# Topological Invariants for Group

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

This process looks like a sort of "inverse process" to what we do in Algebraic Topology especially when we try to associate the Fundamental Group to a given topological space. In Algebraic Topology in general, we are trying to find algebraic invariants of a given topological space whereas here I am trying to find topological invariants for a group.

However, it is clear that the functor which I have defined above is just an example of a functor from $$\mathbf{Grp}$$ to $$\mathbf{Top}$$ and (I believe) is not going to be much useful.

So my question is,

Does there exist any useful topological invariant of a group? More specifically, given any group can we associate a topological space to it (in the same way we did for Fundamental Groups)? If so can some literature be mentioned?

• en.wikipedia.org/wiki/Classifying_space – YCor Apr 26 '19 at 14:27
• @YCor: But $G$ may not be a topological group in general. – user 170039 Apr 26 '19 at 14:28
• Every group is a topological group with the discrete topology. The classifying space is already very interesting for discrete groups. – YCor Apr 26 '19 at 14:32
• In particular, group cohomology can be seen as a topological invariant of a group. Although it has a purely algebraic definition, it is naturally isomorphic to the cohomology of the corresponding classifying space. – Joshua Grochow Apr 26 '19 at 14:52
• The link points to a 1977 paper on the verbal topology by R. Bryant. This is an instance of a group topology (= topology making $G$ a topological group). There are many natural group topologies one can put on a group, such as profinite, etc, which can be trivial or not according to cases. The question seems a bit open-ended at this point... – YCor Apr 27 '19 at 13:02

For any Lie group $$G$$, there is a topological invariant of groups $$\Gamma$$ called the $$G$$-character variety of $$\Gamma$$. It is defined by: $$\mathfrak{X}_G(\Gamma):=\mathrm{Hom}(\Gamma,G)^*/G,$$ where $$\Gamma$$ is given the discrete topology, $$\mathrm{Hom}(\Gamma, G)$$ the compact-open topology, $$\mathrm{Hom}(\Gamma, G)^*$$ is the subspace of homomorphisms that have closed conjugation orbits (polystable), and finally $$\mathrm{Hom}(\Gamma,G)^*/G$$ is the conjugation orbit space of $$\mathrm{Hom}(\Gamma, G)^*$$.
If $$\varphi:\Gamma^\prime\to \Gamma$$ is an isomorphism, then for every polystable $$\rho:\Gamma\to G$$, then $$\rho\circ \varphi$$ is polystable and so defines a point $$[\rho\circ \varphi]\in \mathfrak{X}_G(\Gamma^\prime)$$. This map is invertible since $$\varphi$$ has an inverse. Hence, it is a topological invariant of the group.
One can think of $$\mathfrak{X}_G(\Gamma)$$ as a moduli space of subgroups of $$G$$ arising as the homomorphic image of a fixed group-type $$\Gamma$$.
This invariant of $$\Gamma$$ has applications to many areas of mathematics and mathematical physics (usually where $$\Gamma$$ is finitely generated). If you google "character variety", or do a search in arXiv or MathSciNet (if you have access) you will find many research papers on the topic.
• Actually an even better invariant is $\mathrm{Hom}(\Gamma,G)$ without taking the quotient (but endowed with the conjugation action of $G$). – YCor Apr 27 '19 at 13:03