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Homotopy classification of elliptic operators on stratified manifolds. (English. Russian original) Zbl 1154.58013
Izv. Math. 71, No. 6, 1167-1192 (2007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 71, No. 6, 91-118 (2007).
The authors define a stratified manifold of length $$k>0$$ inductively by stratified manifolds $${\mathfrak M}={\mathfrak M}_k\supset{\mathfrak M}_{k-1}\supset\cdots \supset{\mathfrak M}_0$$ and a bundle structure of a neighborhood $$U\subset {\mathfrak M}$$ of a smooth manifold $${\mathfrak M}_0$$ whose fiber is the cone with base $$\Omega$$ such that $$\Omega$$ is a compact stratified manifold of length at most $$k-1$$, satisfying a proper compatibility condition for the sequence of $${\mathfrak M}_j$$. After modifying notions of the space $$C^{\infty}({\mathfrak M})$$ of smooth functions, the space of $$\Psi(V, {\mathfrak M})$$ of smooth operator families with parameter in $$V$$, and cotangent bundle $$T^*{\mathfrak M}$$, the authors define a pseudo-differential operator $$D(v): L^2({\mathfrak M})\to L^2({\mathfrak M})$$ with parameter $$v\in V$$, by the following conditions:
1. If $$\varphi, \psi\in C^{\infty}({\mathfrak M})$$ and $$\operatorname{supp}\varphi\cap \operatorname {supp}\psi=\emptyset$$, then $$\psi D\varphi$$ is a smooth operator with parameter.
2. The operator $$D$$ is representable modulo the space of smooth operators as
$D=P\biggl(x, r, rv, -ir\frac{\partial}{\partial x}, ir\frac{\partial}{\partial r}+i\frac{n+1}{2}\biggr),$
$$n= \dim\Omega$$, where $$P(x, r,v, \eta, p)\in\Psi(V\times T^*_x{\mathfrak M}_0\times \mathbb R, \Omega)$$ is a pseudo-differential operator with parameters on $$\Omega$$ depending smoothly on the additional parameters $$x\in {\mathfrak M}_0$$ and $$r\in \overline{\mathbb R}+$$ and $$P=0$$ for $$r>r_0$$ where $$r_0$$ is sufficiently small.
The symbol $$\sigma_j(D)$$ of $$D$$ corresponding to the strata $${\mathfrak M}_j\backslash {\mathfrak M}_{j-1}$$ $$(j>0)$$ of $${\mathfrak M}$$ are defined as the symbols of operator $$D$$ regarded as an element of $$\Psi(V, {\mathfrak M}\backslash{\mathfrak M}_0)$$.
Let $$\text{Ell}_*({\mathfrak M},{\mathfrak M}_j)$$ be the group of stable homotopy classes of pseudo-differential operators that are elliptic on $${\mathfrak M}\backslash {\mathfrak M}_j$$. For an elliptic operator $$D$$, taking self-adjoint pseudo-differential operators with $$k-j$$ leading components of the symbol equal to
$(\sigma_k(D)^*\sigma_k(D))^{-1/2},\dots, (\sigma_{j+1}(D)^*\sigma_{j+1}(D))^{-1/2},$
one gets a correspondence $$\varphi: (D)\mapsto [D]$$.
The authors obtain the main theorem,
$\text{Ell}_*({\mathfrak M},{\mathfrak M}_j) \overset\varphi\cong K_*({\mathfrak M}\backslash {\mathfrak M}_j).$
The proof uses an induction for $$j$$, based on the five-lemma for exact sequences of $$\text{Ell}_*$$ and $$K_*$$ corresponding to the pair of spaces $${\mathfrak M}_j\backslash{\mathfrak M}_{j-1}\subset {\mathfrak M}\backslash M_{j-1}$$. Let $${\mathfrak M}\supset X$$ be a stratified pair. As one application, it is mentioned that a necessary and sufficient condition for the existence of a lift of $$a\in \text{Ell}({\mathfrak M}, X)$$ to the group $$\text{Ell}({\mathfrak M})$$ is the nonvanishing of $$\partial\varphi(a)$$ where $$\partial: K_0({\mathfrak M}\backslash X)\to K_1(X)$$ is the boundary homomorphism.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K56 Index theory 19K33 Ext and $$K$$-homology
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