The geometric realisation $|\Gamma|_I$ of a graph $(\Gamma, I)$

Question

Prove that the geometric realisation $|\Gamma|_I$ of a graph $(\Gamma, I)$ is the topological space $$|\Gamma|_I = E\times [0,1]\vert \sim$$where $\sim$ is the equivalence relation generated by the following relations here (Page 12).

Now I want to understand the geometric realisation and want to prove $\sim$ generated by the relations is an equivalence relation.

As far I have understood that the geometric realisation is a $1$ dimensional CW-complex where we will assume each vertex as a point and each edge as unit interval. Also I feel that geometric realisation of graph is embedded in $\mathbb{R}^n$. And the topology coming from the induced topology of $\mathbb{R}^n$. Are my thoughts right?

But I cannot prove that $\sim$ is an equivalence relation, i.e, I want to prove reflexive, symmetric, and transitive.

Please help.

Answer 1

Prove that the geometric realisation $|\Gamma|_I$ of a graph $(\Gamma, I)$ is the topological space $$|\Gamma|_I = E\times [0,1]\vert \sim$$where $\sim$ is the equivalence relation generated by the following relations here (Page 12).

There is nothing to prove because it is a definition. This is based on $\sim$ which is defined as the equivalence relation generated by the following relations:

• $(e, t) ∼ (\bar e, 1 − t)$.
• If $e, f ∈ E$ with $e_− = f_−$ then $(e, 0) ∼ (f, 0)$.
• If $e, f ∈ E$ with $e_+ = f_+$ then $(e, 1) ∼ (f, 1)$.

Each relation $R$ generates an equivalence relation $\sim_R$ (it the intersection of all equivalence relations containing $R$).

But admittedly we should understand what $\sim$ does.

We can visualize a directed graph by drawing its vertices as points and its edges as (directed!) arrows between the vertices. The graph can have multiple edges from a vertex $e_-$ to a vertex $e_+$ and it can have "loops" if $e_- = e_+$. If we have an involution on the graph, we know that edges occur in pairs $e, \bar e$. This means for each arrow from $e_-$ to $e_+$ there exists a "reverse arrow" from $e_+$ to $e_-$.

In other words, we can regard a graph as a set $V$ of vertices, a set $E'$ of (undirected) edges and a function $\phi' : V \to \mathcal P(V) = $ power set of $V$ such that $\phi'(e')$ has one or two elements. The correspondence should be clear: Given a graph in the sense of the author, we associate to each pair $e, \bar e$ of directed edges a single undirected edge $e'$; and given an undirected graph, we associate to each undirected edge $e'$ a pair od directed edges $e, \bar e$ with the obvious involution.

$|\Gamma|_I$ is then a $1$-dimensional CW-complex having $V$ a its $0$-skeleton and for each pair $e, \bar e$ of directed edges (or alternatively for each undirected edge $e'$) an attached $1$-cell whose boundary is a mapped in the obvious way to $V$. Note that if $e'$ connects two distinct vertices of $V$, then there are two distinct attaching maps, but topologically we get the same adjunction space. If $e'$ is a loop, then we have a unique attaching map.