# reference request – Topological invariants for group

Let $$mathbf {Grp}$$ to be the category of groups and $$mathbf {Top}$$ to be the category of topological spaces. To each group $$(G, circ_G)$$we can associate a topological space $$(G, tau_G)$$ the basis of this topology being given by all of the subgroups of $$G$$. Call this topology on $$G$$ to be his Subgroup topology So we get a functor $$mathscr {F}: mathbf {Grp} to mathbf {Top}$$ which associates a given group with its. Also note that any homomorphism $$f: (G, circ_G) to (H, circ_H)$$ induces a continuous function between the correspondent Subgroups of Topological Spaces.

This process looks like a kind of "reverse process" compared to what we do in algebraic topology, especially when we try to associate the fundamental group with a given topological space. In Algebraic Topology in general, we try to find algebraic invariants of a given topological space, while I try to find topological invariants for a group.

However, it is clear that the functor I defined above is only an example of a functor of $$mathbf {Grp}$$ at $$mathbf {Top}$$ and (I think) is not going to be very useful.

So my question is,

Is there a useful topological invariant of a group? More specifically, since any group can be associated with a topological space (as we did for the fundamental groups)? If yes, can we talk about literature?