when C = H o T o p ∗ , the homotopy category of simply-connected pointed topological spaces. To that purpose, we first settle that question in the positive when C = G r a p h s . Then, we construct an almost fully faithful functor from G r a p h s to CDGA , the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when C = CDGA and, as long as we work with finite groups, when C = H o T o p ∗ . Some results on representability of concrete categories are also obtained. [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]

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]

In this paper we prove that the inductive cocategory of a nilpotent CW-complex of finite type X, indcocat X, is bounded above by an expression involving the inductive cocategory of the p-localisations of X. Also, we show that the inductive cocategory is generic for 1-connected H-spaces of finite type. Our arguments are the dualisation of classical results due to Ganea that allow us to improve previous results by Cornea and Stanley on LS category. [ABSTRACT FROM AUTHOR]

Let X be a finite simply-connected CW-complex. If for each prime p, the p-localization of X is co- H-space, then X is a co- H-space. [ABSTRACT FROM AUTHOR]

Let H be an arbitrary group and let ρ : H → Sym (V) be any permutation representation of H on a set V. We prove that there is a faithful H-coalgebra C such that H arises as the image of the restriction of Aut (C) to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant under the H-action and that the composition of the inclusion H ↪ Aut (C) with the restriction Aut (C) → Sym (V) is precisely ρ . We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully. [ABSTRACT FROM AUTHOR]