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Higher Homotopy Operations and The Realizability of Homotopy Groups
Author(s) -
Blanc David
Publication year - 1995
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-70.1.214
Subject(s) - mathematics , homotopy , realizability , n connected , homotopy category , cofibration , regular homotopy , homotopy group , morphism , algebra over a field , homotopy sphere , pure mathematics , eilenberg–maclane space , homotopy hypothesis , realization (probability) , group (periodic table) , statistics , chemistry , organic chemistry , algorithm
We describe an obstruction theory for the realization of a Π‐algebra—that is, a graded group G * with a prescribed action of the primary homotopy operations—as the homotopy groups of some space. The obstructions consist of higher homotopy operations, for which we provide an explicit definition in terms of certain sequences of polyhedra. There is a similar theory for realizing morphisms between Π‐algebras, and thus, in particular, for distinguishing different realizations of a fixed Π‐algebra. As an application we show that, for all primes p , the Π‐algebra π * S r ⊗Z/ p cannot be realized