Borel Equivariant Integration Conventions for the ABBV Formula

Assume the Axiom of Choice. Let $T=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^r$ have the standard product presentation, smooth structure, and period-one basis $e_1,\dots ,e_r$ of the upstream torus definition. Use its selected product Milnor model $ET=(E_{S^1}{)}^r$ with the right diagonal action and $BT=ET\mathrm{/}T$. For a smooth left $T$-manifold $X$, define real Borel equivariant cohomology by $H_T^{\ast }(X;\mathbb{R})=H^{\ast }(ET{\times }_{T}X;\mathbb{R})$ using the quotient convention $(e\cdot t,x)\sim (e,t\cdot x)$. Its coefficient ring is $\mathcal{R}=H^{\ast }(BT;\mathbb{R})\cong {\mathrm{Sym}}_{\mathbb{R}}({\mathfrak{t}}^{\ast })=\mathbb{R}[u_1,\dots ,u_r]$, where $u_i$ is the class corresponding to the dual period-one basis vector $e_i^{\ast }$ and each $u_i$ has degree $2$.

For an integer vector $m=(m_1,\dots ,m_r)$, write the character as ${\chi }_m(t)=\exp (2\pi i{\sum }_i{m}_i{t}_i)$ and define its degree-two class by $u_{{\chi }_m}={\sum }_i{m}_i{u}_i$. With the stated quotient convention, this is the Euler class of the Borel line $ET{\times }_T{\mathbb{C}}_{{\chi }_m}$, where $T$ acts on ${\mathbb{C}}_{{\chi }_m}$ by ${\chi }_m$. Use the complex-orientation convention $c_1(L)=e(L_{\mathbb{R}})$ for a complex line $L$; thus $u_{\chi }$ is also the first Chern class of its Borel character line. Let $S\subset \mathcal{R}$ be the multiplicative set generated by these classes for all nontrivial characters, together with $1$; thus $u_{\chi }\mathrm{\ne }0$ exactly for nontrivial $\chi$. Write $S^{-1}{H}_T^{\ast }(X;\mathbb{R})$ and $S^{-1}\mathcal{R}$ for the fraction localizations: $a\mathrm{/}s=a^{\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }}$ if $t(s^{\mathrm{\prime }}a-s{a}^{\mathrm{\prime }})=0$ for some $t\in S$, and $(a\mathrm{/}s)+(b\mathrm{/}t)=(ta+sb)\mathrm{/}(st)$; the module structure is induced by the coefficient-ring localization. The coefficient-ring identification is used under AC: the upstream classifying-space cohomology proof invokes AC in its Künneth and coefficient arguments.

If $X$ is a compact oriented smooth $T$-manifold without boundary of dimension $n$ and the action preserves its orientation, the Borel projection ${\pi }_X:ET{\times }_{T}X\to BT$ has compact oriented $n$-dimensional fibers. Define its pushforward explicitly as follows. For $N\ge 0$ put $B_N=({\mathbb{C}\mathbb{P}}^N{)}^r$, $X_N=(S^{2N+1}{)}^r{\times }_{T}X$, and $p_N:X_N\to B_N$; the compatible finite Hopf bundles in the chosen Milnor model give $j_N:B_N\to BT$ and $a_N:X_N\to X_T$. Give $X_N$ the orientation of the complex-oriented base followed by that of $X$. For $a\in H^q(X_N;\mathbb{R})$ define $(p_N{)}_{!}a$ by the perfect-duality equation $⟨b⌣(p_N{)}_{!}a,[B_N]⟩=⟨p_N^{\ast }b⌣a,[X_N]⟩$ for every complementary-degree $b$. For $\alpha \in H_T^q(X;\mathbb{R})$ and $d=q-n\ge 0$, choose $N$ with $2N+1\ge d$ and set ${\int }_X\alpha =({\pi }_X{)}_{!}\alpha =(j_N^{\ast }{)}^{-1}(p_N{)}_{!}{a}_N^{\ast }\alpha$; set it to zero for $d<0$. Here $j_N^{\ast }$ is an isomorphism in degree $d$ because it is the polynomial quotient by $(u_1^{N+1},\dots ,u_r^{N+1})$, with each $u_i$ the restricted tautological Euler class, not a newly signed hyperplane class. The construction lemma proves existence and uniqueness of the finite pairing map, independence of $N$, base restriction compatibility, the projection formula, and compatibility with normalized right-Thom Gysin maps. In particular this map lowers degree by $n$, has values in $\mathcal{R}$, and its localized extension is ${\int }_X(\alpha \mathrm{/}s)=({\int }_X\alpha )\mathrm{/}s$. For an oriented $T$-equivariant vector bundle $E\to X$ whose Borel-associated bundle has a normalized Thom class, define its equivariant Euler class $e_T(E)$ as the pullback of that Thom class by the zero section. The Thom class is normalized to restrict on each oriented fiber pair $(D^c,S^{c-1})$ to the positive generator of $H^c(D^c,S^{c-1};\mathbb{R})$. For rank $c=0$, the Thom class and Euler class are $1$. For a complex representation line of character $\chi$, this convention gives Euler class $u_{\chi }$. For a fixed component $F$ of a $T$-action on $M$, the orientation convention is $o(TM{\mathrm{\mid }}_F)=o(TF)\wedge o({\nu }_F)$, where ${\nu }_F$ is its chosen equivariantly oriented normal bundle. This is the Borel convention; no identification with a Cartan, Chow, or quotient-stack cohomology model is assumed here.

Borel and Cartan equivariant cohomology for a compact torus

Real cohomology of the chosen torus classifying space

The Axiom of Choice

Atiyah and Bott, §2, equations (2.1)–(2.9)

Libine, §2.1, Definition 3, and §5.3

Character localization in Borel equivariant cohomology

Construction and Thom Compatibility of Borel Fiber Integration