Flat connections and real characteristic classes

Flat complex connections. Let ${\mathrm{\nabla }}_0$ be the supplied flat complex-linear connection on $E$. By full AC and [F2], choose a Hermitian metric and a compatible Hermitian connection ${\mathrm{\nabla }}_1$. Both preserve the same underlying ${\mathrm{GL}}_r(\mathbb{C})$ reduction. For each $j>0$, [F1] and ${\mathrm{\Omega }}_{{\mathrm{\nabla }}_0}=0$ give $c_j({\mathrm{\nabla }}_0)=0$ pointwise. The exact transgression in [F3] gives $c_j({\mathrm{\nabla }}_1)=d{T}_j$ for a possibly complex-valued $(2j-1)$-form $T_j$. The Hermitian form $c_j({\mathrm{\nabla }}_1)$ is real-valued by [F1], so $c_j({\mathrm{\nabla }}_1)=d\mathrm{Re}(T_j)$ is exact in the real form complex. The comparison theorem [F4], applied to ${\mathrm{\nabla }}_1$, now gives ${\rho }_M(c_j(E))=0$. This step handles a flat connection that is not itself compatible with any selected Hermitian metric; it compares its zero forms to the Hermitian ones rather than applying the comparison theorem outside its hypotheses.

Real Pontryagin classes and even Chern classes. Let $D$ be a flat real connection on $V$. Every positive-degree Pontryagin form is a homogeneous positive-degree polynomial in ${\mathrm{\Omega }}_D$, so [F1] gives $p_j(D)=0$ for $j>0$. The arbitrary-real-connection clause of [F4] yields ${\rho }_M(p_j(V))=0$. The induced connection on $V_{\mathbb{C}}$ is flat. For $k=2j>0$, [F5] gives ${\rho }_M(c_{2j}(V_{\mathbb{C}}))=(-1{)}^j{\rho }_M(p_j(V))=0$; indices above the rank vanish by the stated rank conventions.

Odd Chern classes of a real complexification. Fix a connected component $B$ of $M$. By [F6], choose a homotopy equivalence $h:X\to B$ from a path-connected CW complex. This CW complex is paracompact by [F7], and $h^{\ast }(V_{\mathbb{C}}{\mathrm{\mid }}_B)$ is numerable because $V$ is numerable and numerability pulls back. The bundle is canonically isomorphic to its conjugate. Thus [F8] gives, for odd $k>0$, $c_k(h^{\ast }{V}_{\mathbb{C}})=c_k(\overline{h^{\ast }{V}_{\mathbb{C}}})=-c_k(h^{\ast }{V}_{\mathbb{C}}),$ so $2c_k(h^{\ast }{V}_{\mathbb{C}})=0$. The relation [F10] identifies this class with $h^{\ast }{c}_k(V_{\mathbb{C}}{\mathrm{\mid }}_B)$. Since $h^{\ast }$ is an isomorphism by [F9], $2c_k(V_{\mathbb{C}}{\mathrm{\mid }}_B)=0$. The torsion statement [F11] then gives ${\rho }_B(c_k(V_{\mathbb{C}}{\mathrm{\mid }}_B))=0$. This holds on every component; singular chains split as the direct sum over components, so cochains and cohomology split as products and the global real class is zero.

A closed genus-two hyperbolic surface. In ${\mathbb{H}}^2$ take a regular octagon whose interior angle is $\pi \mathrm{/}4$. Such an octagon exists: if $\mathrm{\ell }$ is its side length and $\theta$ its angle, the right triangle from the center to a vertex and a side midpoint gives $\sin (\theta \mathrm{/}2)=\frac{\cos (\pi \mathrm{/}8)}{\cosh (\mathrm{\ell }\mathrm{/}2)}\mathrm{.}$ Thus $\theta =\pi \mathrm{/}4$ when $\cosh (\mathrm{\ell }\mathrm{/}2)=\cot (\pi \mathrm{/}8)=1+\sqrt{2}$. Identify its sides by the oriented genus-two word $a_1{b}_1{a}_1^{-1}{b}_1^{-1}{a}_2{b}_2{a}_2^{-1}{b}_2^{-1}\mathrm{.}$ The quotient is compact, oriented, and has genus two; all eight vertices become one point and their angles sum to $8(\pi \mathrm{/}4)=2\pi$, so the hyperbolic metric extends smoothly across that point. Call this closed surface $\mathrm{\Sigma }$. By [F16], its universal cover is ${\mathbb{H}}^2$. Its orientation-preserving deck transformations are fractional-linear transformations, giving a homomorphism $\rho :\mathrm{\Pi }={\pi }_1(\mathrm{\Sigma })\to \mathrm{PSL}(2,\mathbb{R})$.

The deck representation's projective-line bundle. Let $\eta =({\mathbb{H}}^2\times {\mathbb{R}\mathbb{P}}^1)\mathrm{/}\mathrm{\Pi }⟶\mathrm{\Sigma },$ where $\mathrm{\Pi }$ acts diagonally through $\rho$ and its projective action. A nonzero tangent direction at $z\in {\mathbb{H}}^2$ determines the endpoint of the unique orthogonal geodesic ray to the ideal boundary ${\mathbb{R}\mathbb{P}}^1$; this map is deck-equivariant. Hence $\eta$ is the unit direction circle bundle of $T\mathrm{\Sigma }$. The tangent Euler number is the Euler characteristic (Milnor–Stasheff; the printed locator is recorded in the source notes), and the genus-two polygon has one 0-cell, four 1-cells, and one 2-cell. Therefore $⟨e(\eta ),[\mathrm{\Sigma }]⟩=\chi (\mathrm{\Sigma })=1-4+1=-2.$ The sign is fixed by the chosen orientation; only its evenness and nonvanishing are used below.

Lift to a flat oriented plane bundle. Write $D(\eta )$ and $S(\eta )$ for the associated disk and circle bundles. The mod-two Thom class $u\in H^2(D(\eta ),S(\eta );{\mathbb{F}}_2)$ maps to the reduction of $e(\eta )$, which is zero because its integer evaluation is $-2$ on the connected oriented surface. By exactness of the pair sequence in [F12], $u$ is the connecting image of a class $a\in H^1(S(\eta );{\mathbb{F}}_2)$. On each fiber, the connecting map $H^1(S^1;{\mathbb{F}}_2)\to H^2(D^2,S^1;{\mathbb{F}}_2)$ takes its nonzero class to the Thom generator; hence $a$ restricts nontrivially to the circle fiber. This degree-one class defines a double cover $\widehat{\eta }\to \eta$ whose restriction to every fiber is the connected degree-two circle cover.

The flat connection and its curvature. Form the associated real two-plane bundle $\widehat{V}=({\mathbb{H}}^2\times {\mathbb{R}}^2)\mathrm{/}\mathrm{\Pi }⟶\mathrm{\Sigma },\gamma \cdot (z,v)=(\gamma z,\widehat{\rho }(\gamma )v)\mathrm{.}$ Its transition matrices lie in $\mathrm{SL}(2,\mathbb{R})$, so the bundle is oriented. In an evenly covered chart, use the standard basis of ${\mathbb{R}}^2$ as a frame. On each overlap the two chosen lifts differ by one deck transformation, so the transition matrix is the constant matrix $\widehat{\rho }(\gamma )$. These constant cocycles define the smooth bundle by [F13]. Put ${\omega }_{\alpha }=0$ in each such frame. Since every transition matrix $A_{\alpha \beta }$ is constant, $A_{\alpha \beta }^{-1}{\omega }_{\alpha }{A}_{\alpha \beta }+A_{\alpha \beta }^{-1}d{A}_{\alpha \beta }=0={\omega }_{\beta }\mathrm{.}$ Thus [F14] glues the zero matrices to a global connection $D$, and [F15] gives ${\mathrm{\Omega }}_D=0$. The unit-circle bundle of $\widehat{V}$ is $\widehat{\eta }$, so its Thom-normalized Euler number is $-1$ by step 1.6. Changing the chosen orientation could reverse that sign, but not its nonvanishing.

Failure without metric compatibility. The real image ${\rho }_{\mathrm{\Sigma }}(e(\widehat{V}))$ evaluates to $-1$ on the real fundamental class, so it is nonzero. If the flat connection $D$ were compatible with any Euclidean metric, the Euler clause of [F4] would identify that real class with the de Rham class of its metric-compatible Pfaffian form. Since ${\mathrm{\Omega }}_D=0$, that form is identically zero by [F1], contradicting the evaluation $-1$. Therefore $D$ is flat but compatible with no Euclidean metric, and the unqualified Euler conclusion is false.

Boundary and rank cases. If $M$ is empty all cohomology groups in clauses 1–3 are zero. For clause 3 in rank $2m>0$, flatness gives $\mathrm{\Omega }=0$, and the degree-$m$ Pfaffian is zero; [F4] therefore gives ${\rho }_M(e(W))=0$. At rank zero the empty Pfaffian and the Euler class in the canonical unit orientation are both $1$, so the real Euler class is the degree-zero unit, not a positive-degree vanishing class. Rank-zero bundles have no positive-degree Chern or Pontryagin classes; for rank one, the determinant/transgression argument of step 1.1 still applies, while positive Pontryagin classes vanish by the rank cutoff. Indices above the complex rank or the real Pontryagin cutoff vanish by definition; degree zero is excluded from the vanishing assertions and is recorded separately in clause 3. The flatness calculations and comparison theorem apply on boundary charts under their stated scopes. The counterexample is a fixed closed surface and has rank two. Full AC is used in [A1] through [F2], [F4], [F6], [F8], and [F12], including the componentwise CW models; after the connections, representation, and frames are supplied, the local matrix and quotient calculations make no further choices. There is no endpoint parameter and no biconditional in this example.