Let $T$ be a collection of triangles in an abstract simplicial complex $A$ such that
$$\partial \left( \sum \limits_{t \in T} \sigma_t \cdot t\right) = 0$$
where $\sigma_t \in \{-1, 1\}$ is some appropriately-chosen collection of coefficients.
I want to know if the following statement is then necessarily true:
Conjecture: One can obtain $T$ by starting with the disjoint union of some polyhedra in $\mathbb{R}^3$ with triangular faces and then "gluing together" pairs of points some number of times (i.e. dragging two distinct points together in space so that they coincide and are equal)). The resulting set of triangular faces, identified by their three nodes, is then isomorphic to $T$.
As followup questions, I'm also interested to know if the analogous statement is true for higher dimensions and for polyhedral complexes.