Annals of K-Theory

Geometric obstructions for Fredholm boundary conditions for manifolds with corners

Paulo Carrillo Rouse and Jean-Marie Lescure

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Abstract

For every connected manifold with corners there is a homology theory called conormal homology, defined in terms of faces and orientation of their conormal bundle and whose cycles correspond geometrically to corner cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), χ cn : = χ 0 χ 1 , is given by the alternating sum of the number of (open) faces of a given codimension.

The main result of the present paper is that for a compact connected manifold with corners X , given as a finite product of manifolds with corners of codimension less or equal to three, we have that:

1) If X satisfies the Fredholm perturbation property (every elliptic pseudodifferential b-operator on X can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of X vanishes, i.e., χ 0 ( X ) = 0 .

2) If the even periodic conormal homology group vanishes, i.e., H 0 pcn ( X ) = 0 , then X satisfies the stably homotopic Fredholm perturbation property (i.e., every elliptic pseudodifferential b-operator on X satisfies the same named property up to stable homotopy among elliptic operators).

3) If H 0 pcn ( X ) is torsion free and if the even Euler corner character of X vanishes, i.e., χ 0 ( X ) = 0 , then X satisfies the stably homotopic Fredholm perturbation property. For example, for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free.

The main theorem behind the above result is the explicit computation in terms of conormal homology of the K-theory groups of the algebra K b ( X ) of b-compact operators for X as above. Our computation unifies the known cases of codimension zero (smooth manifolds) and of codimension one (smooth manifolds with boundary).

Article information

Source
Ann. K-Theory, Volume 3, Number 3 (2018), 523-563.

Dates
Received: 6 July 2017
Revised: 16 February 2018
Accepted: 7 March 2018
First available in Project Euclid: 24 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.akt/1532397767

Digital Object Identifier
doi:10.2140/akt.2018.3.523

Mathematical Reviews number (MathSciNet)
MR3830201

Zentralblatt MATH identifier
06911676

Subjects
Primary: 19K56: Index theory [See also 58J20, 58J22] 58H05: Pseudogroups and differentiable groupoids [See also 22A22, 22E65]

Keywords
Index theory K-theory Manifolds with corners Lie groupoids

Citation

Carrillo Rouse, Paulo; Lescure, Jean-Marie. Geometric obstructions for Fredholm boundary conditions for manifolds with corners. Ann. K-Theory 3 (2018), no. 3, 523--563. doi:10.2140/akt.2018.3.523. https://projecteuclid.org/euclid.akt/1532397767


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References

  • C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L'Enseignement Mathématique 36, L'Enseignement Mathématique, Geneva, 2000.
  • M. F. Atiyah and I. M. Singer, “The index of elliptic operators, I”, Ann. of Math. $(2)$ 87 (1968), 484–530.
  • M. F. Atiyah and I. M. Singer, “The index of elliptic operators, III”, Ann. of Math. $(2)$ 87 (1968), 546–604.
  • M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian geometry, I”, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.
  • U. Bunke, Index theory, eta forms, and Deligne cohomology, Mem. Amer. Math. Soc. 928, Amer. Math. Soc., Providence, RI, 2009.
  • P. Carrillo Rouse, “A Schwartz type algebra for the tangent groupoid”, pp. 181–199 in \Kdsh/theory and noncommutative geometry, edited by G. Cortiñas et al., Eur. Math. Soc., Zürich, 2008.
  • P. Carrillo Rouse, J. M. Lescure, and B. Monthubert, “A cohomological formula for the Atiyah–Patodi–Singer index on manifolds with boundary”, J. Topol. Anal. 6:1 (2014), 27–74.
  • C. Debord and J.-M. Lescure, “Index theory and groupoids”, pp. 86–158 in Geometric and topological methods for quantum field theory (Villa de Leyva, Columbia, 2007), edited by H. Ocampo et al., Cambridge Univ. Press, 2010.
  • C. Debord, J.-M. Lescure, and F. Rochon, “Pseudodifferential operators on manifolds with fibred corners”, Ann. Inst. Fourier $($Grenoble$)$ 65:4 (2015), 1799–1880.
  • A. Douady, “Variétés à bord anguleux et voisinages tubulaires”, pp. [exposé] 1, pp. 1–11 in Topologie différentielle, Séminaire Henri Cartan, 1961/1962 14, Secrétariat mathématique, Paris, 1961/1962.
  • M. Hilsum and G. Skandalis, “Stabilité des $C\sp{\ast}`` $-algèbres de feuilletages”, Ann. Inst. Fourier $($Grenoble$)$ 33:3 (1983), 201–208.
  • L. Hörmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Math. Wissenschaften 274, Springer, 1985.
  • A. Kono and D. Tamaki, Generalized cohomology, Translations of Mathematical Monographs 230, Amer. Math. Soc., Providence, RI, 2006.
  • R. Lauter, B. Monthubert, and V. Nistor, “Pseudodifferential analysis on continuous family groupoids”, Doc. Math. 5 (2000), 625–655.
  • J.-M. Lescure, D. Manchon, and S. Vassout, “About the convolution of distributions on groupoids”, J. Noncommut. Geom. 11:2 (2017), 757–789.
  • P. Loya, “The index of $b$-pseudodifferential operators on manifolds with corners”, Ann. Global Anal. Geom. 27:2 (2005), 101–133.
  • R. B. Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics 4, A K Peters, Wellesley, MA, 1993.
  • R. Melrose and V. Nistor, “\Kdsh/theory of $C^*``$-algebras of $b$-pseudodifferential operators”, Geom. Funct. Anal. 8:1 (1998), 88–122.
  • R. B. Melrose and P. Piazza, “Analytic \Kdsh/theory on manifolds with corners”, Adv. Math. 92:1 (1992), 1–26.
  • R. B. Melrose and P. Piazza, “Families of Dirac operators, boundaries and the $b$-calculus”, J. Differential Geom. 46:1 (1997), 99–180.
  • R. B. Melrose and P. Piazza, “An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary”, J. Differential Geom. 46:2 (1997), 287–334.
  • R. Melrose and F. Rochon, “Index in \Kdsh/theory for families of fibred cusp operators”, \Kdsh/Theory 37:1-2 (2006), 25–104.
  • B. Monthubert, “Groupoids and pseudodifferential calculus on manifolds with corners”, J. Funct. Anal. 199:1 (2003), 243–286.
  • B. Monthubert and V. Nistor, “A topological index theorem for manifolds with corners”, Compos. Math. 148:2 (2012), 640–668.
  • B. Monthubert and F. Pierrot, “Indice analytique et groupoï des de Lie”, C. R. Acad. Sci. Paris Sér. I Math. 325:2 (1997), 193–198.
  • V. Nazaikinskii, A. Savin, and B. Sternin, “Elliptic theory on manifolds with corners, I: Dual manifolds and pseudodifferential operators”, pp. 183–206 in $C^\ast$-algebras and elliptic theory, II (Będlewo, Poland, 2006), edited by D. Burghelea et al., Birkhäuser, Basel, 2008.
  • V. Nazaikinskii, A. Savin, and B. Sternin, “Elliptic theory on manifolds with corners, II: Homotopy classification and \Kdsh/homology”, pp. 207–226 in $C^\ast$-algebras and elliptic theory, II (Będlewo, Poland, 2006), edited by D. Burghelea et al., Birkhäuser, Basel, 2008.
  • V. E. Nazaikinskii, A. Y. Savin, and B. Y. Sternin, “The Atiyah–Bott index on stratified manifolds”, Sovrem. Mat. Fundam. Napravl. 34 (2009), 100–108. In Russian; translated in J. Math. Sci. New York 170:2 (2010), 229–237.
  • V. Nistor, “An index theorem for gauge-invariant families: the case of solvable groups”, Acta Math. Hungar. 99:1-2 (2003), 155–183.
  • V. Nistor, A. Weinstein, and P. Xu, “Pseudodifferential operators on differential groupoids”, Pacific J. Math. 189:1 (1999), 117–152.
  • A. L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics 170, Birkhäuser, Boston, 1999.
  • J. Renault, A groupoid approach to $C\sp{\ast}`` $-algebras, Lecture Notes in Mathematics 793, Springer, 1980.
  • A. Savin, “Elliptic operators on manifolds with singularities and \Kdsh/homology”, \Kdsh/Theory 34:1 (2005), 71–98.
  • C. Schochet, “Topological methods for $C\sp{\ast}`` $-algebras, I: Spectral sequences”, Pacific J. Math. 96:1 (1981), 193–211.
  • S. Vassout, “Unbounded pseudodifferential calculus on Lie groupoids”, J. Funct. Anal. 236:1 (2006), 161–200.