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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 → Y is called a relative phantom map to a map φ: B → 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 → B with φ; (2) a usual phantom map X → 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.
Keywords: relative phantom maps, de Bruijn–Erdos theorem, box complexes, relative triviality
Iriye, Kouyemon 1 ; Kishimoto, Daisuke 2 ; Matsushita, Takahiro 3
@article{10_2140_agt_2019_19_341,
author = {Iriye, Kouyemon and Kishimoto, Daisuke and Matsushita, Takahiro},
title = {Relative phantom maps},
journal = {Algebraic and Geometric Topology},
pages = {341--362},
publisher = {mathdoc},
volume = {19},
number = {1},
year = {2019},
doi = {10.2140/agt.2019.19.341},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.341/}
}
TY - JOUR AU - Iriye, Kouyemon AU - Kishimoto, Daisuke AU - Matsushita, Takahiro TI - Relative phantom maps JO - Algebraic and Geometric Topology PY - 2019 SP - 341 EP - 362 VL - 19 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.341/ DO - 10.2140/agt.2019.19.341 ID - 10_2140_agt_2019_19_341 ER -
%0 Journal Article %A Iriye, Kouyemon %A Kishimoto, Daisuke %A Matsushita, Takahiro %T Relative phantom maps %J Algebraic and Geometric Topology %D 2019 %P 341-362 %V 19 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.341/ %R 10.2140/agt.2019.19.341 %F 10_2140_agt_2019_19_341
Iriye, Kouyemon; Kishimoto, Daisuke; Matsushita, Takahiro. Relative phantom maps. Algebraic and Geometric Topology, Tome 19 (2019) no. 1, pp. 341-362. doi: 10.2140/agt.2019.19.341
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