Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, is it these two singular simplicial complexes associated to the triangulations homologous? Or maybe we can ask the same problem for triangulable space.

Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these two singular simplicial complexes associated to the triangulations homologous? Or maybe we can ask the same problem for triangulable space.

Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, is it these two singular simplicial complexes associated to the triangulations homologous?

Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, is it these two singular simplicial complexes associated to the triangulations homologous? Or maybe we can ask the same problem for triangulable space.