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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]

We study the topology of all possible subsums of the generalized multigeometric series k 1 f (x) + k 2 f (x) + ⋯ + k m f (x) + ⋯ + k 1 f (x n) + ⋯ + k m f (x n) + ⋯ , where k 1 , k 2 , ⋯ , k m are fixed positive real numbers and f runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval. [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]

Previous work of several authors shows that the exceptional Lie group of rank 4, F 4, as a p-compact group, is determined up to isomorphism by the isomorphism type of its maximal torus normalizer for p > 2. This paper considers the case p= 2 proving that F 4 as 2-compact group is also determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows the authors to determine the integral homotopy type of F 4 among connected finite loop spaces with maximal tori. [ABSTRACT FROM AUTHOR]

We prove that the homotopy type of BG 2 is determined by its mod 2 cohomology as well as by its Weyl group data. [ABSTRACT FROM AUTHOR]

We prove that the homotopy type of BG 2 is determined by its mod 2 cohomology as well as by its Weyl group data. [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]