Analytic symbol index, normal embedding compatibility and ordinary pushforward

Apply [F2] to a representative of a compact-support triple. Its continuous bundles and maps are smoothed, radially identified with base bundles, and represented by a homogeneous elliptic symbol by the proved constructions in that supplier. Every relation in the group of [F1] is preserved by [F2], so its additive index descends to $a_{T,M}$. Tensoring by representations proves $R(T)$-linearity. The same supplier proves the claimed operator and complex identifications, using matrix ellipticity and harmonic representatives. Its group restriction and diffeomorphism assertions give the stated naturality. In particular this step constructs an index on symbols; it does not assume an embedding theorem for that index.

First consider the zero section of a real rank-$k$ equivariant metric bundle $N\to Y$. Its orthonormal frame bundle $P\to Y$ is a smooth principal right $O(k)$-bundle; $T$ acts on frames by the differential fiber action and commutes with right multiplication. Its compact normal sphere bundle is $Z=P{\times }_{O(k)}S({\mathbb{R}}^k\oplus \mathbb{R})$. It has an invariant open chart $P{\times }_{O(k)}{\mathbb{R}}^k=N$ around its north section. The vertical compact symbol is the $O(k)$-equivariant extension of $b_{{\mathbb{R}}^k}$ from this chart. By [F2] it has a smooth equivariant homogeneous realization, whose order may be changed to one without changing its class or index. Its finite-dimensional kernel $K$ and adjoint kernel $L$ obey $[K]-[L]=1$ by [F4]. Invariant orthogonal decompositions into irreducible finite-dimensional $O(k)$ modules then give an actual $O(k)$-equivariant isomorphism $K\cong L\oplus \mathbb{C}$: equality of their multiplicities is exactly this virtual equality. The fiber has trivial additional $T$ action, so the associated isomorphism is also $T$-equivariant.

Represent $c$ by an order-one elliptic symbol on $Y$ using [F2], and apply [F3] to this base symbol and the vertical sphere symbol of step 1.2. The product index is the difference of the indices twisted by $K_P$ and $L_P$. Their equivariant bundle isomorphism $K_P\cong L_P\oplus {\mathbb{C}}_Y$, block additivity and quantization independence in [F2] make that difference exactly $a_{T,Y}(c)$. This statement concerns all symbol classes: a virtual difference of representatives is handled additively, and no differential or Dirac representative is required.

In a frame chart of $N$, split a covector on its total space into $(\alpha ;v,\eta )$, with $\alpha \in T^{\ast }Y$ and $(v,\eta )$ the normal base and normal covector. The restriction of the associated product class from step 2.1 to $T^{\ast }N$ is the graded product of $c(\alpha )$ and the normal Koszul map at $v+i{\eta }^{\mathrm{♯}}$. Its possible noninvertibility is the zero normal vector/covector over the compact support of $c$. The tensor differential contracts wherever either factor is invertible, so it has compact support. These local maps patch because real orthogonal frame changes intertwine exterior multiplication and its adjoint. A connection gives the indicated cotangent splitting; varying it is an affine homotopy, and [F3] proves the corresponding elliptic product-symbol homotopy. This is exactly the doubled-normal symbol defining the zero-section $j_{!}$ in [F1]. Thus the associated product is its extension from the normal open chart, rather than an unrelated symbol of the same dimension.

For $f:Y\hookrightarrow M$, use [F5] to identify a disk neighborhood in $N$ equivariantly with a neighborhood of $Y$ in $M$. Fiber compression identifies the compact symbol in step 3.1 with one supported in this disk tube: over the compact support of $c$, choose one uniformly small normal radius, and compress the normal coordinates by a radial diffeomorphism. The same invariant open disk tube embeds in the compact sphere bundle $Z$ and in $M$. Its cotangent identifications have precisely normal complex structure $J(v,\eta )=(-{\eta }^{\mathrm{♯}},v^{\mathrm{♭}})$; the differential of the tube map gives this along the zero section, and the connection splittings are joined by the homotopy just used. The controlled two-cutoff excision in [F2] identifies the kernel and adjoint-kernel representations of the extensions into $Z$ and $M$. Combining this equality with step 2.1 proves $a_{T,M}(j_{!}c)=a_{T,Y}(c)$. Normal ranks constant only componentwise cause no difficulty: compactness gives finitely many components, and apply the argument to each and add. For rank zero the tube is an open-and-closed component and the assertion is direct extension by zero; empty manifolds give zero.

For the trivial group, construct an embedding $Y\hookrightarrow {\mathbb{R}}^N$ by finitely many chart functions: choose bump functions ${\varphi }_i$ supported in coordinate charts whose positive sets cover $Y$, and map to the vector of all ${\varphi }_i$ and ${\varphi }_i{x}_{i,j}$. Equality determines a common positive chart and then its coordinates, proving injectivity; the same coordinate ratios show that the derivative is injective. Compactness makes it an embedding. Regard its target as the open Euclidean chart of $S^N$. Step 4.1 identifies the analytic index of $Y$ with the index of its doubled-normal extension to that sphere, and [F2] computes the latter in the Euclidean chart. By [F6], its class there is the integer $p_{Y,!}(c)$ times $b_{{\mathbb{R}}^N}$. The normalization [F4], restricted to the trivial group, sends this generator to $1$. Hence $a_{\{1\},Y}(c)=p_{Y,!}(c)$. The left-hand side is intrinsic, proving the asserted independence of embedding. This uses only ordinary Bott periodicity; no equivariant representation Bott isomorphism has been presumed.

Suppose $g$ fixes $Y$ pointwise. The compact closure $H=\overline{⟨g⟩}$ then fixes $Y$ and its cotangent bundle. On a compact disk representative, its unitary bundle representations have finitely many weights: character integration gives continuous projection operators onto each weight, whose ranks are locally constant. The finitely many weights spanning a fiber still exhaust its rank on a neighborhood; compactness gives a finite global list. Their projections split the bundles and preserve the symbol. Quantize the finite blocks separately and average over $H$. The resulting elliptic operator represents the restricted symbol, and its kernel and adjoint kernel split into the same weight blocks. Consequently its character at $g$ is ${\sum }_{\lambda }\lambda a_{\{1\},Y}(c_{\lambda })$; quantization independence and group restriction in [F2] identify it with the character of $a_{T,Y}(c)$. Step 5.1 gives the stated formula with $p_{Y,!}$. Finally every index identity is $R(T)$-linear, and evaluation of $s\in R(T)$ at $g$ is a scalar. Localizing where $s(g)\mathrm{\ne }0$ therefore extends the identities by ordinary division, proving the last assertion.