Fixed-inclusion localization for equivariant compact symbol classes

Equivariant bundle maps over closed invariant subsets of a compact $T$-space extend to neighborhoods: extend their finitely many local matrix entries with a partition of unity, and average the resulting section of the Hom bundle over $T$. Averaging preserves a map already equivariant on the closed subset. Invertibility is open, so an extended isomorphism is an isomorphism on an invariant neighborhood. Averaging a metric gives invariant orthogonal complements. Applying this extension to successive short subintervals of a homotopy gives equivariant endpoint bundle isomorphisms: identify the bundle with its slice pullback near each slice, subdivide the compact interval, and compose. These arguments also apply relative to a closed invariant subset where the identification is prescribed.

Closed subgroups $H\subset T$ have the following character properties: restriction $\widehat{T}\to \widehat{H}$ is onto, and if $g\mathrm{\notin }H$ there is $\chi \in \widehat{T}$ trivial on $H$ with $\chi (g)\mathrm{\ne }1$. For completeness, lift $H$ to a closed additive subgroup of ${\mathbb{R}}^r$ containing ${\mathbb{Z}}^r$. Its vector part and discrete lattice part give, by integer changes of basis and Smith reduction of the lattice relations, coordinates in which $H$ is the product of circle factors, finite cyclic subgroups of circle factors, and identity factors. On a circle factor a character has an arbitrary integer exponent; on a cyclic factor every character is the restriction of an integer exponent modulo its order. This proves surjectivity. In these coordinates failure to belong to $H$ is detected either by a nontrivial coordinate on an identity factor or by the appropriate power on a cyclic factor, proving separation. In particular $R(T)\to R(H)$ is onto, because commuting unitary matrices decompose finite-dimensional representations of compact abelian $H$ into characters.

Every equivariant bundle on a compact $T$-space is an equivariant summand of $Y\times V$ for a finite-dimensional representation $V$. To see this without assuming a nonequivariant complement is equivariant, choose finitely many sections spanning every fiber. With an invariant metric, the action on their Banach space of continuous sections is isometric and continuous. Convolve each section with the product Fejér kernels on $T$. The convolutions converge uniformly: uniform continuity makes translations near the identity uniformly close, and the nonnegative kernels have integral one and mass outside each identity neighborhood tending to zero. Each convolution has only finitely many Fourier characters, so its translates span a finite-dimensional representation. Sufficiently close approximations still span every fiber, since the smallest eigenvalue of the original positive evaluation operator has a positive minimum on compact $Y$. Evaluation from the direct sum of these finite-dimensional section representations is an equivariant surjection; its kernel and invariant orthogonal complement give the assertion.

Apply the invariant-metric exponential chart at a point fixed by $g$. In a sufficiently small invariant ball it intertwines the isometry $g$ with its orthogonal differential, so the fixed points are exactly the exponential of the fixed vector subspace. This proves that $F$ is a closed embedded submanifold, $T_{x}F=\ker (I-d{g}_x)$, and its orthogonal normal has no $g$-fixed vector. It is compact and has finitely many components. The cotangent action and the invariant orthogonal splitting imply $X^g=T^{\ast }F$. The normal quotient of $T(T^{\ast }M){\mathrm{\mid }}_{T^{\ast }F}$ is canonically an extension of $p^{\ast }\nu$ by $p^{\ast }{\nu }^{\ast }$. An invariant connection splits this extension; such a connection is obtained by averaging any metric connection. This gives $p^{\ast }(\nu \oplus {\nu }^{\ast })$ and the displayed complex structure. Changing the splitting gives a homotopy of these complex normal identifications. The inclusion is proper because it is a closed embedding.

These facts identify triples with reduced equivariant bundle $K$-theory on the one-point compactification. Indeed choose an invariant compact neighborhood of a triple's support; use step 2.1 there to add complements, and use its symbol isomorphism on the surrounding annulus to glue to a constant representation bundle outside. The two resulting bundles agree at infinity. Conversely, a difference of bundles with a chosen isomorphism at infinity extends that isomorphism over a neighborhood by step 1.1 and restricts to a compact-support triple. Refining the neighborhood or complement adds elementary triples, and a changed extension is joined after stabilization by the same annular clutching construction over $[0,1]$. The constructions are inverse because gluing a bundle with its original transition map recovers that bundle. Thus they also identify suspension groups and relative mapping-cone groups, respecting tensor products and extension from open subsets.

Multiplication by $e$ is locally invertible after localization. Fix an orbit in $F$ with stabilizer $H$, so $g\in H$. On its slice $T{\times }_{H}D(V)$ the bundle ${\nu }_{\mathbb{C}}$ is induced from its fiber representation at the slice center: choose an invariant connection and use radial parallel transport on the slice, which is $H$-equivariant by uniqueness of the parallel-transport equation. This also proves the assertion after shrinking the slice. Thus $e$ there is induced by ${\lambda }_{-1}({\nu }_{\mathbb{C},x}^{\ast })\in R(H)$. By step 1.2 lift this virtual representation to $s_x\in R(T)$. Its value at $g$ is ${\det }_{\mathbb{C}}(I-g{\mathrm{\mid }}_{{\nu }_{\mathbb{C},x}^{\ast }})\mathrm{\ne }0$ by step 2.2; hence $s_x\in S_g$. On the slice, multiplication by $e$ equals multiplication by $s_x$, and has inverse $s_x^{-1}$. The same is true over any invariant subset of the slice and on its cotangent preimage under $p$.

For a based compact $T$-space $Y$ and closed invariant based subset $A$, reduced bundle groups are half-exact at ${\tilde{K}}_T^0(Y)$ in the sequence ${\tilde{K}}_T^0(Y{\cup }_{A}CA)\to {\tilde{K}}_T^0(Y)\to {\tilde{K}}_T^0(A)$. The composite is zero by the cone contraction. If a bundle difference restricts to zero on $A$, step 2.1 first adds a global complement making its subtracted bundle a constant representation bundle, and then stabilizes the restricted equality to an actual isomorphism with a constant representation bundle on $A$. Glue the positive bundle to that constant bundle on $CA$ by this isomorphism, and subtract the corresponding constant bundle on the whole mapping cone. Their difference extends the original class to $Y{\cup }_{A}CA$. This proves the reverse kernel inclusion. Iteratively attach cones; the identifications of two consecutive cone attachments with $SA$ and $SY$ are obtained by collapsing the intervening cone, with the interval coordinates carrying trivial action. The same half-exactness applied after every attachment and every suspension gives the natural long exact sequence extending to all nonpositive degrees. This describes its connecting maps by clutching, rather than postulating them.

Define $j_{!}$ as follows. Pull back a representative on $T^{\ast }F$ to its complex normal $W$, tensor its complex with the Koszul complex of the tautological normal vector, and pass to the even-to-odd map $d+d^{\ast }$. In the graded tensor product its square is the sum of the two positive squares, because the two odd summands anticommute; hence its support is contained in the zero section over the original compact support. Compress the normal fibers into a disk tube, and extend to $X$. Extension is the open-extension map established in step 3.1, so this defines a compact-support class even though the Thom symbol alone has noncompact support when $T^{\ast }F$ is noncompact. Stable sums, homotopies and elementary classes are respected by tensor product. An isotopy of the tubular map and fiber compression gives a homotopy of compact-support symbols: over a compact base support the normal radii can be bounded uniformly. Restricting the Koszul complex on ${\mathrm{\Lambda }}^{\bullet }W$ to the zero section sets its differential to zero and leaves ${\lambda }_{-1}(W)$. The real invariant metric extends to a complex-bilinear equivariant isomorphism ${\nu }_{\mathbb{C}}\cong {\nu }_{\mathbb{C}}^{\ast }$, so this is precisely $p^{\ast }{\lambda }_{-1}({\nu }_{\mathbb{C}}^{\ast })$. Consequently $j^{\ast }{j}_{!}(b)=p^{\ast }{\lambda }_{-1}({\nu }_{\mathbb{C}}^{\ast })b$. This proves self-intersection directly, without invoking an equivariant Thom isomorphism. Suspension gives the same construction in degree one.

The trivial-complex-line Bott isomorphism needed to continue this sequence holds equivariantly. Here is the equivariance check for the explicit ordinary proof [F3]. On $Y\times S^2$ let $T$ act trivially on $S^2$. The hemisphere contractions can be lifted equivariantly by step 1.1, so the clutching automorphism $f(y,z)$ commutes with the fiber action. Laurent coefficients obtained by integration in $z$ therefore intertwine that action. Uniform approximation and the straight-line invertibility homotopy preserve equivariance. Multiplication by powers of $z$, the polynomial linearization block matrices, the Möbius change of the circle variable, and constant frame changes all involve only these equivariant coefficients, scalar functions and identity blocks; hence they are equivariant. The inside and outside generalized eigenspaces are equivariant subbundles, since their polynomial kernels are preserved by every intertwiner. Thus the explicit inverse in [F3] is a map into $K_T^0(Y)\oplus K_T^0(Y)\beta$. Its stabilization, Laurent-shift, direct-sum and homotopy identities hold as equivariant bundle identities by precisely the same matrices; endpoint identifications are equivariant by step 1.1. On $E⊠1$ and $E⊠\beta$ it returns $(E,0)$ and $(0,E)$, respectively. This proves both inverse identities, rather than inferring an equivariant isomorphism merely from an ordinary one. Passing to the kernel of restriction to $Y\vee S^2$ gives product with $\beta$ as an isomorphism for reduced groups; applying the same construction to mapping cones gives the relative version.

Consequently the sequence from step 4.1 continues periodically in both directions. For a closed invariant $A\subset Y$, its relative group equals the compact-support group on $Y\setminus A$: collapse $CA$ in the mapping cone, which does not alter its reduced bundle group. Explicitly a bundle on the contractible cone is a constant representation bundle; extend its chosen trivialization to a neighborhood by step 1.1 and glue to that representation on the collapsed neighborhood. Pullback reverses this operation, including on homotopies. This proves excision. Applied to one-point compactifications and step 3.1, it gives the exact closed/open sequence in both degrees. For an invariant open cover $Y=U\cup V$, compare the closed/open sequences for $(V,V\setminus U)$ and $(Y,Y\setminus U)$, using open extension on $V\to Y$ and $U\cap V\to U$, and the identity on the common closed complement $V\setminus U=Y\setminus U$. A diagram chase yields the cyclic exact sequence $K_{T,c}^q(U\cap V)\to K_{T,c}^q(U)\oplus K_{T,c}^q(V)\to K_{T,c}^q(Y)\to K_{T,c}^{q+1}(U\cap V)$, where the first two maps are the difference of open extensions and their sum. Explicitly lift the image on the common closed complement through either row, subtract its lift in the other row, and use exactness to land in the kernel on the intersection; changing the lift adds exactly the preceding difference map. This proves finite compact-support Mayer–Vietoris. The cone/clutching connecting maps commute with tensoring by a bundle, as do open extension and proper closed restriction. No restriction along a nonproper open inclusion is used. In particular, if both groups vanish on the opens and their intersection, they vanish on the union; if multiplication by a bundle class is invertible on each open and their intersection, the exact sequences give invertibility on the union by lifting and correcting through the adjacent terms.

Put $H=\overline{⟨g⟩}$ in $T$. Since $g$ fixes every point of $F$, continuity makes $H$ act trivially on $F$, hence on $T^{\ast }F$. To obtain the orbit slices needed on $X=T^{\ast }M$ without applying the compact-ambient supplier outside its scope, construct a $T$-invariant metric on $X$: let $m$ be the chosen invariant metric on $M$ and let $\mathrm{\nabla }$ be its Levi-Civita connection. The cotangent lift preserves $\mathrm{\nabla }$ and the dual metric. Its connection splitting identifies a tangent vector at $(x,\xi )\in T^{\ast }M$ with $(u,\alpha )\in T_{x}M\oplus T_x^{\ast }M$; define the Sasaki metric by $q_{(x,\xi )}((u,\alpha ),(v,\beta ))=m_x(u,v)+m_x^{-1}(\alpha ,\beta )$. This is smooth and $T$-invariant. For each $y\in X$, its orbit $O_y=T\cdot y$ is compact as the continuous image of compact $T$, and its stabilizer $H_y$ is closed. The orbit-map theorem [F4], applied to the smooth cotangent action on $X$, gives an injective immersion $T\mathrm{/}{H}_y\to X$; since $T\mathrm{/}{H}_y$ is compact and $X$ is Hausdorff, this map is an embedding, so $O_y$ is a compact embedded submanifold. The normal exponential map of $q$ is defined and is a local diffeomorphism near the zero section of $\nu (O_y)$. Compactness of that zero section gives a uniform disk on which it is defined and locally invertible. It is injective on a sufficiently smaller uniform disk: otherwise there would be distinct normal vectors over points $x_n,y_n\in O_y$, with lengths tending to zero and equal exponential images. Compactness gives a subsequence with $x_n\to x$ and $y_n\to y$; continuity and equality of the images force $x=y$, so both vectors converge in a local bundle chart to $(x,0)$, contradicting local injectivity there. The exponential is therefore a diffeomorphism from a disk neighborhood onto an open tube. Invariance of $q$ makes it equivariant. The normal bundle is the associated bundle $T{\times }_{H_y}{\nu }_y(O_y)$ by $[t,v]\mapsto (t\cdot y,(dt{)}_{y}v)$, and the tube map is $[t,v]\mapsto t\cdot {\exp }_y(v)$. Thus the tube is equivariantly diffeomorphic to $T{\times }_{H_y}{D}_{ϵ}({\nu }_y(O_y))$. If $k\in T$ fixes a point represented by $[t,v]$, uniqueness of tube coordinates implies $t^{-1}kt\in H_y$; since $T$ is abelian, $k\in H_y$. Thus all stabilizers in this slice lie in $H_y$. If $U=T{\times }_{H_y}{D}_{ϵ}({\nu }_y(O_y))$ is a slice neighborhood and $g\mathrm{\notin }{H}_y$, choose $\chi$ as in step 1.2. The character line over $T\mathrm{/}{H}_y$ is equivariantly trivial: the function $t{H}_y\mapsto \chi (t)$ identifies the ordinary trivial line with its $\chi$ twist. Pullback gives the same identification on $U$ and every invariant subspace of $U$. Thus $1-\chi$ annihilates both compact-support groups of every such subspace, and belongs to $S_g$. All those groups therefore vanish after localization. A compact-support representative supported in $X\setminus X^g$ has support covered by finitely many such slices. Intersections of the finitely many slices are invariant subspaces of a slice. Step 6.1 and induction over this finite cover prove that the representative vanishes after localization. Each class has compact support, so no uniform finite cover of the whole noncompact manifold is required.

The open complement $X\setminus X^g$ has no $g$-fixed points, so both its compact-support groups vanish after localization by step 7.1. Localization preserves exactness: if $u\mathrm{/}s$ maps to zero, some $t\in S_g$ kills the image of $u$, and exactness before localization then lifts $tu$, giving a lift of $u\mathrm{/}s$ with denominator $ts$. Apply this observation to the exact closed/open sequence of step 6.1 for $(X,X^g)$. The terms on both sides of restriction vanish, proving that $j^{\ast }$ is an isomorphism in both degrees.

Compactness of $F$ gives a finite such slice cover. The preimages under $p$ cover $T^{\ast }F$, and every intersection lies in a preimage where step 3.2 supplies an inverse. Tensoring with $p^{\ast }e$ gives a morphism of the compact-support Mayer–Vietoris sequences. The last assertion of step 6.1, by finite induction, makes it an isomorphism on $S_g^{-1}{K}_{T,c}^q(T^{\ast }F)$. For $a$ set $b=(p^{\ast }e{)}^{-1}{j}^{\ast }a$. Step 4.2 gives $j^{\ast }{j}_{!}b=j^{\ast }a$, and injectivity from step 8.1 gives $j_{!}b=a$. No claim that the integral Thom map is invertible was needed to derive this localized identity.

The closure of the cyclic group generated by $g$ acts trivially on $F$. On each fiber it is a commuting family of unitary transformations. Character averaging gives its eigenspace projections as continuous bundle endomorphisms. Their ranks are locally constant; near each base point the finitely many projections already spanning the fiber exhaust its rank, so no new character occurs nearby. Compactness therefore gives finitely many eigenbundles globally. The same projections on bundles over $T^{\ast }F$ preserve symbols and compact support. Taking the weighted sum of these bundles is additive and multiplicative under tensor product, because eigenvalues multiply; on the scalar action of $s\in R(T)$ it multiplies by $s(g)$. This proves the asserted extension of ${\mathrm{tr}}_g$ to localized groups and its compatibility with the reconstruction identity. Expanding exterior powers of a direct sum proves the displayed formula for ${\mathrm{tr}}_g(e)$. Fiberwise its rank is the product of $1-\zeta$ over the eigenvalues of $g$ on ${\nu }_{\mathbb{C}}^{\ast }$, hence the displayed determinant. Orthogonality identifies the dual eigenvalue multiset with the original one, and complexification preserves the real characteristic polynomial, giving equality with the real determinant.

The proof applies to every triple and its suspension, without any restriction on its origin. The normal bundle has rank zero when $F=M$, in which case $e=1$ and $j$ is the identity; when $F$ is empty step 7.1 gives zero localized groups. For an isolated fixed point $K^0(F)=\mathbb{Z}$, so the character denominator is its scalar determinant. For a component of positive dimension the preceding proof retains all its exterior-power bundle classes. This completes all the assertions.