Relative phantom maps

Iriye, Kouyemon; Kishimoto, Daisuke; Matsushita, Takahiro

doi:10.2140/agt.2019.19.341

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Relative phantom maps

Kouyemon Iriye, Daisuke Kishimoto and Takahiro Matsushita

Algebraic & Geometric Topology 19 (2019) 341–362

DOI: 10.2140/agt.2019.19.341

Abstract

The de Bruijn–Erdős theorem states that the chromatic number of an infinite graph equals the maximum of the chromatic numbers of its finite subgraphs. Such determination by finite subobjects appears in the definition of a phantom map, which is classical in algebraic topology. The topological method in combinatorics connects these two, which leads us to define the relative version of a phantom map: a map $f : X \to Y$ is called a relative phantom map to a map $φ : B \to Y$ if the restriction of $f$ to any finite subcomplex of $X$ lifts to $B$ through $φ$ , up to homotopy. There are two kinds of maps which are obviously relative phantom maps: (1) the composite of a map $X \to B$ with $φ$ ; (2) a usual phantom map $X \to Y$ . A relative phantom map of type (1) is called trivial, and a relative phantom map out of a suspension which is a sum of (1) and (2) is called relatively trivial. We study the (relative) triviality of relative phantom maps and, in particular, we give rational homology conditions for the (relative) triviality.

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Keywords

relative phantom maps, de Bruijn–Erdos theorem, box complexes, relative triviality

Mathematical Subject Classification 2010

Primary: 55P99

References

Publication

Received: 17 January 2018

Revised: 4 September 2018

Accepted: 5 September 2018

Published: 6 February 2019

Authors

Kouyemon Iriye
Department of Mathematical Sciences Osaka Prefecture University Sakai Japan

Daisuke Kishimoto
Department of Mathematics Kyoto University Kyoto Japan

Takahiro Matsushita
Department of Mathematical Sciences University of the Ryukyus Nishihara Japan

Volume 19, issue 1 (2019)