We define a map f : X → Y to be a phantom map relative 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. We also give a rational homotopy condition for the triviality of relative phantom maps from a non-suspension to a finite Postnikov section.
The matching complex of a simple graph G is a simplicial complex consisting of the matchings on G. Jelić Milutinović et al. [6] studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.
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