What is the contribution of group theory to topology? An answer for a question on MathOverflow.net which asked for some recommendations on textbooks for books in topology received the following comment: 

"It's a great book to introduce applied topology, although it stops just short of using groups."



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*Pedagogically, what is the value of group theory to topology? 

*How does group theory make strengthen topology as a theory?
 A: For a concrete example, the usual proof that $\Bbb R^n$ is not homeomorphic to $\Bbb R^m$ if $n \ne m$ requires group theory (some invariant group (the $n$th homology of 'your space') is associated to a topological space and a homeomorphism induces an isomorphism of such groups and $\Bbb R^n-\{0\}$ and $\Bbb R^m-\{pt\}$ have groups that are obviously not isomorphic). These spaces are somehow the most basic and important examples of topological spaces, yet even showing they are not equivalent as topological spaces requires some knowledge of groups. 
In general, groups (and other algebraic structures) allow us to define a host of such invariants that allow us to distinguish topological spaces. This is extremely important since we often care more about spaces that are path-connected, Hausdorff and in general nice, so a lot of our tools from basic "point-set topology" of showing two spaces are not homeomorphic are lost. Really the goal of any study of topology is to deduce when two spaces are the same (there are weaker notions of "the same" than homeomorphic) and when they are not, and groups are instrumental in the second case.
