Finite tangent index count and inward boundary sum

By [F8] choose a $C^2$ diffeomorphism $\psi :S\to \mathrm{\Sigma }$ to a compact smooth oriented carrier and an excellent Morse function on $\mathrm{\Sigma }$, with finitely many critical points and a smooth gradient section. At an index-$\lambda$ point the gradient's derivative is the Hessian up to a positive metric isomorphism, so its index is $(-1{)}^{\lambda }$ by [F1]. Pull the section back by $D{\psi }^{-1}$ to a $C^1$ tangent section $s_0$ on $S$. At a zero its derivative is conjugate to the old derivative; differentiating the bundle map contributes no extra term because the section value is zero. Hence the same nondegenerate zeros and indices occur. The smooth handle CW model gives ${\sum }_p(-1{)}^{\lambda (p)}=\chi (\mathrm{\Sigma })$ by [F3], and homology functoriality under $\psi$ gives $\chi (\mathrm{\Sigma })=\chi (S)=V-E+F$ for the original topological cellulation. No differentiable handle per original topological cell is asserted.

The ambient and plane orientations coorient the normal line of $E$. Finite chart bumps give a C¹ positive transverse field $V$ near $W$, inward along its boundary, and a positive annihilator $\omega$ of $E$. Approximate their coefficients in finitely many ambient charts by smooth coefficients and patch with smooth bumps. Uniformly small errors preserve transversality and inwardness, giving smooth $V,{\omega }^{\mathrm{\prime }}$ with ${\omega }^{\mathrm{\prime }}(V)>0$. Put $E^{\mathrm{\prime }}=\ker {\omega }^{\mathrm{\prime }}$ and orient it so projection along $V$ gives an orientation-preserving bundle isomorphism $E\to E^{\mathrm{\prime }}$. This is invertible because both planes complement the same line.

Finite smooth bump multiples of local frames of $E^{\mathrm{\prime }}$ span each fiber over $W$. Their parameterized linear combination $s_a$, $a\in {\mathbb{R}}^P$, has a fiber-surjective parameter derivative. Solving for two parameters in each frame shows that its universal zero set over C² $W$ is a C² manifold of dimension $P+1$, with boundary the zero set over $\mathrm{\partial }W$, of dimension $P$. Apply F2 to the parameter projection in interior and boundary charts: C² suffices for dimension difference one, and C¹ suffices on the boundary. Countable chart covers and null unions use the stated countable choice. A common regular parameter gives a section transverse to zero on $W$ and on $\mathrm{\partial }W$. Its zero set $Z$ is a compact oriented C² one-manifold, with boundary $Z\cap \mathrm{\partial }W$. Subdivide a finite oriented interval-chart cover; its paired internal endpoints cancel, giving total signed boundary count zero.

Compare $s_0$ with a C¹ tangent section $s_1$ having isolated nondegenerate zeros. Both zero sets are finite, being closed and discrete in compact $S$. Refine a finite chart cellulation and move its graph slightly to miss both zero sets, with every face inside a tangent trivialization. Give $TS$ a C¹ metric by finite chart bumps, and use oriented orthonormal frames. On the graph the unit-section ratio $r=(s_1\mathrm{/}\mathrm{\mid }{s}_1\mathrm{\mid })(s_0\mathrm{/}\mathrm{\mid }{s}_0\mathrm{\mid }{)}^{-1}$ is a well-defined circle map independent of the frame, since frame changes are rotations. In each face remove small disks about its zeros and cut the remaining region into finitely many disks. Continuous argument increments cancel on paired cut edges, so the boundary winding of each section equals the sum of its local indices. Their difference is the winding of $r$. Summing over all faces cancels every edge increment of $r$ with its reversed occurrence, because $S$ has no boundary. The total index sums agree. This is the circle-degree and lifting calculus of The degree of a based circle loop and Degree defines a function $\mathrm{Deg}:{\pi }_1(S^1,[0])\to \mathbb{Z}$; it requires no C² Sard theorem for a C¹ homotopy.

The signs are uniform on each connected component of $W$: orienting $E$ by the rule that a positive frame of $E$ followed by the ${\omega }^{\mathrm{\prime }}$-positive direction $V$ is a positive frame of $TW$, the inwardness of $V$ gives $V$ a strictly negative outward-normal component at every boundary point, so comparison with the outward-normal-first rule [F7] shows that this induced orientation of $E{\mathrm{\mid }}_C$ is the negative of the boundary orientation of $C$ on every component; on a connected component of $W$ the given orientation of $E$ either agrees with the induced orientation at every point or disagrees at every point, so ${\epsilon }_C$ is one constant over the boundary components of each connected component of $W$.

Combining steps 1.1 and 2.1, every C¹ tangent section of a finitely cellulated closed oriented C² surface with isolated nondegenerate zeros has total index $V-E+F$, and [F3] identifies this count with the Euler characteristic of the cellulation.

On a boundary component $C$ pull $s_a$ back along $E\to E^{\mathrm{\prime }}$ to a C¹ section of $TC$ with nondegenerate zeros. At a zero, differentiating this bundle isomorphism adds no term from the zero section value, so corresponding fiber frames give the same signed determinant. F6 supplies a finite cellulation of closed $C$. Its ordinary tangent-index sum is $\chi (C)$ by step 3.1; with the given fiber orientation of $E$ rather than the boundary tangent orientation, the signed zero count is ${\epsilon }_C\chi (C)$.

At a boundary zero take coordinates with $W=\{x_3\le 0\}$ and outward normal ${\mathrm{\partial }}_3$, and an oriented frame of $E^{\mathrm{\prime }}$. The boundary-coordinate derivative $A$ of the section is invertible. Solving for those two coordinates makes the zero curve a graph over $x_3$; its orientation compares with the positive $x_3$ direction by $\mathrm{sign}\det A$. Its endpoint sign is therefore the signed boundary-section zero count of step 4.1, up to one fixed orientation convention shared by all endpoints. By step 2.2, ${\epsilon }_C=\epsilon$ on every boundary component of one connected component of $W$. Zero signed boundary count gives $0=\epsilon {\sum }_C\chi (C)$ there. Add over the finitely many region components. Empty boundaries contribute zero, and steps 1.1–2.1 give the closed-surface assertion.