Real characteristic forms do not detect integral torsion

Every integral torsion class maps to zero under the real coefficient map. [F2, F12, algebra] Let $x\in H^k(M;\mathbb{Z})$ satisfy $nx=0$ for a nonzero integer $n$. The coefficient map is additive, so $n{\rho }_M(x)={\rho }_M(nx)=0$. By [F12], $H^k(M;\mathbb{R})$ is a real vector space, it has no nonzero element annihilated by $n$; hence ${\rho }_M(x)=0$. This also covers negative $n$ by replacing it with $\mathrm{\mid }n\mathrm{\mid }$.

The complexified tautological real line on ${\mathbb{R}\mathbb{P}}^2$ has first Chern class of exact order two. [F2, F3, F4, F5, F6, F7] Let $j:{\mathbb{R}\mathbb{P}}^2\hookrightarrow {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ be the skeletal inclusion, put $u=w_1(\lambda )$ and $a=c_1({\lambda }_{\mathbb{C}})$. By the odd-class theorem in [F4], $2a=0$. The fiberwise real-linear map $({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}}\to \lambda \oplus \lambda$, $v\otimes (s+it)\mapsto (sv,tv)$, is an isomorphism. Thus [F4] gives ${\rho }_2(a)=w_2(({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}})=w_2(\lambda \oplus \lambda )=u^2\mathrm{.}$ By [F4, F5], $u$ is the polynomial generator and $j^{\ast }(u^2)\mathrm{\ne }0$. Naturality of coefficient change [F2] gives ${\rho }_2(j^{\ast }a)=j^{\ast }{\rho }_2(a)=j^{\ast }(u^2)\mathrm{\ne }0.$ The canonical fiberwise map $(j^{\ast }\lambda ){\otimes }_{\mathbb{R}}\mathbb{C}\to j^{\ast }({\lambda }_{\mathbb{C}})$, $v\otimes z\mapsto v\otimes z$, identifies the complexification $L=(j^{\ast }\lambda {)}_{\mathbb{C}}$ with the pullback of ${\lambda }_{\mathbb{C}}$. The restricted real line is the standard tautological line ${\gamma }^1$ by its fiber description. Naturality [F3] therefore identifies $j^{\ast }a$ with $c_1(L)$. Thus $2c_1(L)=0$ and $c_1(L)\mathrm{\ne }0$, so $c_1(L)$ has exact order two.

Constant sign transitions on normalized frames give a flat Hermitian connection. [F6, F7, F8, F9, F10, F11, algebra] On each standard affine chart $U_i=\{[x_0:x_1:x_2]:x_i\mathrm{\ne }0\}$, let $v_i$ be the unique representative with $i$th coordinate $1$. The transition from $v_i$ to $v_j$ is the smooth nonzero coordinate ratio; the cocycle identity holds because these are rescalings of one vector. Thus [F6] gives the smooth tautological line with fibers $\mathbb{R}x$. Set $s_i=v_i\mathrm{/}\mathrm{\parallel }{v}_i\mathrm{\parallel }$. These are smooth unit frames of ${\gamma }^1$. On each component of $U_i\cap U_j$, the sign of $x_j\mathrm{/}{x}_i$ is constant and the frames satisfy $s_j={\epsilon }_{ij}{s}_i$ for a locally constant ${\epsilon }_{ij}\in \{1,-1\}$. Their complexifications $e_i=s_i\otimes 1$ are unitary frames of $L$ with the same constant transitions. In the underlying real frames $(e_i,i{e}_i)$ the transition matrices are ${\epsilon }_{ij}{I}_2$. Set the real connection matrices ${\omega }_i=0$ in every frame. Since ${\epsilon }_{ij}^{-1}d{\epsilon }_{ij}=0$, the transformation law [F8] holds and the local gluing theorem gives a global real connection. In these frames the complex structure has constant matrix, so the connection commutes with it; its zero matrices also satisfy the Hermitian metric derivative identity. Thus [F9] makes it a complex-linear Hermitian connection. The structure equation [F10] gives $\mathrm{\Omega }=0$, so the determinant normalization [F11] yields $c_1(\mathrm{\nabla })=0$ pointwise.

The comparison theorem shows that the flat form misses this integral torsion. [F1, step 1.1, step 1.2, step 1.3] By step 1.1 the order-two class from step 1.2 maps to zero in real cohomology. Step 1.3 constructs a Hermitian connection with zero first Chern form. The comparison theorem [F1] identifies its de Rham class with that same zero real image, while $c_1(L)$ remains nonzero by step 1.2. This is the promised explicit failure of real characteristic forms to detect integral torsion.

Empty, zero, rank-one, choice, endpoint, and iff cases are covered. [A1, F1, step 1.1, step 1.2, step 1.3, step 2.1, cases] The empty manifold has zero singular and de Rham groups, so the coefficient-map assertion is vacuous there; the witness is the fixed nonempty ${\mathbb{R}\mathbb{P}}^2$. The zero integral class maps to zero, while the exhibited degree-two class has exact order two. The witness is rank one and has a nonzero first Chern class. The exact-order argument in step 1.2 uses the nonzero reduction to exclude the zero class. No parameterized path or endpoint claim occurs. AC is assumed in [A1] and inherited through the characteristic-class suppliers used above; the normalized frames and flat connection in step 1.3 are explicit and choice-free. The remark states no biconditional.