• Corpus ID: 119555894

Deformation theory and rational homotopy type

  title={Deformation theory and rational homotopy type},
  author={M. Schlessinger and Jim Stasheff},
  journal={arXiv: Quantum Algebra},
We regard the classification of rational homotopy types as a problem in algebraic deformation theory: any space with given cohomology is a perturbation, or deformation, of the "formal" space with that cohomology. The classifying space is then a "moduli" space --- a certain quotient of an algebraic variety of perturbations. The description we give of this moduli space links it with corresponding structures in homotopy theory, especially the classification of fibres spaces with fixed fibre F in… 

Infinity structures and higher products in rational homotopy theory

Rational homotopy theory classically studies the torsion free phenomena in the homotopy category of topological spaces and continuous maps. Its success is mainly due to the existence of relatively

A moduli space for rational homotopy types with the same homotopy lie algebra

Since Quillen proved his famous equivalences of homotopy categories in 1969, much work has been done towards classifying the rational homotopy types of simply connected topological places. The

Rational homotopy theory of automorphisms of manifolds

We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy

Rational Homotopy Theory and Deformation Problems from Algebraic Geometry

This paper is a description of research I have been doing over the last four years, applying some of the methods and ideas of rational homotopy theory as developed by Chen, Quillen and Sullivan, to

Operads and Maurer–Cartan spaces

This thesis is inscribed in the topics of operad theory and homotopical algebra. Suppose we are given a type of algebras, a type of coalgebras, and a relationship between those types of algebraic

Homotopy theory of complete Lie algebras and Lie models of simplicial sets

In a previous work, by extending the classical Quillen construction to the non‐simply connected case, we have built a pair of adjoint functors, model and realization, between the categories of

Higher Lie theory

We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original

Operads and Moduli Spaces

This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of


Over 50 years ago, Jim Stasheff [33] solved the classification problem for fibrations with fibre a finite CW complex X. He constructed a universal X-fibration of the form: X → UE → Baut(X) with base

Koszul duality and homotopy theory of curved Lie algebras

This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an



The Lie algebra structure of tangent cohomology and deformation theory

Obstructions to homotopy equivalences

Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

This paper establishes that the homotopy category of rational differential graded commutative coalgebras is equivalent to the homotopy category of rational differential graded Lie algebras which have


The $A(\inft)$-algebra structure in homology of a DG-algebra is constructed. This structure is unique up to isomorphism of $A(\infty)$ algebras. Connection of this structure with Massey products is

Strongly homotopy Lie algebras

The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32,

Rational homotopy theory

1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep things

Cohomology Theory of Lie Groups and Lie Algebras

The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to

Commutative algebras and cohomology

This paper has three purposes: (1) to develop the machinery of a commutative cohomology theory for commutative algebras, (2) to apply this cohomology theory to give a working theory of ring


We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby

Homotopy conditions that determine rational homotopy type