### 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 (US) |
---|---|

Pages (from-to) | 777-817 |

Number of pages | 41 |

Journal | Advances in Mathematics |

Volume | 230 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*230*(2), 777-817. https://doi.org/10.1016/j.aim.2012.02.009

}

*Advances in Mathematics*, vol. 230, no. 2, pp. 777-817. https://doi.org/10.1016/j.aim.2012.02.009

**Higher homotopy operations and André-Quillen cohomology.** / Blanc, David; Johnson, Mark W.; Turner, James M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Higher homotopy operations and André-Quillen cohomology

AU - Blanc, David

AU - Johnson, Mark W.

AU - Turner, James M.

PY - 2012/6/1

Y1 - 2012/6/1

N2 - 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 nΛ) (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.

AB - 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 nΛ) (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.

UR - http://www.scopus.com/inward/record.url?scp=84859097975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859097975&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.02.009

DO - 10.1016/j.aim.2012.02.009

M3 - Article

AN - SCOPUS:84859097975

VL - 230

SP - 777

EP - 817

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -