Complex-oriented diagonal Thom classes realize exceptional exchange on supports

Purity and the supported Thom morphism. Put $Y=M\times M$. By [F1] and [F2], ${\mathrm{\Delta }}^{!}{k}_Y={\mathrm{\Delta }}^{!}{\omega }_Y[-4d]={\omega }_M[-4d]=k_M[-2d]$. The positive orientation identifies the identity of $k_M[-2d]$ with the Thom class in $H_{\mathrm{\Delta }(M)}^{2d}(Y;k)=\mathrm{Hom}(k_M,{\mathrm{\Delta }}^{!}{k}_Y[2d])$. Its adjoint is the purity counit $g_{\mathrm{\Delta }}:{\mathrm{\Delta }}_{\ast }{k}_M[-2d]\to k_Y$, equivalently ${\mathrm{\Delta }}_{\ast }{k}_M\to k_Y[2d]$. In a complex normal ball this is the positive generator of the relative disk pair; complex chart transitions preserve that generator. This fixes a global supported morphism through the orientation isomorphism and its counit, rather than inferring equality of arbitrary global maps from their restrictions to balls.

Pullback and cap product on arbitrary supports. Put $S=A\times B$. Proper base change along $F=(q,u)$ identifies $F^{\ast }{\mathrm{\Delta }}_{\ast }{k}_M[-2d]$ with ${\theta }_{\ast }{k}_T[-2d]$, so $F^{\ast }{g}_{\mathrm{\Delta }}$ gives $g_F:{\theta }_{\ast }{k}_T[-2d]\to k_S$. For a dualizing class $c:k_S\to {\omega }_S[-n]$, compose $c\circ g_F$ and use ${\theta }_{\ast }⊣{\theta }^{!}$ and ${\theta }^{!}{\omega }_S={\omega }_T$. The resulting map $k_T\to {\omega }_T[2d-n]$ is a Borel–Moore class of degree $n-2d$ on $T$. This defines the supported cap operation even if $T$ is singular or has excess dimension; no purity of $\theta$ is asserted. The external product uses ${\omega }_A⊠{\omega }_B={\omega }_S$: on the supplied finite local product cochain models, compact-support cochains tensor, and their finite-dimensional duals tensor as well, giving this canonical comparison and its restriction compatibility.

Comparison with the exchange mate. For a constructible $C$ on $M$, the mate is the adjoint of $b_{!}{a}^{\ast }{q}^{!}C\stackrel{{\mathrm{B}\mathrm{C}}^{-1}}{\to }{u}^{\ast }{q}_{!}{q}^{!}C\stackrel{u^{\ast }{ϵ}_q}{\to }{u}^{\ast }C$, where the arbitrary-map proper-support exchange of [F1] supplies $\mathrm{B}\mathrm{C}:u^{\ast }{q}_{!}\stackrel{\sim }{\to }{b}_{!}{a}^{\ast }$ and ${ϵ}_q$ is the exceptional counit. To identify its dualizing evaluation with step 2.1, factor $q$ as its graph ${\mathrm{\Gamma }}_q:A\hookrightarrow A\times M$ followed by the second projection. The graph is the pullback of $\mathrm{\Delta }$ along $(q,{\mathrm{id}}_M):A\times M\to M\times M$; its supported orientation map is that pullback of $g_{\mathrm{\Delta }}$. Base change of the graph along ${\mathrm{id}}_A\times u$ is exactly $\theta$, so its supported orientation map after this second pullback is $g_F$. For the product projection, tensor–Hom evaluation with the relative dualizing factor ${\omega }_A$ computes the counit; this is the finite product cochain evaluation of step 2.1. Composing the graph counit and that evaluation gives ${ϵ}_q$ by the composition-of-counits identity in [F1]. Cartesian pasting therefore makes its base-changed composite exactly $u^{\ast }{ϵ}_q\circ {\mathrm{B}\mathrm{C}}^{-1}$ above. Taking adjoints and using $q^{!}{k}_M={\omega }_A[-2d]$ proves that the exchange, paired with the second dualizing class by tensor evaluation, is $c\circ g_F$ under ${\theta }_{\ast }⊣{\theta }^{!}$. Every map retains support on $T$ throughout this factorization.

Ordered iterated diagonals. For two ordered smooth middle factors the normal model of the product diagonal is the ordered direct sum of the two complex normal bundles, and on the product of normal coordinate disks the relative cup product of the two positive generators evaluates to $+1$ on the ordered product disk, which is the positive generator of the direct sum; local uniqueness as in step 1.1 and base-change mate pasting as in steps 2.1 and 3.1 therefore give the composed refined operation on every support, in the given order and without an interchange of factors. All real degrees involved are even, so reordering the two complex summands introduces no sign: the only possible sign is $(-1{)}^{(2d_1)(2d_2)}=+1$. Under the additional hypotheses of [F3], the equivariant form holds verbatim, which supplies the same localization, exchange and normalized orientation clauses in $D_{G,c}^b$, with ${\omega }_X^G{\mathrm{\mid }}_X={\omega }_X$ rather than a quotient-stack shift.

Zero sections and Euler classes. Let $V\to X$ have complex rank $c$ and zero section $s$. In a bundle chart the normal pair is the product with $({\mathbb{C}}^c,{\mathbb{C}}^c\setminus \{0\})$, so the same relative cochain calculation gives $s^{!}{k}_V=k_X[-2c]$ and its positive supported counit; here $c$ is the bundle rank, independently of $\dim X$. These local orientations glue because the complex transition matrices preserve the positive fiber generator. On bases in the Thom scope of [F5], this supported class is the normalized Thom class by uniqueness. Pulling its relative-to-absolute image back by $s$ gives $e(V)$ by [F5]. Applying the supported cap construction of step 2.1 after $s_{\ast }$ restricts that class to the zero section; the projection formula thus gives $s^{!}{s}_{\ast }z=e(V)\cap z$. The same counit and base-change identities give its proper-pushforward and refined-pullback compatibility; the equivariant version requires the additional hypotheses of [F3]. AC is inherited through the stated suppliers, including the Thom supplier [F5].