Top de Rham integration equals fundamental-class evaluation

Assume first that $\alpha$ is real and $m\ge 1$. By [F2] choose a smooth singular cycle $z={\sum }_{j=1}^N{a}_j{\sigma }_j$ representing the image of $[M]$ in $H_m(M;\mathbb{R})$, and choose a continuous cocycle $\phi$ representing the class $J_M[\alpha ]=r_M^{-1}[I_M\alpha ]$ with $r_M\phi -I_M\alpha =\delta \psi$ for a smooth cochain $\psi$; evaluating on the smooth cycle $z$ gives $⟨J_M[\alpha ],[M]⟩=\phi (z)=I_M\alpha (z)+\psi (\mathrm{\partial }z)=I_M\alpha (z)={\sum }_{j=1}^N{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha$ by [F1] and the definition of $I_M$. It therefore remains to prove ${\sum }_j{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M\alpha$ for this smooth cycle.

Each ${\sigma }_j$ extends smoothly to an open neighbourhood of ${\mathrm{\Delta }}^m$. Applying [F4] in finitely many charts covering the compact images, the critical values of all ${\sigma }_j$ and the images of all faces of all ${\mathrm{\Delta }}^m$ form a set $N\subseteq M$ that is null in every smooth chart. For $y\mathrm{\notin }N$ every preimage $x\in {\sigma }_j^{-1}(y)$ is an interior regular point of ${\sigma }_j$, and there are only finitely many of them, because a regular point has a neighbourhood on which ${\sigma }_j$ is injective and ${\mathrm{\Delta }}^m$ is compact; put ${\mu }_j(y)={\sum }_{x\in {\sigma }_j^{-1}(y)}\mathrm{sgn}(d{\sigma }_j(x))$ and $\mu (y)={\sum }_j{a}_j{\mu }_j(y)$ for $y\mathrm{\notin }N$.

The orientation normalization can be computed on the standard simplex. For an interior point $x\in {\mathrm{\Delta }}^m$, radial projection from $x$ retracts ${\mathrm{\Delta }}^m\setminus \{x\}$ onto $\mathrm{\partial }{\mathrm{\Delta }}^m$. Since ${\mathrm{\Delta }}^m$ contracts, [F6] identifies $H_m({\mathrm{\Delta }}^m,{\mathrm{\Delta }}^m\setminus \{x\};\mathbb{Z})$ with ${\tilde{H}}_{m-1}(\mathrm{\partial }{\mathrm{\Delta }}^m;\mathbb{Z})$ and sends the identity simplex to its alternating boundary. In the simplicial boundary complex this alternating boundary generates top reduced homology: there are no $m$-simplices, and the cycle equations at each codimension-two face force the signed coefficients of adjacent facets to agree. For $m=1$ the generator is the endpoint difference. The simplicial comparison in [F6] therefore makes the identity simplex a local generator, whose sign fixes the positive Euclidean generator used in [F3]. Fix now $y\mathrm{\notin }N$ and $F_j={\sigma }_j^{-1}(y)$, a finite interior set by step 1.2. The identity simplex is a relative cycle for $({\mathrm{\Delta }}^m,{\mathrm{\Delta }}^m\setminus F_j)$, and ${\sigma }_j$ is a map from this pair to $(M,M\setminus \{y\})$. Excision and the disjoint-union formula [F3, F6], using disjoint small neighbourhoods of the points of $F_j$, identify its relative class with the tuple of positive local generators just computed. The local multiplier formula [F3] consequently sends its class to ${\mu }_j(y){\mu }_y$. If $F_j$ is empty its relative class is zero. Summing with $a_j$ shows that the point restriction of $[z]$ is $\mu (y){\mu }_y$. The restriction of $[M]$ is ${\mu }_y$, so their equality in the one-dimensional real local homology group forces $\mu (y)=1$.

Fix $j$ and let $R_j$ be the regular set in the interior of ${\mathrm{\Delta }}^m$, with its standard orientation. The pullback coefficient of $\alpha$ vanishes on the critical set because it contains the factor $\det D{\sigma }_j$, and the boundary of the simplex has measure zero, so ${\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_{R_j}{\sigma }_j^{\ast }\alpha$. Using a countable Euclidean basis and [F4], cover $R_j$ by connected relatively compact open inverse-branch neighbourhoods $V_i$ and choose a locally finite smooth partition $({\rho }_i)$ subordinate to them by [F5]. Each ${\rho }_i$ has compact support in $V_i$. Put $f_i={\rho }_i\circ ({\sigma }_j{\mathrm{\mid }}_{V_i}{)}^{-1}$ on ${\sigma }_j(V_i)$, extended by zero, and let ${\epsilon }_i$ be the constant differential sign on $V_i$. The change-of-variables formula gives ${\int }_{V_i}{\rho }_i{\sigma }_j^{\ast }\alpha ={\epsilon }_i{\int }_M{f}_i\alpha$. To bound the series, construct a smooth positive top form $\nu$ by patching positive chart volume forms with [F5], and choose $B$ with $\mathrm{\mid }\alpha \mathrm{/}\nu \mathrm{\mid }\le B$ on compact $M$. Put $\beta =B\nu$. Smooth signed change of variables gives ${\int }_M{f}_i\beta ={\epsilon }_i{\int }_{V_i}{\rho }_i{\sigma }_j^{\ast }\beta ={\int }_{V_i}{\rho }_i\mathrm{\mid }{\sigma }_j^{\ast }\beta \mathrm{\mid }$. Monotone convergence [F5] sums the latter nonnegative coefficients to ${\int }_{R_j}\mathrm{\mid }{\sigma }_j^{\ast }\beta \mathrm{\mid }<\mathrm{\infty }$, since the pullback coefficient is bounded on the compact simplex. Therefore ${\sum }_i{\int }_M\mathrm{\mid }{f}_i\alpha \mathrm{\mid }\le {\sum }_i{\int }_M{f}_i\beta <\mathrm{\infty }$. The Riemann–Lebesgue comparison in [F5] identifies the continuous compact-simplex coefficient integrals with these Lebesgue integrals. Thus positive and negative parts may be summed under the integral. For $y\mathrm{\notin }N$, the pointwise signed weight is ${\sum }_i{\epsilon }_i{f}_i(y)={\sum }_{x\in F_j}{\sum }_{i:x\in V_i}\mathrm{sgn}(d{\sigma }_j(x)){\rho }_i(x)={\mu }_j(y)$, since the partition sums to one at each preimage. Therefore ${\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M{\mu }_j\alpha$, with arbitrary values on the null set $N$.

Summing the identity of step 2.2 with the coefficients $a_j$ and using $\mu =1$ outside $N$ from step 2.1 gives, for real $\alpha$ and $m\ge 1$, ${\sum }_{j=1}^N{a}_j{\int }_{{\mathrm{\Delta }}^m}{\sigma }_j^{\ast }\alpha ={\int }_M\mu (y)\alpha (y)={\int }_M\alpha (y),$ which combined with step 1.1 proves $⟨J_M[\alpha ],[M]⟩={\int }_M\alpha$. For complex $\alpha$ apply the real case to $\mathrm{Re}\alpha$ and $\mathrm{Im}\alpha$ and use the complex-linear extension of $J_M$, of the Kronecker pairing and of the integral. For $m=0$ the manifold is a finite set of points with orientation signs $\epsilon (p)$, the fundamental class is $[M]={\sum }_p\epsilon (p)[p]$, the degree-zero comparison sends the class of a function $f$ to the evaluation classes $[f]={\sum }_{p}f(p)[p{]}^{\ast }$, and both sides of the identity are ${\sum }_p\epsilon (p)f(p)$ by the zero-dimensional convention in [F5]; the empty manifold gives $0=0$. The multiplicative consequence follows from the algebra isomorphism property of [F1] applied to the closed representatives. AC enters only through the ${\mathrm{A}\mathrm{C}}_{\omega }$ recorded in the Given, which the de Rham comparison, smooth partitions of unity, smooth-chain comparison, manifold integration and Riemann–Lebesgue comparison require; Sard, excision, change of variables, monotone convergence and the local multiplier computation are choice-free.