Abstract
We consider Lie groupoids of the form \({\mathcal {G}}(M,M_1) := M_0 \times M_0 \sqcup H \times M_1 \times M_1 \rightrightarrows M,\) where \(M_0 = M \setminus M_1\) and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold \(M_1\) in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord. They correspond to (the s-connected version of) the problem of the existence of a holonomy groupoid associated to the singular foliation whose leaves are the connnected components of \(M_1\) and the connected components of \(M_0\). We study the Lie groupoid structure of these groupoids, and verify that they are amenable and Fredholm in the sense recently introduced by Carvalho, Nistor and Qiao. We compute explicitly the K-groups of these groupoid’s \(C^*\)-algebras, we obtain \( K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}\) for \(M_1\) of odd codimension, and \( K _0 (C^* ({\mathcal {G}}(M,M_1))) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}, K _1 (C^* ({\mathcal {G}}(M,M_1))) \cong \{ 0 \}\) for \(M_1\) of even codimension. When M and \(M_1\) are compact we obtain, as an application of our previous K-theory computations, that in the odd codimensional case every elliptic operator (in the groupoid pseudodifferential calculus) can be perturbed (up to stabilization by an identity operator) with a regularizing operator, such that the perturbed operator is Fredholm; and in the even case, given an elliptic operator there is a topological obstruction to satisfy the previous Fredholm perturbation property given by the Atiyah-Singer topological index of the restriction operator to \(M_1\).
Similar content being viewed by others
References
Ammann, B., Lauter, R., Nistor, V.: On the geometry of Riemannian manifolds with a Lie structure at infinity. Int. J. Math. Math. Sci. 2004(4), 161–193 (2004)
Ammann, B., Lauter, R., Nistor, V.: Pseudo-differential operators on manifolds with a Lie structure at infinity. Ann. Math. 165, 717–747 (2007)
Androulidakis, I., Skandalis, G.: The analytic index of elliptic pseudodifferential operators on a singular foliation. J. K-Theory 3, 363–385 (2011)
Androulidakis, I., Skandalis, G.: Pseudodifferential calculus on a singular foliation. J. Noncommut. Geom. 5(1), 125–152 (2011)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I. Ann. Math. 2(87), 484–530 (1968)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III. Ann. Math. 2(87), 546–604 (1968)
Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math. 2(93), 119–138 (1971)
Carrillo Rouse, P.: A Schwartz type algebra for the tangent groupoid. In: \(K\)-theory and Noncommutative Geometry, EMS Series of Congress Reports. European Mathematical Society, Zürich, pp. 181–199 (2008)
Carrillo Rouse, P., Lescure, J.-M.: Geometric obstructions for Fredholm boundary conditions for manifolds with corners. Ann. K-Theory 3(3), 523–563 (2018)
Carvalho, C., Nistor, V., Qiao, Yu.: Fredholm conditions on non-compact manifolds: theory and examples. In: Operator Theory, Operator Algebras, and Matrix Theory, vol. 267 Operator Theory: Advances and Applications, pp. 79–122. Birkhäuser/Springer, Cham (2018)
Côme, R.: The Fredholm property for groupoids is a local property. Results Math. 74(4):Art. ID 160, 33 (2019)
Debord, C.: Holonomy groupoids of singular foliations. J. Differ. Geom. 58(3), 467–500 (2001)
Debord, C.: Local integration of Lie algebroids. In: Banach Center Publications, vol. 54, Institute of Mathematics, Polish Academy of Sciences, Warszawa (2001)
Debord, C., Lescure, J.M., Nistor, V.: Groupoids and an index theorem for conical pseudo-manifolds. J. Reine Angew. Math. 628, 1–35 (2009)
Debord, C., Skandalis, G.: Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps. J. Geom. Phys. 129, 255–268 (2018)
Lauter, R., Monthubert, B., Nistor, V.: Pseudo-differential analysis on continuous groupoids. Documenta Mathematica 4, 625–655 (2000)
Lauter, R., Monthubert, B., Nistor, V.: Spectral invariance for certain algebras of pseudodifferential operators. J. Inst. Math. Jussieua 4(3), 405–442 (2005)
Monthubert, B.: Groupoids and pseudodifferential calculus on manifolds with corners. J. Funct. Anal. 199(1), 243–286 (2003)
Monthubert, B., Nistor, V.: A topological index theorem for manifolds with corners. Compos. Math. 148(2), 640–668 (2012)
Nazaikinskii, V., Savin, A., Sternin, B.: Elliptic theory on manifolds with corners. II. Homotopy classification and \(K\)-homology. \(C^*\)-Algebras and Elliptic Theory II. Trends in Mathematics, pp. 207–226. Birkhäuser, Basel (2008)
Nistor, V.: Groupoids and integration of Lie algebroids. J. Math. Soc. Jpn. 52(4), 847–868 (2000)
Nistor, V.: An index theorem for gauge-invariant families: the case of solvable groups. Acta Math. Hungar. 99(1–2), 155–183 (2003)
So, B.K.: On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth. Adv. Math. 237, 1–32 (2013)
Acknowledgements
The authors would like to thank Victor Nistor for having had the insight of suggesting us to work together in this and future projects. The authors are deeply thankful to the referees that gave very important suggestions, remarks and pointed out some mistakes in previous versions of our work, the article really improved after their intervention.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Carrillo Rouse, P., So, B.K. K-theory and Index Theory for Some Boundary Groupoids. Results Math 75, 172 (2020). https://doi.org/10.1007/s00025-020-01300-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-020-01300-6