Editing Fundamental group (section)

==Edge-path group of a simplicial complex==
When the topological space is homeomorphic to a [[simplicial complex]], its fundamental group can be described explicitly in terms of [[Presentation of a group|generators and relations]].

If ''X'' is a [[connected space|connected]] simplicial complex, an ''edge-path'' in ''X'' is defined to be a chain of vertices connected by edges in ''X''.  Two edge-paths are said to be ''edge-equivalent'' if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in ''X''.  If ''v'' is a fixed vertex in ''X'', an ''edge-loop'' at ''v'' is an edge-path starting and ending at ''v''.  The '''edge-path group''' ''E''(''X'',&nbsp;''v'') is defined to be the set of edge-equivalence classes of edge-loops at ''v'', with product and inverse defined by concatenation and reversal of edge-loops.

The edge-path group is naturally isomorphic to π<sub>1</sub>(|''X''{{hairsp}}|,&nbsp;''v''), the fundamental group of the [[Simplicial set|geometric realisation]] |''X''{{hairsp}}| of ''X''.<ref>{{cite book|last1=Singer|first1=Isadore|author-link1=Isadore Singer|last2=Thorpe|first2=John A.|title=Lecture notes on elementary topology and geometry|url=https://archive.org/details/lecturenotesonel00sing_949|url-access=limited|date=1967|publisher=Springer-Verlag|isbn=0-387-90202-3|page=[https://archive.org/details/lecturenotesonel00sing_949/page/n101 98]}}</ref>  Since it depends only on the [[n-skeleton|2-skeleton]] ''X''<sup>&thinsp;2</sup> of ''X'' (that is, the vertices, edges, and triangles of ''X''), the groups π<sub>1</sub>(|''X''{{hairsp}}|,''v'') and π<sub>1</sub>(|''X''<sup>&thinsp;2</sup>|,&nbsp;''v'') are isomorphic.

The edge-path group can be described explicitly in terms of [[generators and relations]].  If ''T'' is a [[spanning tree|maximal spanning tree]] in the [[n-skeleton|1-skeleton]] of ''X'', then ''E''(''X'',&nbsp;''v'') is canonically isomorphic to the group with generators (the oriented edge-paths of ''X'' not occurring in ''T'') and relations (the edge-equivalences corresponding to triangles in ''X'').  A similar result holds if ''T'' is replaced by any [[simply connected]]&mdash;in particular [[contractible]]&mdash;subcomplex of ''X''.  This often gives a practical way of computing fundamental groups and can be used to show that every [[finitely presented group]] arises as the fundamental group of a finite simplicial complex.  It is also one of the classical methods used for [[Surface (topology)|topological surfaces]], which are classified by their fundamental groups.

The ''universal covering space'' of a finite connected simplicial complex ''X'' can also be described directly as a simplicial complex using edge-paths.  Its vertices are pairs (''w'',γ) where ''w'' is a vertex of ''X'' and γ is an edge-equivalence class of paths from ''v'' to ''w''.  The ''k''-simplices containing (''w'',γ) correspond naturally to the ''k''-simplices containing ''w''.  Each new vertex ''u'' of the ''k''-simplex gives an edge ''wu'' and hence, by concatenation, a new path γ<sub>''u''</sub> from ''v'' to ''u''.  The points (''w'',γ) and (''u'', γ<sub>''u''</sub>) are the vertices of the "transported" simplex in the universal covering space.  The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just ''X''.

It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space.  This was doubtless known to [[Eduard Čech]] and [[Jean Leray]] and explicitly appeared as a remark in a paper by [[André Weil]];<ref>[[André Weil]], ''On discrete subgroups of Lie groups'', [[Annals of Mathematics]] '''72''' (1960), 369-384.</ref> various other authors such as Lorenzo Calabi, [[Wu Wenjun|Wu Wen-tsün]], and Nodar  Berikashvili have also published proofs.  In the simplest case of a compact space ''X'' with a finite open covering in which all [[empty set|non-empty]] finite [[intersection (set theory)|intersections]] of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the [[Nerve of an open covering|nerve of the covering]].