arXiv subjects

A. Murillo

Publications and source records attributed to A. Murillo.

4 recordsLinked to original sources

arXiv2014

Brown representability for exterior cohomology and cohomology with compact supports

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, by proving such a theorem for the so called exterior cohomology in the complete and cocomplete exterior category, we show that the $n$-th cohomology with compact supports of a given countable, locally finite, finite dimensional relative CW-complex $(X,\mathbb{R}_+)$ is naturally identified with the set $[X,K_n]^{\mathbb{R}_+}$ of exterior based homotopy classes from a "classifying space" $K_n$. We also show that this space has the exterior homotopy type of the exterior Eilenberg-MacLane space for Brown-Grossman homotopy groups of type $(R^\infty,n)$, $R$ being the fixed coefficient ring.

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arXiv2012

Abstract sectional category

We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.

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arXiv2007

Homotopy equivalences of localized aspherical complexes

By studying the group of self homotopy equivalences of the localization (at a prime $p$ and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent complex, $\mathcal{E}_{\#}^m (X_{p})$ is in general different from $\mathcal{E}_{\#}^m (X)_{p}$. That is the case even when $X=K(G,1)$ is a finite complex and/or $G$ satisfies extra finiteness or nilpotency conditions, for instance, when $G$ is finite or virtually nilpotent.

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arXiv2000

Subgroups of the group of self-homotopy equivalences

Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex Y. We give a selection of results about certain subgroups of E(Y). We establish a connection between the Gottlieb groups of Y and the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of Y, denoted by E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of a cone decomposition of Y. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent.

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