Higher homotopy operations and André-Quillen cohomology

David Blanc, Mark W. Johnson, James M. Turner

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)777-817
Number of pages41
JournalAdvances in Mathematics
Issue number2
StatePublished - 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).


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

ASJC Scopus subject areas

  • General Mathematics


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