I want to gain intuition about the failure of a smooth stratified space to be a smooth manifold, so I'll keep the question informal. Wiki page: https://en.wikipedia.org/wiki/Topologically_stratified_space
I'm following Momentum Maps and Hamiltonian Reduction, Ortega-Ratiu.
- a stratified space is a topological space that can be decomposed in manifolds, called strata;
- a singular chart for a stratified space is a homeomorphism from an open set of the stratified space to an open subset of an Euclidean space, that behaves nicely on strata (when restricted to strata it gives submanifolds in the Euclidean space );
- there is a notion of $k$-compatibility between singular charts. Different open subsets around the space are homeomorphic to Euclidean spaces of different dimension. Two charts $(\phi,U), (\psi,V)$ are compatible if every point in the intersection of $U,V$ has an open neighborhood $W$ in the intersection of $U,V$, such that the images of $W$ via $\phi,\psi$, once embedded in a big enough Euclidean space, are mapped one to the other by a $k$-diffeomorphism between open subsets of this big Euclidean space containing the images of $U$ and $V$ ;
- a singular $k$-atlas is a collection of singular charts whose domains cover the stratified space, with all the charts in it $k$-compatible for some $k$;
- the stratified space endowed with this atlas is a $C^k$-stratified space;
- if $k = \infty$, we have a smooth stratified space.
I see just two problems for this space not to be a smooth manifold:
- obviously, the varying dimension of the euclidean spaces homeomorphic to open set on the stratified space. Is it the core of the obstruction, or is there something else?
- there is no mention of Hausdorff-ness; is this deep or could it just as well be included?