In a previous work, we associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

In this article we provide sufficient conditions on a space X to verify Ganea conjecture with respect to exterior and proper Lusternik–Schnirelmann category. For this aim we previously develop an exterior version of the Whitehead, cellular approximation, CW-approximation and Blakers–Massey theorems within a homotopy theory of exterior CW-complexes and study their corresponding analogues and consequences in the proper setting.

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex $S$ can be realized as the $k$-skeleton of some elliptic complex as long as $k\,{>}\,\dim S$, or, equivalently, that any simply connected finite Postnikov piece $S$ can be realized as the base of a fibration $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of $S$ are decomposable, then the Anick conjecture holds for $S$. We also relate this conjecture with other finiteness properties of rational spaces.

We consider graded connected Gorenstein algebras with respect to the evaluation map $\text{e}{{\text{v}}_{G\,}}\,=\,\text{Ex}{{\text{t}}_{G}}(k,\varepsilon )\,::\,\text{Ex}{{\text{t}}_{G}}(k,G)\,\to \,\text{Ex}{{\text{t}}_{G}}(k,k)$ . We prove that if $e{{v}_{G}}\,\ne \,0$ , then the global dimension of $G$ is finite.

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by [Escr](X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup [Escr]n#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i[les ]n, and the subgroup [Escr]#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows.