No closed nonorientable piecewise-linear surface embeds in the three-sphere

By [F6], take a finite subdivision of the standard triangulation in which $F$ is a subcomplex. Sum its triangular faces with coefficient $1$ in cellular $2$-chains over ${\mathbb{F}}_2$. Each edge of the closed surface is incident to exactly two of these triangles, so the sum $z$ has zero boundary.

By [F1] and [F2], $[z]=0$ in $H_2(S^3;{\mathbb{F}}_2)$, so $z=\mathrm{\partial }c$ for a finite cellular $3$-chain $c$. Give each tetrahedron its coefficient $0$ or $1$ in $c$.

Orient each tetrahedron by the local orientation of $S^3$ in [F4] and take the integral cellular chain $C$ with the same coefficients $0$ or $1$ as $c$; [F5] identifies these oriented cells with the integral chain generators. For any triangular face outside $F$, its coefficient in $\mathrm{\partial }c$ is zero modulo $2$, so its two incident tetrahedra have equal coefficients. Their induced face orientations are opposite: in an oriented PL coordinate ball at an interior point of that face, the tetrahedra occupy opposite half-balls, and the outward-normal-first rule [F7] reverses the face sign. Their contributions to $\mathrm{\partial }C$ therefore cancel. For a face in $F$, the coefficient of $\mathrm{\partial }c$ is one modulo $2$, so exactly one incident tetrahedron has coefficient one; its contribution to $\mathrm{\partial }C$ is $+1$ or $-1$. Thus $w=\mathrm{\partial }C$ is supported on $F$ with coefficient $+1$ or $-1$ on each triangle.

By [F8], $\mathrm{\partial }w={\mathrm{\partial }}^{2}C=0$. Hence along each edge of $F$ the two incident oriented triangles induce opposite edge orientations. The triangles around any vertex form a cycle because $F$ is a triangulated surface; these compatible edge orientations therefore give one local orientation on its star. Interior points of triangles, edges and vertex stars cover $F$, and the local generators agree on their overlaps. They define the continuous orientation section of [F4], contradicting nonorientability.