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Coarse cohomology and index theory on complete Riemannian manifolds. (English) Zbl 0780.58043

Mem. Am. Math. Soc. 497, 90 p. (1993).
The main tasks of the book are the development of a new theory of ‘cohomology’ (coarse cohomology) for complete metric spaces in which closed bounded sets are compact (proper metric spaces) and the establishment of a version of the Atiyah-Singer Index Theorem for complete noncompact Riemannian manifolds.
The coarse cohomology is a kind of Alexander-Spanier cohomology defined as follows: Let \(M\) be a proper metric space. The coarse complex \(CX^*(M)\) is defined by taking as \(CX^ q(M)\) the space of Borel functions \(\varphi:M^{q+1}\to\mathbb{R}\) which are bounded on every compact subset and for each \(\varepsilon>0\), \(\text{supp}(\varphi)\cap\{x\in M^{q+1}| d(x,\Delta)<\varepsilon\}\) is relatively compact in \(M^{q+1}\). The coboundary operator is defined by \(\delta\varphi(x_ 0,\ldots,x_{q+1})=\sum^{q+1}_{i=0}(-1)^ i\varphi(x_ 0\ldots,\hat x_ i,\ldots,x_{q+1})\). The author defines the coarse cohomology \(HX^*(M)\) of \(M\) to be the cohomology of the complex \(\bigl(CX^*(M),\delta\bigr)\).
The main features of the coarse cohomology are: (i) There is a natural map \(c\) (the character map) from \(HX^*(M)\) into the Alexander-Spanier cohomology with compact supports of \(M\), \(H^*_ c(M)\). (ii) The coarse cohomology is contravariantly functorial on the category of proper metric spaces and uniformly bornologous proper Borel maps \((f:M\to N\) is uniformly bornologous if for every \(R>0\), there exists \(S>0\) such that \(d(x,y)<R\Rightarrow d(f(x),f(y))<S)\). (iii) If \(M\) is compact, \(HX^*(M)=HX^*(pt)+R(\dim=0)\) or \(0(\dim>0)\). (iv) If \(M\) is noncompact \(HX^ 0(M)=0\). (v) \(HX^ 1(M)\) is determined, in the case that \(M\) is a locally compact noncompact space, by the ends of \(M\) defined by the Freudenthal compactification of \(M\). (vi) The coarse cohomology has a natural secondary product structure. (vii) Bornotopy maps induce the same homomorphisms on coarse cohomology. Bornotopy- equivalent spaces have isomorphic coarse cohomology. (Let \(f,g:M\to N\) be morphisms. The author says that \(f\) and \(g\) are bornotopic, if there is a constant \(R>0\) such that \(d(f(x),g(x))<R\) for all \(x\in M)\).
Thus the coarse cohomology is not a generalized cohomology theory in the usual sense and measures the behaviour at infinity of a space.
The author relates the coarse cohomology to a compactly supported version of anti-Čech cohomology (the open coverings used are locally finite and its members are relatively compact, they are coarse and coarse and one passes to an inverse limit). With the aid of this relation he computes the coarse cohomology of several spaces and establishes that the character map \(c\) is an isomorphism in some interesting cases. Then, the coarse cohomology is used, by the author, to construct higher indices for elliptic operators on noncompact complete Riemannian manifolds. Let \(M\) be a proper metric space. An \(M\)-module is a pair \((H,\rho)\), where \(H\) is a Hilbert space and \(\rho\) is a representation of the \(C^*\)-algebra of continuous functions on \(M\) vanishing at infinity into \(L(H)\). Given an \(M\)-module \((H,\rho)\), the author considers a special algebra of operators on \(H\), \(B_ H\), and constructs a Connes character map \(\chi:HX^*(M)\to HC^*(B_ H)\), where \(HC^*\) denotes the cyclic cohomology theory of A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 257-360 (1985; Zbl 0592.46056)]. Let \(M\) be a proper metric space, \(H\) a graded \(M\)-module, \(D\) a graded elliptic operator on \(H\) and \([\varphi]\in HX^{2q}(M)\). Then \(D\) has an index, \(\text{Ind}(D)\), in \(K_ 0(B_ H)\) and the author defines \(\text{Ind}_ \varphi(D)\) to be the complex number \(\langle\text{Ind}(D),\chi[\varphi]\rangle\), where \(\langle,\rangle\) is the pairing defined by Connes [op. cit.] between \(K_ 0(B_ H)\) and \(HC^{2q}(B_ H)\), up to the multiplicative constant \((2\pi i)^ qq!\).
The author uses the work of A. Connes and H. Moscovici [Topology 29, No. 3, 345-388 (1990; Zbl 0759.58047)] to calculate the above index in topological terms, and establishes the main results of the book: “Let \(M\) be a complete Riemannian manifold of dimension \(2m\) (resp. \(2m-1)\) and let \(D\) be a graded (resp. ungraded) generalized Dirac operator over \(M\). Let \([\varphi]\in HX^{2q}(M)\) (resp. \([\varphi]\in HX^{2q-1}(M))\) be a coarse cohomology class. Then \(\text{Ind}_ \varphi(D)={q!\over(2q)!(2\pi i)^ q}\langle I_ D\smile c[\varphi],[M]\rangle\) (resp. \(={q!\over(2q-1)!(2\pi i)^ q}\langle I_ D\smile c[\varphi],[M]\rangle\), where \(I_ D\in H^*(M)\) is the index class”.
Vanishing theorems for the index paired with coarse classes are studied by the author constructing an analogue, for proper metric spaces, of Higson’s compactification [On the relative \(K\)-homology theory of Baum and Douglas, J. Funct. Anal. (to appear)]. In the last chapter (6) of the book, some geometric and analytic applications of the above results to the study of complete Riemannian manifolds of nonpositive curvature are given. The book concludes with a discussion of the coarse analogue of the Novikov conjecture on the homotopy invariance of the higher signatures.
The main results of this book have been announced in a previous paper of the author [Bull. Am. Math. Soc., New Ser. 23, No. 2, 447-453 (1990; Zbl 0719.58036)], where the coarse cohomology is designed by ‘exotic’ cohomology.

MSC:

58J22 Exotic index theories on manifolds
55N35 Other homology theories in algebraic topology
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K56 Index theory
51K99 Distance geometry
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