TITLE: Under what conditions the quotient space of a manifold is a manifold?
QUESTION [68 upvotes]: There are many operations we can do with topological spaces that when we apply to topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of that.
Another common operation is to take the quotient by some equivalence relation. Now, the definition of manifold I'm used to is the following: a topological manifold is a topological space $M$ such that $M$ is Hausdorff, has an enumerable basis for the topology and is locally euclidean of dimension $n$.
Now, we know that the quotient of a Hausdorff space is not necessarily Hausdorff. So, under what conditions if we have a manifold $M$ and an equivalence relation $\sim$ on $M$ the quotient space $M/\sim$ will be a manifold?

REPLY [10 votes]: There is quite a lot on this problem already known. The study of the Hausdorff (usually metrizable) spaces gotten from taking quotients of manifolds is called decomposition space theory and has a wonderful history.
A special case of this which has gathered a lot of attention is when is the quotient of a manifold $M$ actually homeomorphic to $M$. Bing in the 1950's and 1960's studied cases of this problem extensively and showed a certain quotient of $S^3$ is actually homeomorphic to $S^3$ and used such a decomposition to construct involutions of $S^3$ not conjugate to any smooth ones.  Brown proved the generalized Schoenflies theorem using similar techniques. In the 80's Freedman used this theory to prove to 4-dimensional Poincare conjecture.
In your case, if the quotient map has only one nontrivial set $K$ being identified to a point and $K$ is compact, then the quotient space is homeomorphic to a manifold if and only if $K$ is a countably infinite intersection of nested closed balls ($K$ is cellular according to the literature). Such a characterization also works perfectly if there are only finitely many nontrivial sets identified to a point. 
If there are infinitely many such sets, things become more complicated. Realizing when a "generalized manifold" is a manifold to my knowledge is not completely understood though there are a lot of partial results.
Daverman gives a nice account of most of this story in his book here including some of the mentioned partial results near the end.