## Abstract

There are two main approaches to the problem of realizing a Π-algebra (a graded group Λ equipped with an action of the primary homotopy operations) as the homotopy groups of a space X. Both involve trying to realize an algebraic free simplicial resolution G _{.} of Λ by a simplicial space W _{.}, and proceed by induction on the simplicial dimension. The first provides a sequence of André-Quillen cohomology classes in H ^{n+2}(Λ;Ω ^{n}Λ) (n≥1) as obstructions to the existence of successive Postnikov sections for W _{.} (cf. Dwyer et al. (1995) [27]). The second gives a sequence of geometrically defined higher homotopy operations as the obstructions (cf. Blanc (1995) [8]); these were identified in Blanc et al. (2010) [16] with the obstruction theory of Dwyer et al. (1989) [25]. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of Λ. In this paper we. (a)provide an explicit construction of the cocycles representing the cohomology obstructions;(b)provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to "long Toda brackets"); and(c)show that these two constructions correspond under an evident map.

Original language | English |
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Pages (from-to) | 777-817 |

Number of pages | 41 |

Journal | Advances in Mathematics |

Volume | 230 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2012 |

### Bibliographical note

Funding Information:We would like to thank the referee for his or her careful reading of the paper. This research was supported by BSF grant 2006039, and the third author was supported by a Calvin Research Fellowship (SDG).

## Keywords

- André-Quillen cohomology
- Higher homotopy operations
- Homotopy-commutative diagram
- K-Invariants
- Obstruction theory
- Primary
- Secondary

## ASJC Scopus subject areas

- General Mathematics