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Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds. (English) Zbl 1183.58005

The author is interested in studying Poincaré duality for pseudomanifolds, i.e., manifolds \(X\) with a set \(\Sigma(X)\subset X\) of singular points. The approach followed is the same as in a previous work by the same author on similar subjects. The author claims that different techniques have previously been used, to produce results close to those in the present paper, by A. Savin, V. Nazaiknskii and B. Sternin [see in References].
The first step in this construction is to characterize the tangent space for such pseudomanifolds \(X\) like a groupoid that contains as a subgroupoid the tangent space in the regular parts \(X\setminus\Sigma(X)\). The philosophy here is mediated by the so-called tangent groupoid considered by A. Connes. The author denotes this tangent space \(T^qX\) (or \(T^cX\)) and he recognizes on \(T^qX\) a natural structure of a noncommutative smooth groupoid. Then, the duality between the algebra \(C(X)\) of continuous functions on \(X\) and the \(C^*\) algebra \(C^*(T^qX)\) are defined in terms of bivariant \(K\)-theory and imply the existence of an isomorphism \(\Sigma^q:K_0(X)\cong K_0(C^*(T^qX))\). The main result of this paper is the characterization of this isomorphism by means of a noncommutative symbol map, extending in this way the well-known smooth case. (In fact, when \(\Sigma(X)=\emptyset\), the isomorphism \(K_0(X)\cong K_0(C^*(TX))\cong K^0(T^*X)\), is the principal symbol map \([P]\mapsto[\sigma(P)]\), sending the class of an elliptic pseudodifferential operator (a basis cycle of the \(K\)-homology of \(X\)), to the class of its principal symbol (a basic cycle of the \(K\)-theory of \(T^*X\) with compact supports).) The author considers noncommutative symbols on \(X\) as pseudodifferential operators on \(T^qX\).
The paper, after a detailed introduction, splits into three more sections. 2. Equivalence of noncommutative tangent spaces and Dirac elements. 3. Cycles of the \(K\)-theory of the noncommutative tangent space. 4. Poincaré dual of elliptic noncommutative symbols.
Remark. The central point in the construction considered in this paper is the definition of the tangent space \(T^qX\) of a pseudomanifold \(X\). Even if the groupoid structure is considered fundamental, this aspect is not a peculiarity for the tangent space of a pseudomanifold (since also the tangent space of a smooth manifold has a natural structure of a groupoid, even if in this case, it is commutative). Instead it appears that the very important point is the embedding of the tangent bundle, \(T(X\setminus\Sigma(X))\), of the regular part \(X\setminus\Sigma(X)\), just into \(T^qX\). This approach is completely different from that usually used in algebraic geometry to characterize singularities. In fact, \(T^qX\) is defined by means of a set-like approach embedding \(X\) into a larger smooth environment, that, in some sense, solves the singularities of \(X\).
Let us also emphasize that the same philosophy has been followed by the reviewer of this paper in order to characterize singular PDEs and their solutions in the geometry of PDEs. However the techniques used in this paper are completely different. In fact, in geometry of PDEs there is a canonical smooth environment into which a singular PDE can be embedded. This is just a suitable jet-derivative space, say \(J^k_n(W)\). On \(J^k_n(W)\) there is also a canonical distribution (the Cartan distribution), \({\mathbf E}_k(W)\subset TJ^k_n(W)\)) defined, contained in the tangent space \(TJ^k_n(W)\), that allows us to characterize integral manifolds and hence singular solutions. But in these cases tangent spaces can be identified with commutative groupoids.
Thus, another important aspect that may be related to this paper, is the identification of the extraordinary cohomology theory for any smooth manifold \(X\) and the cobordism theory of pseudomanifolds contained in \(X\). In fact, the author states that a motivation for this paper is to validate the choice of a tangent space which is \(K\)-dual to the singular manifold. Does this choice agree with cobordism theory? Does the natural isomorphism \(\Omega_\bullet\otimes_{\mathbf L}{\mathbf Z}[\beta,\beta^{-1}]\cong K_0\otimes_{\mathbf Z}{\mathbf Z}[\beta,\beta^{-1}]\) over smooth (or singular) schemes, where \(K_0\) is the Grothendieck group of algebraic vector bundles, identify a new algebraic cobordism \(\Omega_\bullet\)?

MSC:

58B34 Noncommutative geometry (à la Connes)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K35 Kasparov theory (\(KK\)-theory)
58H05 Pseudogroups and differentiable groupoids
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References:

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