## Abstract

A II-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R(A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with π*(X)=A, of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R(A), in which the space at the bottom is BAut(A) and the successive fibres are determined by II-algebra cohomology. (This cohomology is the analog for II-algebras of the Hochschild cohomology of an associative ring or the André-Quillen cohomology of a commutative ring.) It seems clear that the deformation theory can be applied with little change to study other moduli problems in algebra and topology.

Original language | English (US) |
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Pages (from-to) | 857-892 |

Number of pages | 36 |

Journal | Topology |

Volume | 43 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2004 |

## Keywords

- Classification
- Moduli
- Realization

## ASJC Scopus subject areas

- Geometry and Topology