Differential Spin-c orientation of a proper submersion

Let $\pi :X\to B$ be a smooth proper submersion of finite-dimensional Hausdorff second-countable manifolds without boundary, with compact fibers of constant dimension $n$. Its vertical tangent bundle is $T^{v}X=\ker (d\pi )$. A Riemannian structure on $\pi$ is a smooth positive-definite metric $g^{T^{v}X}$ on $T^{v}X$ and a smooth horizontal distribution $T^{H}X\subset TX$ complementary to $T^{v}X$. A smooth metric $g^{TB}$ exists on $B$ by a smooth partition of unity and is used only as an auxiliary choice. Make $T^{H}X$ orthogonal to $T^{v}X$ and give it the pulled-back metric, obtaining $g^{TX}=g^{T^{v}X}\oplus {\pi }^{\ast }{g}^{TB}$. If ${\mathrm{\nabla }}^{TX}$ is its Levi-Civita connection and $P^v:TX\to T^{v}X$ is orthogonal projection, define ${\mathrm{\nabla }}_V^{T^{v}X}W=P^v({\mathrm{\nabla }}_V^{TX}W)$ for vector fields $V$ on $X$ and vertical vector fields $W$. The Koszul formula shows that this connection is independent of $g^{TB}$ and restricts to the Levi-Civita connection on each fiber.

A Spin-c structure $s$ is a principal ${\mathrm{Spin}}^c(n)$ lift $P_c$ of the oriented orthonormal frame bundle $P_{SO}(T^{v}X)$. Write ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))\mathrm{/}⟨(-1,-1)⟩$, let $\rho :\mathrm{Spin}(n)\to SO(n)$ be the vector representation, and let $\chi ([q,z])=z^2$. Its determinant line is $L=P_c{\times }_{\chi }\mathbb{C}$. The homomorphism $[q,z]\mapsto (\rho (q),z^2)$ to $SO(n)\times U(1)$ has invertible derivative. The metric connection ${\mathrm{\nabla }}^{T^{v}X}$ and a unitary connection ${\mathrm{\nabla }}^L$ on $L$ therefore determine a unique principal connection on $P_c$ projecting to their product. The associated spinor representation is $[q,z]\mapsto z{\mathrm{\Delta }}_n(q)$, where ${\mathrm{\Delta }}_n$ is the complex spin representation with Clifford convention $c(v)c(w)+c(w)c(v)=-2g(v,w)$. It gives the induced spinor connection, whose central curvature is $F_{{\mathrm{\nabla }}^L}\mathrm{/}2$ because the determinant character is $z^2$.

In the Freed–Lott geometric model, a differential Spin-c orientation is the tuple $(s,g^{T^{v}X},T^{H}X,{\mathrm{\nabla }}^L)$ with these conventions. On each fiber, the vertical Dirac operator is $D_b={\sum }_{j}c(e^j){\mathrm{\nabla }}_{e_j}^S$ in a local oriented orthonormal frame, using the induced spinor connection; a twisting bundle adds its stated unitary connection by tensor product. The normalized Todd form is

$\mathrm{Todd}({\mathrm{\nabla }}^{T^{v}X},{\mathrm{\nabla }}^L)=\widehat{A}({\mathrm{\nabla }}^{T^{v}X})\wedge \exp \text{\hspace{0.17em}⁣}(-\frac{F_{{\mathrm{\nabla }}^L}}{4\pi i})\mathrm{.}$

Odd fiber dimension uses the source suspension convention. No boundary condition or equivalence relation between different tuples is part of this definition, and no separate odd-form correction is included. A differential pushforward also needs the chosen differential extension and an analytic transgression construction; those are not implied by the geometric tuple alone.

Differential extension of a generalized cohomology theory

Smooth manifolds and their smooth charts

Smooth partitions of unity exist on manifolds

The A-hat form and relative Chern character

Given Spin and Spin-c structures and their spinor bundles

Complex Spin-c Thom orientation from the equivariant spinor symbol

The horizontal distribution identifies $TX$ with $T^{v}X\oplus T^{H}X$ and therefore specifies the geometric family connection. The Todd normalization uses $c_1({\mathrm{\nabla }}^L)=-F_{{\mathrm{\nabla }}^L}\mathrm{/}(2\pi i)$ and the Spin-c factor $e^{c_1\mathrm{/}2}$. This is the convention $\mathrm{ch}(\mathrm{\nabla })=\mathrm{tr}{e}^{-F_{\mathrm{\nabla }}\mathrm{/}(2\pi i)}$; the central Spin-c weight is one, so its spinor curvature contributes $F_{{\mathrm{\nabla }}^L}\mathrm{/}2$ before the minus-sign exponential. A separate added odd form changes a differential-orientation convention and is not included in this fixed Freed–Lott convention.