# Deformation theory and rational homotopy type

@article{Schlessinger2012DeformationTA, title={Deformation theory and rational homotopy type}, author={M. Schlessinger and Jim Stasheff}, journal={arXiv: Quantum Algebra}, year={2012} }

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…

## 91 Citations

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## References

SHOWING 1-10 OF 74 REFERENCES

### Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

- Mathematics
- 1978

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…

### ON THE HOMOLOGY THEORY OF FIBRE SPACES

- Mathematics
- 1980

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

- Mathematics
- 1994

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

- Mathematics
- 2011

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

- Mathematics
- 1948

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

- Mathematics
- 1962

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…

### FORMAL SOLUTION OF THE MASTER EQUATION VIA HPT AND DEFORMATION THEORY

- Mathematics
- 1999

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…