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?

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    $\begingroup$ en.wikipedia.org/wiki/Classifying_space $\endgroup$ – YCor Apr 26 at 14:27
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    $\begingroup$ @YCor: But $G$ may not be a topological group in general. $\endgroup$ – user 170039 Apr 26 at 14:28
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    $\begingroup$ Every group is a topological group with the discrete topology. The classifying space is already very interesting for discrete groups. $\endgroup$ – YCor Apr 26 at 14:32
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    $\begingroup$ 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. $\endgroup$ – Joshua Grochow Apr 26 at 14:52
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    $\begingroup$ 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... $\endgroup$ – YCor Apr 27 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.

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    $\begingroup$ Actually an even better invariant is $\mathrm{Hom}(\Gamma,G)$ without taking the quotient (but endowed with the conjugation action of $G$). $\endgroup$ – YCor Apr 27 at 13:03

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