Phantom Maps and Finiteness Conditions

James Schwass

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arXiv · 1512.00357

Phantom Maps and Finiteness Conditions

James Schwass

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Abstract

A phantom map is a potentially nontrivial map which induces the zero map on every homology theory and on homotopy groups. Zabrodsky has shown that in the presence of particular finiteness conditions on spaces $X$ and $Y$ every map $X\to Y$ is a phantom map. More specifically, Zabrodsky essentially requires $Y$ to be a finite CW complex and $X$ to be a Postnikov space. We show Zabrodsky's observations hold under less restrictive finiteness conditions on the spaces $X$ and $Y$, making use of the Zabrodsky lemma and the machinery of resolving classes. As an application we identify, up to extension, the group of self-homotopy equivalences of spaces belonging to a particular family.

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James Schwass. 2016-03-31. Phantom Maps and Finiteness Conditions. https://arxiv.org/abs/1512.00357

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