Ordinary differential-cohomology integration with explicit target degrees

Suppose $k>n$ and put $j=k-n\ge 1$. For a degree-$k$ character $h$ define ${\widehat{\pi }}_{!}h(z)=(-1{)}^{n}h(T_{\pi ,j-1}z)$. The signed chain identity makes $Tz$ a $(k-1)$-cycle, and the formula is a homomorphism because $T$ and $h$ are homomorphisms. For a smooth integral $j$-chain $c$ on $M$, that identity gives $T(\mathrm{\partial }c)=(-1{)}^n\mathrm{\partial }(Tc)$. Hence ${\widehat{\pi }}_{!}h(\mathrm{\partial }c)=(-1{)}^{n}h(T\mathrm{\partial }c)=h(\mathrm{\partial }Tc)={\int }_{Tc}R(h)={\int }_c{\pi }_{!}R(h)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z}),$ where the last equality is [F3]. For every $j$-cycle $c$, $Tc$ is a $k$-cycle, so this evaluation identity also proves integral periods of ${\pi }_{!}R(h)$. On local smooth $(j+1)$-simplices, Stokes and $\mathrm{\partial }T=(-1{)}^{n}T\mathrm{\partial }$ show that its exterior derivative integrates to zero, hence vanishes by shrinking simplices. It is therefore a permitted curvature. If $\tilde{h}$ is a real lift of $h$, then $(-1{)}^n\tilde{h}\circ T$ is a lift of the pushforward, and its characteristic cocycle is $\mathrm{Int}(R(h))\circ T-(\delta \tilde{h})\circ T$. The Gysin clause of [F3] identifies its class with ${\pi }_{!}I(h)$.

If $k=n\ge 1$, then ${\pi }_{!}I(h)\in H^0(M;\mathbb{Z})$ is locally constant integer-valued and represents the characteristic map by [F5]. Let $v=\mathrm{Int}(R(h))-\delta \tilde{h}$ be its integral characteristic cocycle. For a point generator $[m]$, the transfer identity gives $\mathrm{\partial }{T}_{\pi ,0}[m]=0$, so $v(T_{\pi ,0}[m])={\int }_{T_{\pi ,0}[m]}R(h)={\int }_{W_m}R(h)$ by [F3]. The left side is exactly the degree-zero Gysin cocycle. Hence the fiber integral equals the same locally constant integer function pointwise, proving both compatibilities.

If $k=n=0$, the derivative of $\pi$ is an isomorphism at every point, so $\pi$ is a local diffeomorphism and its image $S$ is open. To see that $S$ is closed, let $m_j\in S$ converge to $m$ and choose $w_j$ over $m_j$; the set $K=\{m\}\cup \{m_j:j\ge 1\}$ is compact, so properness makes ${\pi }^{-1}(K)$ compact, and a convergent subsequence gives a point over $m$. Each fiber is discrete and compact, hence finite. For $m\in S$, take disjoint local-diffeomorphism neighborhoods about its finitely many preimages. If no smaller neighborhood of $m$ has preimage contained in their union, a sequence of points outside that union over a sequence converging to $m$ has, by properness, a subsequence converging to a point of the fiber, contradicting that the neighborhoods cover that fiber. Shrinking to the common image neighborhood therefore gives an evenly covered set. Thus ${\pi }^{-1}(S)\to S$ is a finite covering, while fibers over $M\setminus S$ are empty. The sign ${ϵ}_x$ is locally constant on each sheet, so the signed sum is locally constant on $S$ and extends by zero to the locally constant function on $M$ in the Statement. The degree-zero transfer normalization in [F3] gives exactly this sum as the integral Gysin map, and integration of $f$ over the oriented zero-dimensional fiber has the same value. Hence $I$ and $R$ compatibility holds, including for empty fibers.

If $k<n$, the convention makes the target differential group zero; the ordinary Gysin target has negative degree and is zero, and fiber integration of a $k$-form over an $n$-dimensional fiber has negative form degree and is zero. Thus both compatibility identities hold.

For a transverse pullback square and $k>n$, naturality follows by substituting the naturality identity for $T$ into the defining evaluation. For $k=n\ge 1$, it is naturality of the integral Gysin map; for $k=n=0$, a pullback covering square bijects the sheets over each lifted point and preserves the pulled-back orientation signs, so the signed sums commute with pullback. If $k<n$, both maps have zero target by step 1.4.

Let $\pi :W\to M$ and $\kappa :M\to B$ have fiber dimensions $n$ and $m$. If $k>n+m$, both stages use step 1.1; composition of the transfers and $(-1{)}^{n+m}=(-1{)}^n(-1{)}^m$ gives the composite formula. If $k=n+m$ and $n,m>0$, the first push has positive degree $m$ and the second is the top-degree Gysin branch of step 1.2, so compatibility is composition of the transfer-induced and integral Gysin maps in [F3]. If $n=0<m$, the first stage is the degree-positive transfer along a finite covering and the second is top-degree Gysin; [F3] identifies their composite with the composite-submersion transfer. If $m=0<n$, the first stage is top-degree Gysin and the second is the signed finite sum of step 1.3; the Gysin and finite-cover composition clauses of [F3] give the composite map. If $n=m=0$, two signed finite sums compose by rearranging the finite sum over the composite fiber and multiplying the two orientation signs, which is the composite orientation sign. Finally, if $k<n+m$, then either the first push already has zero target, or it yields degree $k-n<m$ and the second push is zero; this agrees with the zero composite target in step 1.4. These cases exhaust all dimensions and input degrees.