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Groupoids and an index theorem for conical pseudo-manifolds. (English) Zbl 1169.58005

Atiyah and Singer defined a topological and an analytic index. These are maps \(K^0(T^*V) \to \mathbb{Z}\) for a closed manifold \(V\). The Atiyah-Singer index theorem says that both maps agree. In the present paper, the authors generalize the theorem to pseudomanifolds with isolated singularities. For such a pseudomanifold \(X\), the first two authors have defined a groupoid \(T^SX\) playing the role of the tangent space in [J. Funct. Anal. 219, No. 1, 109–133 (2005; Zbl 1065.58014)]. For the definition of the analytic index, the authors follow Connes’ approach by constructing a tangent groupoid for \(T^SX\). For the definition of the topological index, the authors adapt the original approach of Atiyah and Singer to the singular setting. They define the notion of a conical bundle and, for such a bundle \(E\) over \(X\), they construct an “inverse Thom map” from \(K_*(C^*(T^SE))\) to \(K_*(C^*(T^SX))\). For the conical bundle coming from the tubular neighbourhood of an embedding of \(X\) into a space \((\mathbb{R}^n)^S\), which they define, this map is proved to be an isomorphism. It yields the generalization of the Thom isomorphism. The other maps appearing in the definition of the topological index generalize rather directly from the classical situation. The authors prove the equality of the analytic and the topological index using deformation groupoids. Their proof is new also in the classical setting. As all their constructions, it relies heavily on methods from \(KK\)-theory.
In the end, based on results in [J.-M. Lescure, Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds, http://arxiv.org/abs/math/0609328], the authors give a more concrete description of the analytical index as a Fredholm index of elliptic operators in Melrose’s \(b\)-pseudodifferential calculus. They also explain the relation to Fuchs type operators.

MSC:

58J22 Exotic index theories on manifolds
22A22 Topological groupoids (including differentiable and Lie groupoids)
19K56 Index theory

Citations:

Zbl 1065.58014
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References:

[1] DOI: 10.4007/annals.2007.165.717 · Zbl 1133.58020 · doi:10.4007/annals.2007.165.717
[2] DOI: 10.1017/S0305004100049410 · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[3] DOI: 10.2307/1970715 · Zbl 0164.24001 · doi:10.2307/1970715
[4] Baaj Saad, C. R. Acad. Sci. Paris Sér. I Math. 296 (21) pp 875– (1983)
[5] DOI: 10.1007/s00211-005-0588-3 · Zbl 1116.65119 · doi:10.1007/s00211-005-0588-3
[6] DOI: 10.1016/S0022-1236(03)00268-4 · Zbl 1048.47052 · doi:10.1016/S0022-1236(03)00268-4
[7] DOI: 10.1016/0022-1236(84)90101-0 · Zbl 0538.58033 · doi:10.1016/0022-1236(84)90101-0
[8] DOI: 10.1016/0022-1236(84)90106-X · Zbl 0556.58027 · doi:10.1016/0022-1236(84)90106-X
[9] DOI: 10.1016/0022-1236(90)90098-6 · Zbl 0696.53021 · doi:10.1016/0022-1236(90)90098-6
[10] DOI: 10.1007/BF01210930 · Zbl 0657.58037 · doi:10.1007/BF01210930
[11] DOI: 10.1007/BF00136813 · Zbl 0733.57010 · doi:10.1007/BF00136813
[12] DOI: 10.2307/2374646 · Zbl 0664.58035 · doi:10.2307/2374646
[13] da Silva A. Cannas, Berkeley Math. Lect. Notes Ser. pp 10– (1999)
[14] DOI: 10.1007/s10977-005-3109-3 · Zbl 1117.58009 · doi:10.1007/s10977-005-3109-3
[15] Cheeger J., J. Di{\currency}. Geom. 18 pp 575– (1983)
[16] DOI: 10.1007/BF01895667 · Zbl 0960.46048 · doi:10.1007/BF01895667
[17] DOI: 10.2977/prims/1195180375 · Zbl 0575.58030 · doi:10.2977/prims/1195180375
[18] DOI: 10.4007/annals.2003.157.575 · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575
[19] Debord C., J. Di{\currency}. Geom. 58 pp 467– (2001)
[20] DOI: 10.1016/j.jfa.2004.03.017 · Zbl 1065.58014 · doi:10.1016/j.jfa.2004.03.017
[21] Fedosov B., S.) 5 (4) pp 467– (1999)
[22] Freed D., Asian J. Math. 3 (4) pp 818– (1999)
[23] DOI: 10.1016/0040-9383(80)90003-8 · Zbl 0448.55004 · doi:10.1016/0040-9383(80)90003-8
[24] Gorokhovsky A., Math. 560 pp 151– (2003)
[25] Hilsum M., Ann. Sci. Ec. Norm. 20 (4) pp 325– (1987)
[26] Kasparov G. G., Ser. Math. 44 pp 571– (1980)
[27] Lauter R., Doc. Math. 5 pp 625– (2000)
[28] Lescure J.-M., Bull. Soc. Math. France 129 (4) pp 593– (2001)
[29] Mackenzie K., London Math. Soc. Lect. Notes pp 124– (1987)
[30] Melo S. T., Math. 561 pp 145– (2003)
[31] Melrose R. B., Math. Res. Lett. 2 (5) pp 541– (1995)
[32] DOI: 10.1016/0001-8708(92)90059-T · Zbl 0761.55002 · doi:10.1016/0001-8708(92)90059-T
[33] DOI: 10.1090/S0002-9939-99-04850-9 · Zbl 0939.35202 · doi:10.1090/S0002-9939-99-04850-9
[34] Muhly P., J. Oper. Th. 17 (1) pp 3– (1987)
[35] Nazaikinskii V. E., AP/051 pp 2006– (2025)
[36] Nazakinski V. E., Mat. Sb. 196 (9) pp 23– (2005)
[37] DOI: 10.1006/jfan.1996.0135 · Zbl 0868.58076 · doi:10.1006/jfan.1996.0135
[38] Nistor V., Doc. Math. 2 pp 263– (1997)
[39] DOI: 10.2969/jmsj/05240847 · Zbl 0965.58023 · doi:10.2969/jmsj/05240847
[40] DOI: 10.2140/pjm.1999.189.117 · Zbl 0940.58014 · doi:10.2140/pjm.1999.189.117
[41] DOI: 10.1007/s10977-005-1515-1 · Zbl 1087.58013 · doi:10.1007/s10977-005-1515-1
[42] DOI: 10.1023/A:1006521714633 · Zbl 0914.58030 · doi:10.1023/A:1006521714633
[43] DOI: 10.1016/j.jfa.2005.12.027 · Zbl 1105.58014 · doi:10.1016/j.jfa.2005.12.027
[44] DOI: 10.1007/BF02329732 · Zbl 0526.53039 · doi:10.1007/BF02329732
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