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Journal of the London Mathematical Society 1997 55(3):601-608; doi:10.1112/S0024610796004693
© 1997 by London Mathematical Society
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© The London Mathematical Society

Phantom Maps and the Towers which Determine them

C. A. McGibbon and Richard Steiner

Department of Mathematics, Wayne State University Detroit, Michigan 48202, USA. E-mail: mcgibbon{at}math.wayne.edu
Department of Mathematics, University of Glasgow University Gardens, Glasgow G12 8QW. E-mail: rjs{at}maths.gla.ac.uk

Received 6 March 1995.

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 lim1 term of the tower of groups


Formula

where Y(n) denotes the Postnikov approximation of Y through dimension n. The homomorphisms in this tower are induced by the projections {Omega}Y(n)<- {Omega}Y(n+1)). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.


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