Appendix A. Simplicial integration and the ordinary geometric Chern character

Written by GPT-6.1 Sol (OpenAI), October 2026, at Ultra. Not yet reviewed. Public domain (CC0).

This appendix identifies the differential forms in the geometric examples with ordinary cohomology classes. The comparison uses integration over the actual oriented simplices, including for relative classes and compact supports. It also specifies the odd suspension convention: the positive unitary convention is the negative of Blackadar's displayed suspension map.

A1. The comparison theorem and its conventions

Let be a finite simplicial complex with a total order on its vertices, and let be a subcomplex. An increasing vertex list gives a simplex its preferred orientation. All coefficient groups for the differential-form comparison are complex; an integral or rational topological characteristic class is mapped to complex cohomology by change of coefficients.

The topological first Chern class uses the complex orientation. In particular the tautological line on , with its usual complex orientation, has Chern number . Higher Chern classes use the same normalization. The ordinary even character is where is the Newton polynomial giving the -th power sum in elementary symmetric functions. On a split bundle this is the sum of the exponentials of the first Chern classes of its line summands. Ordinary Chern classes, their naturality and Whitney sum formula are the topological foundations used here; their construction and proof are in Allen Hatcher, Vector Bundles and K-Theory, Theorem 3.2 and the ensuing proof, printed pp. 78–83. Definition (A.1) and the split calculation appear on printed pp. 109–110. The sign left as a choice in that construction is calibrated explicitly in A6 below.

For a compatible simplexwise smooth idempotent , put and use rank in degree zero. This is the curvature convention . For an identity-relative unitary , put The geometric components in (A.2) and (A.3) are times the corresponding cyclic differential-form components in the connections lesson. The raw current pairings retain the normalization of the n-trace lesson's (7.2)–(7.3).

Theorem A.1. Let on , where is a constant matrix idempotent, and let carry that specified relative trivialization. Actual oriented simplex integration gives In degree zero both sides are the relative rank difference. If is compatible and smooth on simplices, with , then Here the positive ordinary odd character has the explicit definition where is Blackadar's displayed stable suspension isomorphism and is the ordinary cohomology suspension with the positively oriented circle factor first. A9 constructs both maps with their actual frames; A10 proves the formula in every odd degree. Definition (A.6) specifies the identification of stable-unitary with suspended topological K-theory used in this appendix. An abstract equivalence with no displayed endpoint convention is not used to select its sign.

The same equalities hold in compact-support cohomology of a locally finite simplicial complex, with functions and representatives smooth on a sufficiently fine subdivision. Each individual compact-support K-class has only finitely many nonzero degree components; a uniform dimension bound on the entire complex is unnecessary. No integral torsion is recovered by a curvature form. The integral cocycle in A6 represents the full first Chern class before change of coefficients.

We prove the theorem in A2–A11. The ordinary singular-cohomology foundations imported at specific steps are stated with accessible proof locators. The curvature/topological equality itself will follow from the line calculation, a finite splitting model and the relative cone argument.

A2. Compatible forms and relative simplicial integration

A form in consists of a smooth form on each closed affine simplex, with identical tangential restrictions on common faces. Smoothness means extension to a neighborhood in the affine span of the simplex. A degree exceeding a face's dimension has zero pullback to that face. Set This is a tangential relative condition; it imposes no vanishing normal derivatives. The simplicial relative complex consists of cochains vanishing on simplices of , with the usual alternating-face differential .

Lemma A.2. Restriction of compatible forms to a subcomplex is onto in every degree. The actual integration map is a chain map and induces an isomorphism

Proof. For boundary extension on an affine simplex, write in barycentric coordinates. On , retract to that face by A smooth cutoff , equal to one near zero and zero for , extends a face form by , and by zero off that region. The support avoids the pole and the extension has the required neighborhood smoothness. Extend the faces in a fixed order. On the next face subtract the current restriction from the desired form. The residual vanishes on intersections with earlier faces. On an earlier face , its retraction under lies in , so the new correction has zero tangential restriction there. It changes no previously completed face. This extends a compatible boundary form. Starting with the prescribed forms on , extend over new vertices, edges and higher simplices in turn. This proves surjectivity.

Stokes and face compatibility give , also in the relative complexes. On a single simplex choose a vertex , let , and define Writing , the interval coefficient of its differential is . Integration gives The last pullback is vertex evaluation in degree zero and zero in higher degrees. Thus the form cohomology of a simplex is in degree zero and zero otherwise. The augmented vertex-cone contraction of its simplicial chains gives the same cohomology calculation. Integration sends the constant one to one, proving the absolute simplex comparison.

Induct on dimension. Surjectivity provides the short exact sequences for forms on , and integration maps them to the cochain sequences. The simplex result and the boundary induction, applied to their long exact sequences, prove the cell-pair comparison by the five lemma. For an arbitrary finite pair put and . Both relative complexes for are finite direct sums of the cell-pair complexes, one per new -simplex: all common-face restrictions are already zero. Restriction in is onto by the boundary extension, preserving the zero restriction on . Comparing long exact sequences and applying the five lemma gives (A.8), by induction over the finitely many layers.

The normalized Whitney form for an ordered face is Its restriction to a face omitting a specified vertex is zero; otherwise it restricts to the same formula. For a cochain , sum its face coefficients times these forms to define . Differentiation gives . Using and , this is , so . On its own -face, . That simplex has volume ; on every other -face the restriction is zero. Therefore Together with Lemma A.2, this makes the inverse of (A.8).

For a barycentric subdivision , restriction of forms and the signed chain subdivision satisfy the exact equality This is additivity of integration over the oriented subdivision. For completeness, construct subdivision chains recursively by coning the subdivided boundary to the simplex barycenter. The cone boundary formula proves that this is a chain map. Send the barycenter of each face to its largest vertex to obtain a simplicial map . Both and the identity are carried by each original simplex. Fill their discrepancy on a simplex, after subtracting the already constructed face homotopy, by a vertex cone in that simplex. The discrepancy is a cycle by the lower-dimensional identities. This proves one chain homotopy. For the reverse composite use the same induction inside the subdivision of the largest original face; that subdivision is the cone from its barycenter over its subdivided boundary. These homotopies preserve subcomplexes and finite carriers. Thus (A.13) uses the ordinary subdivision cohomology identification for pairs.

We interpret simplicial cohomology as ordinary cohomology using the characteristic-simplex comparison. Its foundational proof is Hatcher, Algebraic Topology, Theorem 2.27, printed pp. 128–130. Over , algebraic duality preserves the homology comparison: functionals on cycles or boundaries extend by a vector-space basis. A5 supplies the carrier homotopies needed for its naturality under the maps used here. Formulae (A.7) and (A.13), rather than a renamed abstract de Rham map, remain the integration maps throughout.

A3. Integration carries wedge to the ordinary cup product

Under the characteristic-simplex comparison in A2, restriction of a singular Alexander–Whitney cup cochain to each preferred affine simplex gives the simplicial formula below: its front and back singular faces are exactly the affine simplicial faces. Thus the cohomological product compared here is the ordinary singular-cohomology cup product.

For cochains in the fixed vertex order use the Alexander–Whitney formula The shared vertex matters. A degree-zero left factor evaluates at the first vertex, and a degree-zero right factor at the last.

Lemma A.3. There is a linear cochain homotopy , lowering total degree by one, with It is relative if either input is relative, and compactly supported if either input is compactly supported on a locally finite complex. Consequently preserves ordinary cup products and the corresponding relative module products.

Proof. On , let , and set , , . The differential of (A.14) is . Indeed deletions before the overlap give the first terms, and deletions after it give the second. The two remaining overlap terms have coefficients and and cancel. Thus , and .

On an ordered simplex , use the radial contraction of (A.10) to its first vertex, and let be the vertex-evaluation projection, zero in positive degree. Define Then The cross terms cancel with signs and . The remaining terms are , giving (A.16). Neither this contraction nor the radial contraction is claimed to commute with every face restriction.

Construct functionals inductively. On vertices set them to zero; there . Suppose the proper-face functionals satisfy . For an -simplex, , set, on input degree , The chain-map identity and the face induction give For original deleted indices , the two orders have coefficients and ; their restrictions and face functionals agree, so they cancel. Now set . On degree , the projection in (A.16) vanishes, and (A.17) annihilates . Hence , proving the induction identity. Define on simplices of dimension . This gives (A.15), including total degree zero where both sides vanish.

Every local functional is bilinear in the restricted inputs. If either input is zero on a whole simplex, its value is zero there. This proves relativity on simplices of , and proves the stronger carrier assertion that a nonzero value requires both input restrictions to be nonzero. A compact support meets finitely many simplices in a locally finite complex, so the output has finite support. The recursion also commutes with extension by zero across finite relative stages. For closed inputs (A.15) reduces to an ordinary coboundary, proving the product assertion.

Figure A.1. On the exact ordered triangle , , while . The preferred edge values of the homotopy are , ; thus , as required by (A.15). The front edge is , the back edge , and their shared vertex is . Reproducible source: workflow-wedge-cup-triangle.py. The contraction and Whitney-form context is Ezra Getzler, Lie Theory for Nilpotent L-infinity Algebras, Section 3; the smooth relative and product identities used here have been proved in A2–A3.

A4. Curvature, connection independence and bundle operations

For a compatible connection, its local matrix form has curvature . The covariant differential on a matrix form of degree is . Direct expansion gives the Bianchi identity . Graded trace cyclicity annihilates the commutator, so These trace forms are invariant under changes of frame, hence are global compatible forms.

For two connections on the same bundle, interpolate linearly on . In local frames independent of the parameter the product connection has no parameter-direction component; its curvature is . Closedness and the interval decomposition used in (A.10) give The trace makes this a global primitive. Equivalently it is interval contraction of the closed product curvature form; therefore its face restrictions agree. If the connections agree on , then , and the primitive is relative. This proves absolute and relative connection independence with the required support whenever the homotopy is confined to a common finite stage.

For block sums, curvature and its powers are block diagonal, so the geometric class is additive. For a tensor connection its curvature is ; the even-degree summands commute. Taking the exponential and trace gives the wedge product of the two character forms. Equation (A.19) compares this connection with a chosen projection connection, and Lemma A.3 turns wedge into ordinary cup. Thus the integrated geometric construction is additive and multiplicative. On a fixed finite polyhedron the exponential is truncated by dimension.

A5. Continuous maps, smooth representatives and ordinary naturality

Lemma A.4. For every continuous complex bundle on a finite polyhedron, the integrated geometric class is well defined using a smooth projection on a subdivision. It is independent of that choice and is natural under every continuous map of finite polyhedra.

Proof: smoothing. Choose finitely many trivializations, a continuous Hermitian metric and subordinate continuous weights . A finite shrinking and distances to closed complements construct such weights with supports inside the trivializing opens. Orthonormalize each local frame using the inverse positive square root of its Gram matrix. Mapping a fiber vector to its local coordinates multiplied by gives an isometric embedding in a finite trivial bundle: the squared component norms add to the fiber norm, and the zero extensions are continuous. Its range has a continuous Hermitian projection .

Choose and subdivide until for each vertex of the simplex containing . The affine interpolation Its spectrum lies in . To verify this, invert by factoring through and using the Neumann series whenever ; the spectrum is real because is Hermitian. Define with positive contour orientation. This is the orthogonal spectral projection onto the cluster near one. The uniformly separated contour makes the resolvent smooth on a neighborhood of every closed simplex, and restrictions agree. Applying the same formula to gives a continuous projection homotopy from to . Endpoint range bundles are isomorphic: divide a projection path into finitely many uniformly close steps and use on each the polar unitary described below in A9. Thus represents . If is a constant reference on , vertex interpolation, the spectral cutoff and this homotopy preserve that reference exactly on .

Proof: choice independence. If smooth projections have isomorphic range bundles, polar-deform a continuous fiber isomorphism to a unitary and extend it by zero, obtaining a rectangular matrix with , . The continuous block projection connects with . Choose constant on endpoint collars. Triangulate the product by ordered prisms, subdivide, and approximate this projection by Hermitian vertex interpolation within . To retain its exact smooth endpoints, replace the approximation by , where the disjoint collar cutoffs are supported where already equals the endpoint projection. All endpoint terms are pulled back from . This is compatible and smooth on the product triangulation and remains within of . Formula (A.20) gives a smooth product projection with the specified endpoints. We do not differentiate the continuous matrix .

Here is the precise endpoint comparison for this triangulated product. For an ordered -simplex define its prism chain Diagonal faces in consecutive terms cancel. The remaining side faces give the prism of the boundary, while the horizontal faces give top minus bottom. Hence If subdivides the product triangulation, use and in this formula. An original endpoint simplex need not itself be a simplex of the subdivision. For a closed cochain on the subdivided product, satisfies . Apply this to the integrated closed curvature forms. Equation (A.13) identifies the endpoint cochains with integration on the original endpoints, proving their cohomology classes equal. In degree zero the endpoint ranks agree directly. This proves independence of the embedding, bundle isomorphism, smooth representative and subdivision. Stabilization by a trivial bundle changes only rank; additivity extends to virtual bundles.

Proof: naturality. For a simplicial map , direct affine change of variables gives A collapsed positive-dimensional simplex pulls a top-degree form back to zero. An uncollapsed simplex contributes the orientation sign of the target vertex permutation. Degree zero is evaluation.

The characteristic-simplex comparison with singular chains is natural on cohomology, including collapsed or reordered maps. Let be that comparison. Induct on a simplex to compare and . After the face homotopy is constructed, the discrepancy is a singular cycle carried by the convex target simplex. Cone it to a vertex of that simplex to define . Its boundary is the cycle. Vertices have zero discrepancy, so no augmentation issue remains. The induction proves . The same carrier induction compares old and subdivided characteristic chains. This supplies the ordinary interpretation of (A.24) and (A.13).

For continuous , take inverse images of the target open vertex stars. A Lebesgue number and sufficiently fine subdivision make every closed domain vertex star lie in one member. Assign the domain vertex to that target vertex. The resulting map is simplicial: for an interior point of a source simplex, its image under has positive barycentric coordinates at all assigned vertices, so they span a target face. At a boundary point use only source vertices with nonzero coordinates. Thus and lie in a common simplex, and their straight homotopy is continuous and remains in the realization. This is the finite simplicial-approximation construction; compare Hatcher, Algebraic Topology, Theorem 2C.1 and its proof, printed pp. 177–179.

Homotopic continuous bundle pullbacks are isomorphic: embed the bundle on the compact product and use the close-projection endpoint argument. Ordinary cohomology pullback is homotopy invariant by the singular prism identity. Apply (A.24) only to the simplicial approximation, whose pulled-back projection and curvature are smooth. Choice independence therefore gives No derivative of the merely continuous map is taken.

Figure A.2. The spectral example uses the exact bound , giving clusters and with cutoff at . These are proved bounds, not sampled eigenvalues. For lower edge vertices and upper vertices , the interval-first edge prism is . Its boundary is : top minus bottom minus right plus left. Locators: (A.20)–(A.23). The drawing's N1 and N3 labels denote the smoothing and prism steps, respectively, of A5. Reproducible source: draw_character_naturality.py.

A6. A line bundle: descent, normalized integration and the topological sign

Lemma A.5. For a simplexwise smooth line bundle with curvature , In fact a sufficiently fine subdivision carries an integral obstruction cocycle and a compatible one-form such that

Proof: smooth embedding, frames and logarithms. A given compatible smooth line bundle and its connection can be represented by a smooth range bundle without changing the smooth category. Choose a fine subdivision whose closed vertex stars lie in bundle trivializations. Smooth compatible nonnegative cutoffs , supported in those opens and with no common zero, are obtained from the barycentric cutoff construction (A.31) below, grouping vertex cutoffs inside their assigned trivializing opens. Weighted local metrics give a compatible smooth Hermitian metric. In smooth unitary local frames, map a fiber vector to its coordinate tuple multiplied by . Zero extensions are smooth because the cutoffs are supported inside their charts; squared norms sum to the original fiber norm. This is an actual smooth isometric embedding, and its range has a compatible smooth Hermitian projection . The given connection transports through this smooth isomorphism; A4 compares it with the projection connection if necessary. This step preserves the given smooth connection and does not rely on a merely continuous bundle isomorphism from A5.

Subdivide until throughout each closed vertex star. This is possible because its diameter is at most twice the mesh and is uniformly continuous. For a unit vector , use the closed-star frame . The denominator is at least , and the frame has compatible neighborhood-smooth restrictions.