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Phantom Maps and the Towers which Determine them
Author(s) -
McGibbon C. A.,
Steiner Richard
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610796004693
Subject(s) - mathematics , tower , combinatorics , homomorphism , abelian group , imaging phantom , dimension (graph theory) , class (philosophy) , algebraic number , discrete mathematics , mathematical analysis , physics , artificial intelligence , computer science , civil engineering , optics , engineering
Let X and Y be pointed spaces. A phantom map from X to Y is a map whose restriction to any finite skeleton of X is null‐homotopic. Let Ph ( X , Y ) denote the set of homotopy classes of phantom maps from X to Y . As a pointed set it is isomorphic to the lim 1 term of the tower of groups[ X , Ω Y ( 1 )] ← [ X , Ω Y ( 2 )] ← ··· ← [ X , Ω Y ( n )] ← ··· ,where Y ( n ) denotes the Postnikov approximation of Y through dimension n . The homomorphisms in this tower are induced by the projections Ω Y ( n ) ← Ω Y ( n +1) ). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.