Suppose $G$ be a discreat group acting on $\mathbb R^n$ freely via two different actions $\rho_1$ and $\rho_2$. Suppose that $\mathbb R^n/\rho_1$ is homeomorphic to $\mathbb R^n/\rho_2$. However the smooth structures on these two space might be different. What kind of invariant tells you the difference between smooth structures on the quotient space? Lets assume $n\ne 4$, hence $\mathbb R^n$ has a unique smooth structure.
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$\begingroup$ What you ask is not clear : the quotient also depends on the action as a topological manifold. $\endgroup$– abxDec 5, 2013 at 12:26
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$\begingroup$ @abx Thanks, I've edit the post to make it clear. $\endgroup$– J. GEDec 5, 2013 at 12:38
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$\begingroup$ Consider the simplest case of $(S^1)^n$, it has many exotic smooth structures (for $n$ large) and its universal cover is diffeomorphic to $\mathbb R^n$ provided $n \neq 4$. These smooth structures are relateable to exotic smooth structures on spheres via smoothing theory.... so you could probably cook up invariants to distinguish some of these using signatures of carefully-crafted bounding manifolds. $\endgroup$– Ryan BudneyDec 5, 2013 at 12:45
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$\begingroup$ @Ryan, Is there any way to 'read' the smooth structur from the actions? $\endgroup$– J. GEDec 5, 2013 at 12:53
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$\begingroup$ I don't know any direct ways that wouldn't go through the kind of construction I mention above. $\endgroup$– Ryan BudneyDec 5, 2013 at 13:00
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