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The localization problem in index theory of elliptic operators. (English) Zbl 1297.58001
This monograph is an exposition of the localization principle in index theory of elliptic operators, also called superposition principle. After recalling the classical form of this principle, the authors focus on recently published generalized versions, of which they have been coauthors, having as applications well known theorems.
The simplest form of the superposition principle is as follows. Let \(M\) be a closed smooth Riemannian manifold, \(E\) a Riemannian or Hermitian vector bundle over \(M\), and \(D\) an elliptic operator on the space \(C^\infty(M;E)\) of \(C^\infty\) sections. Let \(A\) and \(B\) be disjoint closed subsets of \(M\). By performing a surgery in \(A\), we can produce a new closed smooth manifold \(M_A\) with another elliptic operator \(D_A\) extending \(D|_{M\setminus A}\). Then we have the index increment \(\Delta_A=\text{ind}D_A-\text{ind}D\). Similarly, let \(M_B\) be a smooth closed manifold with an elliptic operator \(D_B\) obtained by modifying \(M\) and \(D\) in \(B\), giving rise to an index increment \(\Delta_B\). Also, let \(M_{A\cup B}\) and \(D_{A\cup B}\) be the smooth closed manifold and elliptic operator obtained by performing both modifications in \(A\) and \(B\), simultaneously, with the corresponding index increment \(\Delta_{A\cup B}\). Then the superposition principle states that \(\Delta_{A\cup B}=\Delta_A+\Delta_B\). This is a simple consequence of the (local) index formula of Atiyah-Singer.
The first part of the book is devoted to more general versions of this principle that are not so easy to prove. In the first one, they consider the concept of a collar space, which is a separable Hilbert space with a continuous action of the algebra \(C^\infty([-1,1])\), where the constant function \(1\) acts as the identity. This generalizes the above setting by considering a \(C^\infty\) function \(\chi:M\to[-1,1]\) with \(\chi^{-1}(-1)=A\) and \(\chi^{-1}(1)=B\), and considering the induced action of \(C^\infty([-1,1])\) on \(L^2(M;E)\) by multiplication via \(\chi\). There is an obvious definition of the support \(\text{supp}h\) of each \(h\in H\) as a closed subset of \([-1,1]\). For any closed \(F\subset[-1,1]\), let \(H_F\) be the closure of the subspace of elements of \(H\) with support in \(F\), which is \(C^\infty([-1,1])\)-invariant, and therefore itself is a collar space. Two collar spaces, \(H\) and \(G\), are said to coincide on \([-1,1]\setminus F\) if there is an isomorphism \(j:H_F\to G_F\); in this case, \((H,G,j)\) is called a surgery of collar spaces over \(F\).
Using the above concept of support of elements of collar spaces, it is possible to define the support \(\text{supp}A\) of an operator between collar spaces, \(A:H\to G\), as a closed subset of \([-1,1]\times[-1,1]\). Let \(\Delta_\epsilon\) denote the \(\epsilon\)-neighborhood of the diagonal \(\Delta\) in \([-1,1]\times[-1,1]\). A family of linear bounded operators between collar spaces, \(A=\{\,A_\delta:H\to G\mid\delta>0\,\}\), is called a proper operator \(H\to G\) if \(A_\delta\) depends continuously on \(\delta\) in the operator norm, and, for all \(\epsilon>0\), there is some \(\delta_0>0\) so that \(\text{supp}A_\delta\subset\Delta_\epsilon\) if \(\delta>\delta_0\) (this is the idea of locality used in this setting). A proper operator \(A:H\to G\) is called a collar Fredholm operator if \(A_\delta\) is Fredholm for all \(\delta\), and there is a proper operator \(B_\delta:G\to H\) so that each \(B_\delta\) is inverse of \(A_\delta\) modulo compact operators. In particular, all operators \(A_\delta\) have the same index, called the index \(\text{ind}A\) of \(A\).
Let \(A_i:H_i\to G_i\) (\(i=1,2\)) be collar Fredholm operators. It is said that \(A_1\) and \(A_2\) are obtained from each other by surgery on a closed \(F\subset[-1,1]\) if there are collar space surgeries \((H_1,H_2,j_H)\) and \((G_1,G_2,j_G)\) over \(F\) such that, for all closed \(K\subset[-1,1]\setminus F\), there is some \(\delta_0>0\) such that \(j_GA_{1\delta}u=A_{2\delta}j_Hu\) whenever \(\text{supp}u\subset K\) and \(\delta>\delta_0\).
Now the generalized superposition principle states that, if \(D\) is a collar Fredholm operators, \(D_+\) and \(D_-\) are obtained from \(D\) by surgeries over \(\{1\}\) and \(\{-1\}\), respectively, and \(D_\pm\) is obtained with the corresponding simultaneous surgery over \(\{\pm1\}\), then \(\text{ind}D-\text{ind}D_-=\text{ind}D_+-\text{ind}D_\pm\).
For elliptic operators, finer stable homotopy invariants than the numerical index are defined in the \(K\)-theory of the manifold. Passing to the non-commutative setting, we can consider the \(K\)-theory \(K^*(A)\) of a \(C^*\) algebra \(A\). An ungraded Fredholm module over \(A\) is a triple \(x=(H,\rho,F)\), where \(H\) is a Hilbert space, \(\rho:A\to\mathfrak{B}(H)\) is a representation of \(A\) on \(H\), and \(F\in\mathfrak{B}(H)\), so that \([F,\rho(\phi)]\sim0\) for all \(\phi\in A\) (locality), \(F\approx F^*\) and \(F^2\approx1\). Here, for \(F,G\in\mathfrak{B}(H)\), \(F\sim G\) means that \(F-G\) is a compact operator, and \(F\approx G\) means that \(\rho(\phi)(F-G)\) and \((F-G)\rho(\phi)\) are compact operators for all \(\phi\in A\). If moreover \(H\) has a \(\mathbb{Z}_2\)-grading so that \(\rho\) is even and \(F\) is odd, then \(x\) is called a graded Fredholm module over \(A\). It is said that \(x\) is degenerate when the above relations of the definition of Hilbert module are equalities. There are obvious definitions of homotopy, unitary equivalence and direct sum of Fredholm modules over \(A\), and two Fredholm modules \(x\) and \(x'\) are called equivalent when there is another Fredholm module \(x''\) such that \(x\oplus x''\) and \(x'\oplus x''\) are unitarily equivalent to homotopic Fredholm modules. The groups \(K^0(A)\) (respectively, \(K^1(A)\)) consist of the equivalence classes \([x]\) of graded (respectively, ungraded) Fredholm modules \(x\), with the operation induced by the direct sum, the zero element being represented by the degenerate Fredholm modules.
The role of the two disjoint closed subsets is played now by two ideals \(J_1\) and \(J_2\) of \(A\) such that \(J_1+J_2=A\). For two Fredholm modules \(x\) and \(\tilde x\) over \(A\), the condition of agreeing over \(J_0=J_1\cap J_2\) can be easily defined. In this case, a new Fredholm module \(x\diamond\tilde x\) is defined by pasting together along \(J_0\) the part of \(x\) corresponding to \(J_1\) with the part of \(\tilde x\) corresponding to \(J_2\). Then the superposition principle for \(K\)-theory states that \([x\diamond\tilde x]-[x]=[\tilde x]-[\tilde x\diamond x]\) in \(K^*(A)\).
Going one step further with this kind of ideas, a superposition principle is also proved for the \(KK\)-theory of a pair of \(C^*\) algebras, involving a procedure of cutting and pasting Kasparov modules.
The second part of the book describes examples where the above versions of the superposition principle are applied. For instance, one chapter is devoted to obtain the relative index theorem of Gromov-Lawson on complete Riemannian manifolds, and its Bunke \(K\)-theoretic version. Another chapter deals with applications to boundary value problems, obtaining the Agranovich-Dynin theorem, the Agranovich theorem and the Bojarski theorem, as well as an index formula for boundary value problems with symmetric conormal symbol. The last chapter deals with applications to the spectral flow of families of Dirac type operators with classical boundary conditions. The authors derive a formula for the spectral flow, showing that it is equal to the index of an elliptic operator.

MSC:
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J22 Exotic index theories on manifolds
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