Piecewise linear ball ($B^n$)

A piecewise linear ball is a simplicial complex admitting a homeomorphism to an $n$-simplex that is linear on the simplexes of suitable finite simplicial subdivisions. Its boundary is a piecewise linear sphere.

Two piecewise linear balls of dimension three glued along a common boundary disk form another piecewise linear ball. The Schoenflies theorem identifies each boundary pair with a sphere and a hemisphere. Coning the resulting boundary homeomorphisms extends them over the balls, which then become the two halves of a standard ball. The same argument shows that removing a ball attached along one boundary disk leaves a ball.