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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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