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The center of a graded connected Lie algebra is a nice ideal

Yves Félix (1996)

Annales de l'institut Fourier

Let ( 𝕃 ( V ) , d ) be a free graded connected differential Lie algebra over the field ℚ of rational numbers. An ideal I in the Lie algebra H ( 𝕃 ( V ) , d ) is called nice if, for every cycle α ∈ 𝕃 ( V ) such that [ α ] belongs to I , the kernel of the map H ( 𝕃 ( V ) , d ) → H ( 𝕃 ( V ⊕ ℚ x ) , d ) , d ( x ) = α , is contained in I . We show that the center of H ( 𝕃 ( V ) , d ) is a nice ideal and we give in that case some informations on the structure of the Lie algebra H ( 𝕃 ( V ⊕ ℚ x ) , d ) . We apply this computation for the determination of the rational homotopy Lie algebra L X = π * ( Ω X ) ⊗ ℚ of a simply connected space X . We deduce that the...

The cohomology algebra of certain free loop spaces

Toshihiro Yamaguchi, Katsuhiko Kuribayashi (1997)

Fundamenta Mathematicae

Let X be a simply connected space and LX the space of free loops on X. We determine the mod p cohomology algebra of LX when the mod p cohomology of X is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of LX when the rational cohomology of X is a graded complete intersection algebra. The key to both of these results is the identification of an important subalgebra of...

The Gray filtration on phantom maps

Lê Minh Hà, Jeffrey Strom (2001)

Fundamenta Mathematicae

This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition...

The rational homotopy of Thom spaces and the smoothing of isolated singularities

Stefan Papadima (1985)

Annales de l'institut Fourier

Rational homotopy methods are used for studying the problem of the topological smoothing of complex algebraic isolated singularities. It is shown that one may always find a suitable covering which is smoothable. The problem of the topological smoothing (including the complex normal structure) for conical singularities is considered in the sequel. A connection is established between the existence of certain relations between the normal Chern degrees of a smooth projective variety and the question...

The rational homotopy type of configuration spaces of two points

Pascal Lambrechts, Don Stanley (2004)

Annales de l’institut Fourier

We prove that the rational homotopy type of the configuration space of two points in a 2 -connected closed manifold depends only on the rational homotopy type of that manifold and we give a model in the sense of Sullivan of that configuration space. We also study the formality of configuration spaces.

The top cohomology class of certain spaces.

Aniceto Murillo (1991)

Extracta Mathematicae

In this abstract we present an explicit formula for a cycle representing the top class of certain elliptic spaces, including the homogeneous spaces. For thet, we shall rely on the connection between Sullivan's theory of minimal models and Rational homotopy theory for which [3], [6] and [10] are standard references.

Variations on a conjecture of Halperin

Gregory Lupton (1998)

Banach Center Publications

Halperin has conjectured that the Serre spectral sequence of any fibration that has fibre space a certain kind of elliptic space should collapse at the E 2 -term. In this paper we obtain an equivalent phrasing of this conjecture, in terms of formality relations between base and total spaces in such a fibration (Theorem 3.4). Also, we obtain results on relations between various numerical invariants of the base, total and fibre spaces in these fibrations. Some of our results give weak versions of Halperin’s...

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