FTP, name this topological construct, that is defined as the group of equivalence classes of loops in a space, and denoted "pi sub one."
The group of deck transformations of a universal cover of a topological space is isomorphic to this.  When it is Abelianized, the result is the associated space's first homology group.  It can be calculated by Van Kampen's theorem if its value for some subspaces that cover the space in question is known.  Examples of it include the group Z of integers for a circle and Z*Z ["z star z"] for a torus, and any space for which it is trivial is said to be simply connected.  FTP, name this topological construct, that is defined as the group of equivalence classes of loops in a space, and denoted "pi sub one."
The group of deck transformations of a universal cover of a topological space is isomorphic to this.  
fundamental group