A combinatorial definition for the fundamental group is to introduce generators $g_{ij}$ for each pair of vertices $v_i,v_j$ for which $i<j$. And $g_{ij}g_{jk}=g_{ik}$ whenever $v_i,v_j,v_k$ span a 2-simplex of $K-L$ (where $L$ is a spanning tree of $K$), and $i<j<k$. (For instance, see Armstrong Basic Topology pg. 133-135)
A priori, the definition seems to depend on the spanning tree $L$.
What are the ways to show that the fundamental group is in fact independent of choice of $L$?
Following Armstrong's book, he proved that $G(K,L)$ is isomorphic to $E(K,v)$, the edge group, which is known to be independent of $v$. Hence, this is probably one way to show the independence of $L$.
Are there other ways?
Thanks a lot.
I assume at some point they prove the definition is equivalent to the standard one. Then it is implied by the fact quotienting out by contractible spaces is a homotopy equivalence, and homotopy equivalent spaces having isomorphic fundamental groups.