Equivariant cohomology of the nonfixed locus vanishes after character localization

If $F\mathrm{\ne }\mathrm{\varnothing }$, use [F1] to choose an equivariant tubular disk $D_{\rho }({\nu }_F)$ and let $K=M\setminus D_{\rho \mathrm{/}2}({\nu }_F)$, where $D_{\rho \mathrm{/}2}$ denotes the image of the smaller open disk. This is closed in compact $M$, hence compact by [F6]. Every $x\in K$ is nonfixed, so its stabilizer $H_x⊊T$ is a proper closed subgroup. If $F=\mathrm{\varnothing }$, set $K=M$; it is compact and every point is nonfixed. If $F=M$, then $K=\mathrm{\varnothing }$. For each $x\in K$, [F1] gives an open invariant orbit tube $V_x$ with all stabilizers contained in $H_x$; compactness gives a finite subcover $V_1,\dots ,V_m$ of $K$. For $F\mathrm{\ne }\mathrm{\varnothing }$, the sphere bundle $Q=S_{\rho \mathrm{/}2}({\nu }_F)\subset K$ is compact: cover the compact base by finitely many relatively compact bundle charts; over each compact chart closure the sphere fiber is compact, and the finitely many resulting compact sets cover $Q$.

Fix a tube $V_j$ with proper orbit stabilizer $H_j$. By [F2], choose a nontrivial character ${\chi }_j:T\to S^1$ that is trivial on $H_j$. Consider any invariant subspace $W\subseteq V_j$, with its subspace topology. Its inverse image in the slice has the form $T{\times }_{H_j}{W}_0$ for an $H_j$-invariant subspace $W_0\subseteq D(V_j^0)$. The Borel identification is the homeomorphism $[e,[t,w]]\mapsto [e\cdot t,w]$ from $ET{\times }_T(T{\times }_{H_j}{W}_0)$ to $ET{\times }_{H_j}{W}_0$, with inverse $[e,w]\mapsto [e,[1,w]]$; the quotient relations show both maps are well-defined. Write $B_H=ET\mathrm{/}{H}_j$. The map $B_H\to BT$ is the associated quotient bundle $ET{\times }_T(T\mathrm{/}{H}_j)\to BT$, and [F4] gives its compact-fiber CW-type and paracompactness properties. Since $ET\to BT$ is numerable, this associated quotient bundle is numerable as well: its local product charts over each Milnor chart retain the same subordinate partition. By [F4], $ET\to B_H$ is locally a principal $H_j$-bundle: over a Milnor $T$-chart it has form $U\times T\to U\times (T\mathrm{/}{H}_j)$, and local sections of the Lie quotient $T\to T\mathrm{/}{H}_j$ give the principal $H_j$-charts. The projection $q:B_H\to BT$ pulls $L_{{\chi }_j}$ back to the associated line $ET{\times }_{H_j}{\mathbb{C}}_{{\chi }_j{\mathrm{\mid }}_{H_j}}$; the explicit trivialization $[e,z]\mapsto ([e{]}_{H_j},z)$ is well-defined because ${\chi }_j{\mathrm{\mid }}_{H_j}=1$. Thus this is a numerable trivial line bundle, using its global chart and the constant partition $1$. The space $B_H$ is path-connected as a continuous image of contractible $ET$, so [F5] applies and its first Chern class is zero. Under the Borel identification, the projection $ET{\times }_{H_j}{W}_0\to BT$ factors through $B_H$, and the character class on $ET{\times }_{H_j}{W}_0$ is the pullback of this zero class. It therefore acts as zero on $H_T^{\ast }(W;k)$. As ${\chi }_j$ is nontrivial, this class becomes invertible in ${\mathcal{S}}_T(k)$, and the localized module is zero.

Let $W_j\subseteq V_j$ be any finite family of invariant open subsets, with each $V_j$ an orbit tube as above. We prove ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }({\bigcup }_{j=1}^m{W}_j;k)=0$ by induction on $m$. It holds for $m=1$ by step 1.2. For $m>1$, put $A={\bigcup }_{j<m}{W}_j$ and $B=W_m$. By induction the localized cohomology of $A$ is zero, and step 1.2 gives zero for $B$. The intersection $A\cap B\subseteq V_m$ is invariant and open, so step 1.2 also gives zero for it. The Borel quotients of invariant open sets are open, since the quotient map is open (the saturation of an open set is the union of its $T$-translates). Thus Mayer–Vietoris applies to the Borel open cover. To see that localization preserves its long exactness, split the sequence at each image/kernel into short exact module sequences and apply [F8] to each; exactness of the localized pieces then gives exactness at every Mayer–Vietoris term. The segment from the localized $H^q(A\cap B;k)$ through $H^{q+1}(A\cup B;k)$ to the localized $H^{q+1}(A;k)\oplus H^{q+1}(B;k)$ has zero end terms, forcing the middle term to vanish in every degree.

If $Q=\mathrm{\varnothing }$, then $ET{\times }_{T}Q=\mathrm{\varnothing }$ and all its singular cochain groups are zero by [F3], so the claim holds. Otherwise let $Q_j=Q\cap V_j$ with the relative topology. These are invariant open subsets of $Q$ and cover it, since the $V_j$ cover $K$ and $Q\subseteq K$. Under the slice diffeomorphism $V_j\cong T{\times }_{H_j}D(V_j^0)$, $Q_j$ has the form $T{\times }_{H_j}{Z}_j$, where $Z_j=\{w\in D(V_j^0):[1,w]\in Q_j\}$ is an $H_j$-invariant subspace. The generalized local calculation in step 1.2 gives ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }(Q_j;k)=0$. The Borel images of the $Q_j$ are open in $ET{\times }_{T}Q$: each is the image of the open invariant subspace $ET\times Q_j$ under the orbit quotient, which is open. Repeat the finite-cover Mayer–Vietoris induction of step 2.1 on this relative-open cover. At each stage the new intersection is an invariant subspace of the newest orbit tube, so step 1.2 makes its localized equivariant cohomology zero; localization preserves exactness by [F8]. Hence ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }(Q;k)=0$.

Apply step 2.1 with $W_j=V_j$ and then with $W_j=V_j\cap D_{\rho }({\nu }_F)$; this shows the open union $V={\bigcup }_j{V}_j$ and $V\cap D_{\rho }({\nu }_F)$ have zero localized equivariant cohomology. The punctured tube $D_{\rho }({\nu }_F)\setminus F$ equivariantly deformation retracts onto $Q$ by radial interpolation $v\mapsto ((1-t)\parallel v\parallel +t\rho \mathrm{/}2)v\mathrm{/}\parallel v\parallel$. The paths stay in the disk because both endpoint radii are strictly below $\rho$. By step 3.1, ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }(Q;k)=0$; [F7] and the equivariant deformation retraction therefore give ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }(D_{\rho }({\nu }_F)\setminus F;k)=0$. Now $M\setminus F=V\cup (D_{\rho }({\nu }_F)\setminus F)$, and their intersection is covered by the open sets $V_j\cap (D_{\rho }({\nu }_F)\setminus F)\subseteq V_j$; step 2.1 makes this intersection localized-acyclic as well. A final localized Mayer–Vietoris sequence gives ${\mathcal{S}}_T(k{)}^{-1}{H}_T^{\ast }(M\setminus F;k)=0$. If $F=\mathrm{\varnothing }$, the finite tube union covers all $M$ and step 2.1 applies directly; if $F=M$, the complement is empty.