Hi!


I apologize in advance if this question is better fit for <https://math.stackexchange.com/>.


Out of curiosity I'm interested in a particular case of the problem of what properties of a manifold is given by combinatorial information associated to the gluing of the manifold from pieces of $\mathbb{R}^n$.


Let $X$ be a manifold and $K$ be a covering of $X$ by subsets of $X$ homeomorphic to $\mathbb{R}^n$. We may produce a simplicial object $N\_\cdot K$ called the Čech nerve of $K$. If the Hausdorff condition on $X$ is a constraint on the allowable gluings of $X$ from pieces of $\mathbb{R}^n$, we might expect to find this as a constraint on the set of nerves $N\_\cdot K$ arising from manifolds (contra non-Hausdorff "premanifolds").


So my question becomes: Does the Hausdorff condition affect the set of possible nerves arising from such coverings? If so by what criterion? Is this criterion both necessary and sufficient? I would be interested in a characterization of such "separatedness" of simplicial objects.


Sincerely,


Eivind



Gluing of manifolds and the Hausdorff condition.
First a comment. You don't need the full Cech nerve of the cover. All the information in it is encoded in the Cech Lie groupoid. So the question boils down to: When does a Lie groupoid have a Hausdorff quotient?


Secondly, if $M\_K$ denotes this Lie groupoid, it is etale, meaning all of its structure maps are local diffeomorphisms. If $M$ were a Hausdorff manifold, then in particular, it would be an orbifold, and since $M\_K$ is Morita equivalent to $M,$ $M\_K$ would be a proper Lie groupoid. Recall that a proper Lie groupoid $\mathcal{G}$ is one such that source and target map put together $$\left(s,t\right):\mathcal{G}\_1 \to \mathcal{G}\_0 \times \mathcal{G}\_0$$ is a proper map, i.e. the pre-image of compact subsets are compact.


Conversely, if $M\_K$ is a proper etale Lie groupoid, it is an orbifold groupoid and it follows that the quotient, which is diffeomorphic to $M,$ must be Hausdorff.


So, the quotient is Hausdorff if and only if the map $$\coprod U\_\alpha \cap U\_\beta \to \coprod U\_\alpha \times \coprod U\_\alpha$$ which sends a point $x$ in the intersection of $U\_\alpha$ and $U\_\beta$ to its copy in $U\_\alpha$ and its copy in $U\_\beta$, i.e. $$\left(x \in U\_\alpha \cap U\_\beta\right) \mapsto \left(x \in U\_\alpha, x \in U\_\beta\right),$$ is a proper map. If you would like, you can reinterpret this result in terms of the simplicial nerve.


