Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps
Let \(G\rightarrow [r]{s}M\) be a Lie groupoid with range and source \(M\), \(F\) a closed saturated submanifold \(M\), and \(W=M\setminus F\). Then \(G=G_W\sqcup G_F\), where \(G_W\) and \(G_F\) are subgroupoids of \(G\) having \(W\) and \(F\) as ranges and sources. From this partition, we have the exact sequence of \(C^\ast\) algebras \[ 0\to C^\ast(G_W)\to C^\ast(G)\to C^\ast(G_F)\to 0. \] In this paper, derived exact sequences of cohomology groups from this sequence mainly focusing on the associated index maps and full index maps are studied (cf. \textit{C. Debord} and \textit{G. Skandalis}, ``Blowup constructions for a Boutet de Monvel type calculus'', Preprint, \url{arXiv:1706.09588}]). To define index maps, the full symbol algebra \[ \Sigma^W(G)=C(\mathbb{S}\mathfrak{A}^\ast G)\times_{C(\mathbb{S}\mathfrak{A}^\ast G_F)}\Psi^\ast(G_F), \] where \(\mathbb{S}\mathfrak{A}^\ast G\) is the cosphere bundle of the algebroid \(\mathfrak{A}G\) of \(G\), and \(\Psi^\ast(G_F)\) is the \(C^\ast\)-algebra of order 0 pseudo differential operators on \(G_F\), is introduced, and the exact sequence \[ 0\to C^\ast(G_W)\to \Psi^\ast(G)\rightarrow{\sigma_{full}}\Sigma~W(G)\to 0 \] is presented (\S2.4). The authors remark the full symbol map has two components; the usual commutative symbol \(\Psi^\ast(G)\to C(\mathbb{S}\mathfrak{A}^\ast G)\), and the boundary (or non commutative) symbol \(\sigma_\partial: \Psi^\ast(G)\to \Psi^\ast(G_F)\). In \S2.4, the index map \(\mathrm{ind}^W_G\in KK(C^\ast(\mathfrak{A}^{nc}_W G), C^\ast(G_W))\) is defined. Also from (1) and (2), \(\partial^W_G\in KK^1(C^\ast(G_F),C^\ast(G_W))\) and \(\widetilde{\mathrm{ind}}^W_{full}(G)\in KK^1(\Sigma^W(G), C^\ast(G_W0)\) are defined (\S4.1. [\textit{G. G. Kasparov}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571--636 (1980; Zbl 0448.46051)]), provided \(G_F\) is amenable. Then \(\widetilde{\mathrm{ind}}^W_{full}\) is shown to be the Kasparov product of the Connes-Thom element \([th]\in KK^1(\Sigma^W(G), \Sigma^W(G)\rtimes \mathbb{R})\), the inclusion \(i:\Sigma~W(G)\rtimes\mathbb{R}\hookrightarrow C^\ast(\mathfrak{A}^{nc}_W G)\) and \(\mathrm{ind}^W_G\) (Prop.4.2). Adopting these constructions, assuming \(G_W=W\times W\), an answer to the question when an elliptic symbol \(\sigma\), which defines an element of \(K_1(C_0(\mathbb{S}\mathfrak{A}^\ast G))\), can be lifted to a Fredholm operator, is given (Proposition 4.3). \S4 concludes to compute \(\partial^{G,F}\) and \(\tilde{\mathrm{ind}}^{G,F}\) in terms of index element of \(G_W\), assuming the index of \(G_F\) is an invertible element of \(KK(C_0(\mathfrak{A}^\ast G_F), C^\ast(G_F))\) (Prop.4.4 -4.8). Authors say the assumption \(\mathrm{ind}_{G_F}\) is invertible is strong, but satisfied if \(G_F\) is a bundle of simply connected solvable Lie group. In \S5, the last Section, assume \(\mathbb{R}_+^\ast\) act smoothly and properly on \(G\), the Connes-Thom element \(\alpha=(H,D)\in KK^1(C_0(M), C_0(M/\mathbb{R}_+^\ast)\), where \(H\) is a Hilbert module obtained as a completion of \(C_0(M)\) and \(D=\dfrac{1}{i}\dfrac{\partial}{\partial t}\), and its inverse \(\beta\) are introduced. Then assuming \(G\) is amenable, and \(W\) is an open saturated subset in \(M\) for the actions of \(G\) and \(\mathbb{R}_+^\ast\), it is shown \(\beta^{G,W}_\Sigma\) is invertible and \[ \beta^{G,W}_\Sigma\otimes\tilde{\mathrm{ind}}^W_{full}(G)=-\tilde{\mathrm{ind}}~{W/\mathbb{R}_+^\ast}_{full}(G/\mathbb{R}_+^\ast)\otimes\beta^{G_W} \] (Proposition 5.4. cf. Proposition 5.3). The authors say the Connes-Thom invariance of the index map is a main tool in Connes' proof of the Atiyah-Singer index theorem [\textit{A. Connes}, Noncommutative geometry. San Diego, CA: Academic Press (1994; Zbl 0818.46076)] and is also central in the generalization for manifolds with boundaries [\textit{P. Carrillo Rouse} et al., J. Topol. Anal. 6, No. 1, 27--74 (2014; Zbl 1346.19007)]. As preliminaries, exact sequences and commutative diagrams of cohomology groups, coming from (1) and (2), and related topics, are explained in \S2 and 3.
scientific article; zbMATH DE number 6867194
scientific article