Higher homotopy operations and André-Quillen cohomology

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

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 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 (US)
Pages (from-to)777-817
Number of pages41
JournalAdvances in Mathematics
Volume230
Issue number2
DOIs
StatePublished - Jun 1 2012

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Obstruction
Homotopy
Cohomology
Obstruction Theory
Homotopy Groups
Brackets
Cocycle
Proof by induction
Algebra

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Blanc, David ; Johnson, Mark W. ; Turner, James M. / Higher homotopy operations and André-Quillen cohomology. In: Advances in Mathematics. 2012 ; Vol. 230, No. 2. pp. 777-817.
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Higher homotopy operations and André-Quillen cohomology. / Blanc, David; Johnson, Mark W.; Turner, James M.

In: Advances in Mathematics, Vol. 230, No. 2, 01.06.2012, p. 777-817.

Research output: Contribution to journalArticle

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