cut and glue technique for higher dimensional complexes My understanding is that with surfaces, it is possible to start from a 2D sheet and use cut and glue to construct all possible surfaces up to Homeomorphism. That is cut and glue is as expressive as 2D simplicial complexes in its expressive power.
What type of higher dimensional complexes can we express using cut and glue in higher dimensions? 
Are there any good references for this?
 A: This construction I am sure is written somewhere, most likely, in the book H.Seifert, W.Threlfall, "A Textbook of Topology" (I just do not have it with me now):
Let $M$ be a compact connected triangulated $n$-dimensional manifold (for simplicity, without boundary). Let $G$ be the graph dual to the triangulation $\Delta$: The vertices of $G$ are the barycenters of the facets (n-dimensional simplices), two vertices are connected by an edge $e$ iff the corresponding facets have a common codimension 1 face denoted $e^*$ (a "panel"). Let $T\subset G$ be a maximal subtree. Next, "cut open" $M$ along the panels $e^*$ for the edges $e$ of $G$ which are not in $T$. The result is a finite simplicial complex $\Delta_T$. One then verifies that $T$ is isomorphic to a triangulated $n$-dimensional ball. With more work one proves that some  subdivision of $\Delta_T$ is isomorphic to a convex simplicial (i.e. its faces define a triangulated sphere) polyhedron in $R^n$. My suggestion is to work this out in the case of surfaces, say, of a triangulated 2-dimensional sphere. 
Now, one can reverse this process and glue appropriate faces of $\Delta_T$ (the ones which have the same image in $M$). The result is your triangulated manifold $M$. 
Edit. I finally checked: Seifert and Threlfall have the above construction in section 60 of their book, but only for 3-dimensional manifolds. The same works in all dimensions, just things become messier. 
