(path)Connectedness of a simplicial complex Vs it's $1$-skeleton

by user   Last Updated September 12, 2019 04:20 AM

Let $\Delta$ be an abstract simplicial complex https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex and $|\Delta|$ be its geometric realization.

Let $\Delta ^{(1)} :=\{F $ is a face of $\Delta : |F|\le 2 \}$ . So $\Delta ^{(1)}$ is a simplicial complex of dimension $1$ , so in particular it is a simple graph. Now $\Delta ^{(1)}$ is connected as a graph iff its geometric realization $|\Delta ^{(1)}|$ is path connected. Now consider the following two statements:

(1) $\Delta ^{(1)}$ is connected as a graph.

(2) $|\Delta |$ is path connected as a topological space.

My question is: are (1) and (2) equivalent ? If not then at least, does one of them imply the other ?

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