Compact twisted signature operators give additive bundle indices and finite-character signatures

Smooth bundle metrics and unitary connections needed for every bundle exist by finite formulas. Choose finitely many precompact trivializations and smooth nonnegative bumps with positive sum, normalize the bumps, and sum the local positive metrics. Gram–Schmidt on the finitely many local frames gives smooth unitary frames: all denominators are positive smooth square roots. The ordinary derivative in each such frame defines a local unitary connection. Their finite weighted sum is a global connection, since the weights sum to one in the Leibniz rule; metric compatibility is retained by the same sum. Thus this construction applies to arbitrary finite-rank smooth bundles, including every auxiliary complement occurring in a Grothendieck relation, not only flat lines. A difference of two connections is a smooth endomorphism-valued one-form. The formal adjoint identity on a closed oriented four-manifold is $d_{\mathrm{\nabla }}^{\ast }=-\ast d_{\mathrm{\nabla }}\ast$, obtained by integrating $d⟨u\wedge \ast v⟩$ and using unitarity and Stokes. It does not require ${\mathrm{\nabla }}^2=0$.

In an oriented orthonormal coframe, $\ast$ sends a basis wedge to its complementary wedge with the permutation sign, so it is an isometry and ${\ast }^2=(-1{)}^{k(4-k)}$ on $k$-forms. The coefficients of $\tau$ in degrees $0,1,2,3,4$ are $(-1,-1,1,1,-1)$ times $\ast$; substitution gives ${\tau }^2=1$, hence self-adjointness. Using step1.1, the identities $a_{k+1}=(-1{)}^k{a}_k$ and $a_{k-1}(-1{)}^k=-a_k$ for this coefficient list give $D_E\tau +\tau D_E=0$ on every degree, including the zero endpoint terms. The $\pm 1$ projection ranges are smooth bundles, since $(1\pm \tau )\mathrm{/}2$ are smooth projections of constant rank on each component. The principal symbol of $d_{\mathrm{\nabla }}$ is exterior multiplication $\epsilon (\xi )$ and that of its formal adjoint in the differential-symbol convention is $-{\iota }_{{\xi }^{\mathrm{♯}}}$. The wedge/deletion identity $\epsilon (\xi ){\iota }_{{\xi }^{\mathrm{♯}}}+{\iota }_{{\xi }^{\mathrm{♯}}}\epsilon (\xi )=\mathrm{\mid }\xi {\mathrm{\mid }}^2$ shows that $P=D_E^2$ has positive-frequency symbol $\mathrm{\mid }\xi {\mathrm{\mid }}^{2}I$. Curvature terms and connection coefficients are lower order. This $P$ is formally symmetric and nonnegative, since $(Pu,u)=\mathrm{\parallel }{D}_{E}u{\mathrm{\parallel }}^2$ on smooth forms. It preserves the two chirality bundles. Therefore [F1] applies to $P_{+}$ and $P_{-}$, with one common sufficiently large $\lambda$.

Put $B_{\pm }=(P_{\pm }+\lambda {)}^{-1}$. On smooth sections, $D_E$ commutes with $P$, so uniqueness of the shifted inverses gives $B_{-}{D}_E^{+}=D_E^{+}{B}_{+}$, and the analogous reversed identity. Let $Q=D_E^{-}{B}_{-}:L^2({\mathrm{\Omega }}^{-})\to H^1({\mathrm{\Omega }}^{+})$. This is bounded by [F1] and the local first-order coefficient estimates. Direct multiplication first on smooth sections gives $D_E^{+}Q=I-\lambda B_{-}$ on $L^2({\mathrm{\Omega }}^{-})$ and $Q{D}_E^{+}=I-\lambda B_{+}$ on $H^1({\mathrm{\Omega }}^{+})$. Smooth density extends them to those spaces. The errors are compact: $B_{-}:L^2\to H^2$ followed by inclusion into $L^2$ is compact, and $B_{+}:H^1\to H^3$ followed by inclusion into $H^1$ is compact. Hence [F2] proves that the displayed first-order map is Fredholm with exactly the claimed domain and target.

Give a finite principal cover $Y\to X$ with its specified group of order $h$ the pullback metric and orientation. It is a compact closed smooth manifold: finitely many smaller evenly covered neighborhoods with compact closures cover $X$, and their finitely many compact lift closures cover $Y$. Each deck map is an orientation-preserving isometry, so commutes with $d,d^{\ast },\ast ,D,\tau$ and the uniquely characterized harmonic projection of [F3]. For a character $\chi$, projection $h^{-1}{\sum }_g\overline{\chi (g)}{g}^{\ast }$, where $h$ is also the covering degree, therefore splits all these complexes and harmonic spaces. It also splits their cohomology: a projected closed form or primitive represents exactly the corresponding cohomology eigenspace. Forms in this eigenspace are sections of a flat line bundle with locally constant unitary transition scalars among the character values, by choosing one sheet on each evenly covered chart. With the convention $g^{\ast }\omega =\chi (g)\omega$ its holonomy is the inverse character: changing the chosen sheet multiplies the coefficient by $\chi (g)$, and parallel transport around the loop compares the endpoint sheet by the inverse. Either way its finite tensor power is canonically trivial, because all transition scalars have that finite order. The local differential, adjoint and chirality are precisely those already used for that flat bundle; covering integration multiplies both the $L^2$ pairing and middle intersection pairing by the same positive integer $h$. Thus kernel dimensions and all inertia signs agree in the two descriptions. This also works for a disconnected principal cover, since its finitely many sheets and components enter the same finite projection.

We need an actual regularity check for its cokernel, not an assumed first-order domain theorem. If $v\in L^2$ and $Pv=f\in L^2$ distributionally, put $w=B(f+\lambda v)\in H^2$. Then $u=v-w\in L^2$ satisfies $(P+\lambda )u=0$ distributionally. For any smooth $h$, $\varphi =Bh$ is smooth and $(P+\lambda )\varphi =h$; formal symmetry therefore gives $(u,h)=0$. Smooth density implies $u=0$, so $v\in H^2$, and higher regularity makes it smooth if $f$ is smooth. In particular a distributional $L^2$ solution of $D_E^{-}v=0$ satisfies $P_{-}v=0$ and is smooth. A vector annihilates the range of $D_E^{+}:H^1\to L^2$ exactly when it solves that distributional equation, by testing on smooth sections and using density. Its cokernel is thus its smooth opposite-chirality kernel. The same argument applies to its kernel. Consequently its index is the finite number $\dim \ker (D_E{\mathrm{\mid }}_{{\mathrm{\Omega }}^{+}})-\dim \ker (D_E{\mathrm{\mid }}_{{\mathrm{\Omega }}^{-}})$.

For fixed bundle, two unitary connections, even for different smooth Hermitian metrics, have exterior covariant derivatives differing by a smooth zeroth-order wedge map. Their adjoints both have the formula of step1.1, so the two total $D_E$ operators also differ by a smooth zeroth-order map. Viewed from $H^1$ to $L^2$ this difference is compact by [F1]'s compact inclusion; all smooth positive fiber metrics give equivalent norms on these spaces by finite compact chart bounds. The chirality uses only the fixed base metric. Thus [F2] gives equal indices without a curvature-zero premise or a uniform family regularity assertion. A smooth bundle isomorphism, with transported metric and connection, conjugates the operators by bounded invertible maps on $H^1$ and $L^2$ and preserves the dimensions of kernel and cokernel. Connection/metric independence removes the transported choices. Direct sums give block-sum operators and the direct sums of their kernels and cokernels, so indices add. The explicit pair relations defining $K_{\mathrm{s}\mathrm{m}}^0(X)$ now give a well-defined additive map: a witnessing stable bundle isomorphism gives equality after adding the same integer index, which can be cancelled in $\mathbb{Z}$.

On $Y$, $d^2=0$ gives $D^2=d{d}^{\ast }+d^{\ast }d$ and $\ker D=\ker d\cap \ker d^{\ast }$ by the energy identity (the cross term is $⟨d^{2}u,u⟩=0$). By [F3], this is its harmonic cohomology. The projection in step3.2 commutes with that identification. Chirality exchanges degrees $k$ and $4-k$; for $k\mathrm{\ne }2$ its involution on these paired harmonic spaces has equally many positive and negative eigenvectors, by $v\mapsto v\pm \tau v$. Hence they contribute zero to the index in step4.1. On harmonic degree two, $\tau =\ast$ and ${\ast }^2=1$. The form ${\int }_Y\alpha \wedge \bar{\beta }$ is $⟨\alpha ,\ast \beta {⟩}_{L^2}$, so it is positive on the $+\ast$ harmonic space and negative on the $-\ast$ space, with no cross terms. By [F4]'s multiplicative integration and its naturality under the finite deck maps, this is exactly the middle cup/intersection form on the corresponding topological cohomology/homology block. Complexification is legitimate here: real and imaginary parts split both the form and singular cochain complexes into two real copies, so the real comparison complexifies directly. To check the normalization, sum the oriented top simplices of the finite compatible smooth triangulation in [F4]. This is an actual smooth fundamental cycle, and integration on its finitely many simplices sums to integration over $Y$. Thus the evaluation of the cup class on that cycle is exactly the displayed integral, without an unknown scalar or sign. Cap duality gives the same form under the middle cohomology-to-homology identification. If opposite left/right conventions conjugate the character, complex conjugation on these real-defined cellular and de Rham complexes conjugates the pairing matrix and preserves both inertia counts. This proves the claimed index-to-character-signature identification, with no general flat-coefficient de Rham comparison assumed.

For the trivial line, step5.2 with the trivial cover gives $I_X(1)=\mathrm{sign}(X)$, with the positive sign because ${\tau }_2=+\ast$. For each finite-character line $L$, its finite tensor-order trivialization from step3.2 and [F5] show that $[L]-1$ is additive torsion in $K_{\mathrm{s}\mathrm{m}}^0(X)$. Applying the integer-valued additive map of step5.1 gives $I_X([L])=I_X(1)$. Together with step5.2 this proves equality of every actual finite-character signature with $\mathrm{sign}(X)$, including conjugate characters and the trivial character. The zero bundle, empty kernel and disconnected finite covers are included by the same finite direct sums. Every analytic application was to the compact smooth base or a finite compact cover and to the actual shifted Laplace-type domains; no Atiyah–Singer, Chern character, eta invariant, general $L^2$ index or boundary Fredholm theorem has been imported.