TY - JOUR

T1 - Relative phantom maps

AU - Iriye, Kouyemon

AU - Kishimoto, Daisuke

AU - Matsushita, Takahiro

N1 - Funding Information:
Acknowledgement The authors were supported respectively by JSPS KAKENHI (No. 26400094), JSPS KAKENHI (No. 25400087) and JSPS KAKENHI (No. 28-6304). The authors are grateful to Jérôme Scherer and the anonymous referee for useful comments.
Publisher Copyright:
© 2019, Mathematical Sciences Publishers. All rights reserved

PY - 2019/2/6

Y1 - 2019/2/6

N2 - The de Bruijn–Erd?os 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 dimensional skeleton 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 from a suspension, and in particular, we give rational homotopy conditions for the (relative) triviality.

AB - The de Bruijn–Erd?os 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 dimensional skeleton 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 from a suspension, and in particular, we give rational homotopy conditions for the (relative) triviality.

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U2 - 10.2140/agt.2019.19.341

DO - 10.2140/agt.2019.19.341

M3 - Article

AN - SCOPUS:85062854784

VL - 19

SP - 341

EP - 362

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 1

ER -