Controlled equivariant symbol realization, homotopy and excision

The elementary change used below is the following bundle rotation. For two isomorphic summands, the matrix $R_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ is invertible with inverse $R_{-\theta }$. If their identification is an invertible map $h:A\to B$, its expression on $A\oplus B$ is $\left(\begin{array}{cc}\cos \theta & -\sin \theta h^{-1}\\ \sin \theta h& \cos \theta \end{array}\right)$. The formula is used only where $h$ is invertible, and can be joined to the identity using a scalar cutoff $\theta$ supported in that region. Applying the corresponding rotations to source and target commutes with a diagonal map $h\oplus I$. Invariant scalar cutoffs exist by averaging ordinary smooth cutoffs over the compact group.

On a compact invariant base, a continuous equivariant bundle embeds in a trivial finite-dimensional representation bundle. Choose finitely many continuous local generators, extended by cutoffs, whose evaluations span every fiber. Average a continuous nonnegative bump on $G$, of integral one and supported near its identity, against the translates of each section; the resulting section is uniformly close to it. Approximate this bump uniformly by a representative function using [F3]. The resulting averaged sections are still uniformly close and each belongs to a finite-dimensional translation-invariant space, since the left translates of its coefficient function span such a space. Surjectivity of a finite family of fiber evaluations is stable under a sufficiently small uniform perturbation, by a positive minimum singular value on the compact base. Their finite orbit spans therefore give an equivariant surjection $V\times Z\to A$. Haar-average the bundle metric; its adjoint embeds $A$ equivariantly in $V\times Z$, and the orthogonal image projection is continuous. This proof also works on a neighborhood of a compact invariant subset, and for a bundle trivial outside a compact subset, retaining the finite exterior generators.

Realize a smooth $c$ by [F1] and Haar-average the realization, giving an invariant elliptic operator $P$. Two realizations with the same symbol differ by a compact operator on the relevant Sobolev spaces. A path of invariant Fredholm operators has constant virtual index: split domain as $\ker P\oplus (\ker P{)}^{\perp }$ and target as $\mathrm{ran}P\oplus (\mathrm{ran}P{)}^{\perp }$. Nearby operators have an invertible infinite block by a Neumann series. Invertible equivariant triangular row and column operations reduce them to that block and a map $S$ between the fixed finite-dimensional kernel and cokernel spaces. Invariant orthogonal splitting of $S$ gives $[\ker S]-[\mathrm{coker}S]=[\ker P]-[\mathrm{coker}P]$. This proves local constancy in $R(G)$, hence constancy along a continuous path. In particular compact perturbations preserve it, since the whole compact-perturbation path is Fredholm by [F1]. The same argument applies to homogeneous symbols of any order and their simultaneous-closure realizations, using [F1] at one Sobolev level.

A finite-dimensional continuous representation of a compact Lie group is smooth in the needed matrix coordinates. Indeed choose a smooth bump $h$ near the identity so that $C=\int h(g)\rho (g)dg$ is invertible. Then $\rho (k)C=\int h(k^{-1}g)\rho (g)dg$ is smooth in $k$, by differentiating the smooth bump on the compact group; multiplication by $C^{-1}$ proves the assertion. For a continuous equivariant orthogonal projection $p:Z\to \mathrm{End}(V)$ over a compact smooth base, approximate its entries uniformly by smooth functions in local charts and patch with a partition; symmetrize and average over $G$. The smooth equivariant selfadjoint result $q$ can be made arbitrarily close to $p$. If $\mathrm{\parallel }q-p\mathrm{\parallel }<1\mathrm{/}4$, the contour projection $p^{\mathrm{\prime }}=(2\pi i{)}^{-1}{\int }_{\mathrm{\mid }z-1\mathrm{\mid }=1\mathrm{/}2}(z-q{)}^{-1}dz$ is smooth and equivariant, has the same rank, and is arbitrarily close to $p$ as the approximation improves. The restriction $p^{\mathrm{\prime }}{\mathrm{\mid }}_{\mathrm{im}p}$ is an isomorphism when $\mathrm{\parallel }{p}^{\mathrm{\prime }}-p\mathrm{\parallel }<1$, proving that the original bundle is equivariantly isomorphic to a smooth one. The resolvent identity gives the claimed closeness; the spectral projection and its rank follow by diagonalizing the finite selfadjoint matrix and integrating $(z-\lambda {)}^{-1}$ around the circle. Apply the same construction to a homotopy on a compact base times an interval, with approximation relative to smooth endpoints by using a parameter cutoff. Small perturbations of a bundle map preserve invertibility on any fixed compact set where its least singular value is positive.

Here is the precise compact-local replacement, including the stabilization needed to change the exterior bundles. Write $Z=T^{\ast }X$ or $T^{\ast }U$, and choose invariant relatively compact open sets $O_1⋐O_2\subset Z$ containing the support, so $b$ is invertible on $Z\setminus {\bar{O}}_1$. By step 1.2 choose a finite representation $V$ and a complement $H$ to $B$ over a neighborhood of ${\bar{O}}_2$, identifying $B\oplus H$ with $V$ there. Glue $A\oplus H$ on $O_2$ to the trivial $V$-bundle outside ${\bar{O}}_1$ by $b\oplus I_H$ on their overlap. Call this bundle $A^{\mathrm{\prime }}$, and let $c:A^{\mathrm{\prime }}\to V$ be $b\oplus I_H$ inside and the identity outside. The triple $(A^{\mathrm{\prime }},V,c)$ has trivial exterior bundles. It is stably isomorphic to the original triple: compare $(A\oplus V,B\oplus V,b\oplus I_V)$ with $(A^{\mathrm{\prime }}\oplus B,V\oplus B,c\oplus I_B)$. Inside $O_2$, write $v=v_B+h$ in $V=B\oplus H$. At $\theta =0$ identify the source by $(a,v)\mapsto ((a,-h),v_B)$ and the target by $(z,v)\mapsto ((z,-h),v_B)$. In the annulus rotate the two $B$ coordinates $(ba,v_B)$ in the source and $(z,v_B)$ in the target through a cutoff angle from zero to $\pi \mathrm{/}2$, keeping the $H$ coordinate $-h$. Step 1.1 makes these invertible identifications and makes the triple maps commute. At $\theta =\pi \mathrm{/}2$, the gluing identifies the source map with $(a,v)\mapsto (-v,ba)$ and the target map with $(z,v)\mapsto (-v,z)$, both defined throughout the exterior without any complement there. Thus the identifications glue globally and the two stabilized triples are isomorphic. No exterior bundle was presumed extendible. The same construction proves locality under changing the exterior space. The new bundles are trivial outside a compact set, so step 1.2 gives global finite representation embeddings; their exterior evaluation maps, and hence their projections, may be held constant. Step 2.1 then smooths them and the map relative to that exterior. All choices are uniform over a compact homotopy parameter. This proves reduction of continuous compact-support triples to smooth ones without a prior smooth structure on their original bundles.

Choose smooth connections by patching the chart connections with a partition and Haar-average them. Parallel transport along $t\mapsto (x,t\xi )$ identifies a smooth equivariant bundle over $T^{\ast }X$ with the pullback of its zero-section bundle. The linear transport ODE on each compact segment has a unique invertible solution, depends smoothly on its parameters, and preserves equivariance because the averaged connection and the radial paths do. One may obtain this solution directly by the convergent iterated-integral series in each finite chart segment; on a compact segment its $k$th term is bounded by $C^k\mathrm{/}k!$, and differentiated series on compact parameter sets converge after increasing $C$. This also proves independence of subdivisions and the parameter-homotopy assertion. For a noncompact $U$ with controlled exterior data, choose the connection to be the trivial one there before averaging, so the radial identification is the identity off the same compact base subset.

In these pullback identifications choose $R$ so large that $b$ is invertible for $\mathrm{\mid }\xi \mathrm{\mid }\ge R$ over the compact base region in question. Compose $b$ with a radial map which is the identity for $\mathrm{\mid }\xi \mathrm{\mid }\le R$, has radius $2R$ for $\mathrm{\mid }\xi \mathrm{\mid }\ge 2R$, and keeps radii at least $R$ for $\mathrm{\mid }\xi \mathrm{\mid }\ge R$. Interpolation between its radius and $\mathrm{\mid }\xi \mathrm{\mid }$ is an admissible homotopy, because the whole exterior remains invertible. The resulting exterior is the homogeneous degree-zero invertible map $c(x,\xi )=b(x,2R\xi \mathrm{/}\mathrm{\mid }\xi \mathrm{\mid })$. Replace the interior by any cutoff extension of $c$; interpolate linearly inside the disk, holding the exterior fixed. Noninvertibility is allowed there, so this is another admissible homotopy. Smooth approximation of a continuous $c$ on the compact sphere bundle, followed by averaging, preserves invertibility and gives a path through invertible sphere symbols; its least singular value has a positive minimum. Exterior identity data can be kept fixed throughout. These operations prove homogeneous presentation, also for homotopies.

Quantization of a smooth symbol homotopy is norm continuous by [F1]. Continuous homotopies were reduced to such paths in steps 2.1–5.1. For bundles varying with the parameter, embed into a common representation over the compact parameter space as in step 1.2. Their projection matrices vary uniformly continuously; subdivide the interval until successive projections are less than one apart, so projection restriction identifies successive fibers invertibly. For smooth projections the connection transport of step 4.1 gives the same endpoint identifications through a smooth family. Thus the index is independent of smoothing, complements, radial choices and homotopies. Block sums give sums of kernels and cokernels. An everywhere-invertible triple can be kept invertible during radial collapse to its zero-section map, giving a base bundle isomorphism; multiplication by that isomorphism has index zero. Isomorphic triples give conjugate kernels. These facts prove that elementary stabilizations and every comparison in the Statement preserve the index. Changing the order multiplies the symbol by a positive scalar power of the elliptic $\mathrm{\Lambda }$ in [F1], whose realization is an isomorphism and has zero index.

For the noncompact statement the compression in step 3.1, followed by radial transport, makes $E,F$ the common trivial representation bundle outside a compact base subset, and $c$ equals its identity there. Extend both bundles to the supplied compact ambient by those exterior trivializations, and use [F1] to choose any invariant realization $P_0$ of $c$. Choose invariant smooth cutoffs ${\eta }_1,{\eta }_2$ compactly supported in $U$, with ${\eta }_1=1$ on a neighborhood of the base subset and ${\eta }_2=1$ on a neighborhood of $\mathrm{supp}{\eta }_1$. Let $h:E\to F$ be the exterior identity, defined wherever $1-{\eta }_1$ is nonzero, and set $P={\eta }_2{P}_0{\eta }_1+h(1-{\eta }_1)$. The last term is extended by zero where its multiplier vanishes. Its principal symbol is ${\eta }_2{\eta }_{1}c+(1-{\eta }_1)h=c$: where the first cutoff is not one, $c=h$, and where it is nonzero the second cutoff is one. The nonlocal part of the kernel is supported in $L\times L$, $L=\mathrm{supp}{\eta }_2$; the remaining term is multiplication by the exterior identity. In particular outside $L$, in either the source or target variable, $P$ is precisely that identity. Thus no global isomorphism between the possibly different interior bundles $E,F$ is asserted. Averaging preserves this control since the cutoffs and exterior data are invariant.

Conjugation by an equivariant diffeomorphism transports every operator and its kernels. Restricting the group action changes neither operator nor vector spaces; any already invariant realization remains a valid realization after restriction. Tensoring by a finite-dimensional representation tensors kernels and adjoint kernels, since the operator acts as $P\otimes I$. On a point an arbitrary linear equivariant map has virtual kernel-minus-cokernel equal to domain-minus-target by finite-dimensional orthogonal splitting. These establish the stated naturality and coefficient assertions without imposing any Bott normalization.

Let $d$ be a first-order elliptic differential complex and put $D=d+d^{\ast }$. At nonzero $\xi$, write $a=\sigma (d)$; then $a^2=0$ and exactness implies $a+a^{\ast }$ is invertible, since its squared norm on $v$ is $\mathrm{\parallel }av{\mathrm{\parallel }}^2+\mathrm{\parallel }{a}^{\ast }v{\mathrm{\parallel }}^2$ and exactness gives $\ker a\cap \ker a^{\ast }=\{0\}$. Thus the Hodge operators ${\mathrm{\Delta }}_j$ and $D$ have elliptic realizations by [F1], even when their principal symbols are matrices. For ${\mathrm{\Delta }}_j$, symmetry identifies its cokernel with its smooth kernel; the Fredholm range is their orthogonal complement. Its inverse on that complement is bounded from $L^2$ to $H^2$ by [F4] and the elliptic estimate, which follows from a parametrix. Denote this inverse by $G_j$, zero on the harmonic kernel, and its projection by ${\mathrm{\Pi }}_j$. The differential commutes with $\mathrm{\Delta }$ on smooth sections. Harmonic sections are killed by both $d$ and $d^{\ast }$ by the nonnegative squared-norm identity. Hence $dGu$ and $Gdu$ are both orthogonal to harmonic sections and solve the same equation with source $du$; uniqueness gives $dG=Gd$ on smooth sections, and gives $I-\mathrm{\Pi }=d{d}^{\ast }G+d^{\ast }dG$. A closed smooth section therefore equals its harmonic projection plus $d(d^{\ast }Gu)$. Exact harmonic sections vanish by integration by parts. This proves unique harmonic representatives, finite-dimensional cohomology, and $\ker D^{+}={⨁}_{j\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}\ker {\mathrm{\Delta }}_j$, $\ker (D^{+}{)}^{\ast }={⨁}_{j\mathrm{o}\mathrm{d}\mathrm{d}}\ker {\mathrm{\Delta }}_j$. To check the symbol-class identification explicitly, exactness gives the orthogonal bundle decompositions $E^j=\mathrm{im}{a}_{j-1}\oplus \mathrm{im}{a}_j^{\ast }$ over the nonzero cotangent vectors; each restriction $a_j:\mathrm{im}{a}_j^{\ast }\to \mathrm{im}{a}_j$ is an equivariant bundle isomorphism. In the even-to-odd block of $a+a^{\ast }$, these isomorphisms and their adjoints occur once each, after only permuting the image summands. Hence the folded block represents the compactly supported symbol class of the exact complex, not merely an operator with the same numerical index. The harmonic-representative argument up to this point applies to the arbitrary positive density in the Statement and does not require its invariance. To compute the representation-valued index, repeat it with the positive smooth invariant density ${\mu }_G={\int }_G{g}^{\ast }\mu dg$ supplied by [F1] and [F2]. With the invariant bundle metrics, $U_g$ is now unitary. Since $d$ commutes with $U_g$, taking adjoints gives $d^{\ast }{U}_g=U_g{d}^{\ast }$; thus $D$, $\mathrm{\Delta }$, their harmonic projections and inverses on orthogonal complements commute with $G$. Sending a harmonic section to its cohomology class is consequently an equivariant isomorphism. The principal symbol of a formal adjoint depends on the bundle metrics but not on the density, since density derivatives contribute only lower-order terms. Hence this invariant-density $D^{+}$ realizes the same folded symbol class, and its even and odd harmonic kernels give ${\sum }_j(-1{)}^j[H^j]$ in $R(G)$ by step 1.3. The arbitrary-density harmonic representatives already proved remain valid, though their subspaces need not be preserved by $G$. This applies in particular to de Rham forms.

Extend this controlled operator to either compact ambient. Outside $L$ its equation is $u=0$, so its smooth kernel consists exactly of solutions on $U$ supported in $L$. Such a solution extends smoothly by zero, since $L$ is compactly contained in $U$. Restriction and zero extension therefore identify the two ambient kernel representations. The adjoint has the same support property, because its nonidentity kernel is obtained by transposing a kernel supported in $L\times L$. The adjoint kernels, and hence cokernel representations by [F1], are likewise identical. This proves ambient independence and excision. All choices and triple comparisons have already been shown to preserve the index, so this controlled calculation computes the originally given compact-support data rather than just one preferred representative. Empty manifolds, zero bundles and rank-zero symbols give zero spaces; compact zero-manifolds give finite direct sums of the point calculation in step 7.1. This completes the proof.