The boundary of a bounded contractible Lipschitz polyhedron is connected

Put $M=\bar{\mathrm{\Omega }}$. It is compact by boundedness and closedness. Near each boundary point a Lipschitz graph chart $x_3=\varphi (x^{\mathrm{\prime }})$ straightens $M$ by the explicit homeomorphism $(x^{\mathrm{\prime }},x_3)\mapsto (x^{\mathrm{\prime }},x_3-\varphi (x^{\mathrm{\prime }}))$ to a closed halfspace neighborhood. Interior points use Euclidean balls. These charts show that $M$ is a Hausdorff second-countable three-manifold with topological boundary exactly $\mathrm{\partial }\mathrm{\Omega }$ and interior exactly $\mathrm{\Omega }$. Its orientation is the standard Euclidean orientation on its interior; this is precisely the orientation convention for manifolds with boundary in [F2]. By [F1], $\mathrm{\Omega }\hookrightarrow M$ is a homotopy equivalence. Contractibility of $\mathrm{\Omega }$, [F3], and factorization through a point give $H^2(M;\mathbb{R})=0$: the point has one singular simplex in each degree, and its real cochain differentials are successively zero and identity, so its positive cohomology is zero. Also $M$ is path connected: its interior is path connected by contractibility, and a half-chart path joins any boundary point to its interior. The augmentation identifies $H_0(M;\mathbb{R})$ with $\mathbb{R}$, because any difference of points is the boundary of a path and every zero-chain of coefficient sum zero is a finite linear combination of such differences. The boundary is nonempty: a ray from any interior point eventually leaves the bounded set and its first exit point is a boundary point.

By [F2], $H_1(M,\mathrm{\partial }M;\mathbb{R})=0$. The exact sequence in [F3] then makes $H_0(\mathrm{\partial }M;\mathbb{R})\to H_0(M;\mathbb{R})$ injective. The boundary is a two-manifold by the graph charts, hence locally path connected, so its connected components are path components. If there were two, choose one point in each. Their difference is nonzero in $H_0(\mathrm{\partial }M;\mathbb{R})$: the sum of coefficients on the first path component defines a linear functional on zero-chains which kills every one-simplex boundary, yet sends this difference to one. Its image in $H_0(M;\mathbb{R})$ is zero since the two points are joined by a path in $M$, contradicting injectivity. Therefore the nonempty boundary is connected. No finite-element incidence matrix is used to infer this continuum topological claim.