The signature package on Witt spaces

Pierre Albin; Éric Leichtnam; Rafe Mazzeo; Paolo Piazza

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 2, page 241-310
  • ISSN: 0012-9593

Abstract

top
In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of  X —is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C r * Γ Mishchenko bundle associated to any Galois covering of  X with covering group Γ , we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K -theory of  C r * Γ . We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of  X , defined through the homology L -class of  X , whenever the rational assembly map K * ( B Γ ) ⊗ ℚ → K * ( C r * Γ ) ⊗ ℚ is injective.

How to cite

top

Albin, Pierre, et al. "The signature package on Witt spaces." Annales scientifiques de l'École Normale Supérieure 45.2 (2012): 241-310. <http://eudml.org/doc/272219>.

@article{Albin2012,
abstract = {In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold $X$ which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of $X$—is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a $C^*_r\Gamma $ Mishchenko bundle associated to any Galois covering of $X$ with covering group $\Gamma $, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the $K$-theory of $C^*_r\Gamma $. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of $X$, defined through the homology $L$-class of $X$, whenever the rational assembly map $K_* (B\Gamma )\otimes \mathbb \{Q\}\rightarrow K_*(C^*_r \Gamma )\otimes \mathbb \{Q\}$ is injective.},
author = {Albin, Pierre, Leichtnam, Éric, Mazzeo, Rafe, Piazza, Paolo},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {stratified pseudomanifold; Witt condition; iterated conic metrics; signature operator; index class; higher signatures; stratified homotopy invariance; assembly map},
language = {eng},
number = {2},
pages = {241-310},
publisher = {Société mathématique de France},
title = {The signature package on Witt spaces},
url = {http://eudml.org/doc/272219},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Albin, Pierre
AU - Leichtnam, Éric
AU - Mazzeo, Rafe
AU - Piazza, Paolo
TI - The signature package on Witt spaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 2
SP - 241
EP - 310
AB - In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold $X$ which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of $X$—is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a $C^*_r\Gamma $ Mishchenko bundle associated to any Galois covering of $X$ with covering group $\Gamma $, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the $K$-theory of $C^*_r\Gamma $. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of $X$, defined through the homology $L$-class of $X$, whenever the rational assembly map $K_* (B\Gamma )\otimes \mathbb {Q}\rightarrow K_*(C^*_r \Gamma )\otimes \mathbb {Q}$ is injective.
LA - eng
KW - stratified pseudomanifold; Witt condition; iterated conic metrics; signature operator; index class; higher signatures; stratified homotopy invariance; assembly map
UR - http://eudml.org/doc/272219
ER -

References

top
  1. [1] P. Albin & R. B. Melrose, Resolution of smooth group actions, in Spectral Theory and Geometric Analysis, Northeastern University, 2009. Zbl1218.58009MR2560748
  2. [2] B. Ammann, R. Lauter & V. Nistor, Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math.165 (2007), 717–747. Zbl1133.58020MR2335795
  3. [3] S. Baaj & P. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les C * -modules hilbertiens, C. R. Acad. Sci. Paris Sér. I Math.296 (1983), 875–878. Zbl0551.46041MR715325
  4. [4] M. Banagl, Topological invariants of stratified spaces, Springer Monographs in Math., Springer, 2007. Zbl1108.55001MR2286904
  5. [5] M. Banagl, The signature of singular spaces and its refinement to generalized homology theory, in Proceedings of the MSRI Workshop “Topology of Stratified Spaces”, 2008. Zbl1236.55001
  6. [6] B. Blackadar, K -theory for operator algebras, second éd., Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press, 1998. Zbl0913.46054MR1656031
  7. [7] J.-P. Brasselet, G. Hector & M. Saralegi, Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom.9 (1991), 211–243. Zbl0733.57010MR1143404
  8. [8] J.-P. Brasselet & A. Legrand, Un complexe de formes différentielles à croissance bornée sur une variété stratifiée, Ann. Scuola Norm. Sup. Pisa Cl. Sci.21 (1994), 213–234. Zbl0839.55005MR1288365
  9. [9] J. Brüning & R. Seeley, An index theorem for first order regular singular operators, Amer. J. Math.110 (1988), 659–714. Zbl0664.58035MR955293
  10. [10] S. Chang, On conjectures of Mathai and Borel, Geom. Dedicata106 (2004), 161–167. Zbl1066.57034MR2079840
  11. [11] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A.76 (1979), 2103–2106. Zbl0411.58003MR530173
  12. [12] J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, in Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., 1980, 91–146. Zbl0461.58002MR573430
  13. [13] J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom.18 (1983), 575–657. Zbl0529.58034MR730920
  14. [14] S. J. Curran, Intersection homology and free group actions on Witt spaces, Michigan Math. J.39 (1992), 111–127. Zbl0755.57017MR1137893
  15. [15] T. Eppelmann, Signature homology and symmetric L-theory, Thèse, University of Heidelberg, 2007. Zbl1137.57305
  16. [16] S. C. Ferry, A. Ranicki & J. Rosenberg, A history and survey of the Novikov conjecture, in Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 226, Cambridge Univ. Press, 1995, 7–66. Zbl0954.57017MR1388295
  17. [17] G. Friedman, Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata, Topology Appl.134 (2003), 69–109. Zbl1032.55004MR2009092
  18. [18] G. Friedman & J. McClure, The symmetric signature of a Witt space, preprint, arXiv:1106.4798. Zbl1276.55010MR3062949
  19. [19] J. B. Gil & G. A. Mendoza, Adjoints of elliptic cone operators, Amer. J. Math.125 (2003), 357–408. Zbl1030.58012MR1963689
  20. [20] M. Goresky & R. MacPherson, Intersection homology theory, Topology19 (1980), 135–162. Zbl0448.55004MR572580
  21. [21] M. Goresky & R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983), 77–129. Zbl0529.55007MR696691
  22. [22] N. Higson, K-homology and operators on non compact manifolds, http://web.me.com/ndh2/math/Unpublished.html, 1989. 
  23. [23] M. Hilsum, Signature operator on Lipschitz manifolds and unbounded Kasparov bimodules, in Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math. 1132, Springer, 1985, 254–288. Zbl0602.46069MR799572
  24. [24] M. Hilsum, Fonctorialité en K -théorie bivariante pour les variétés lipschitziennes, K -Theory 3 (1989), 401–440. Zbl0702.57008MR1050489
  25. [25] M. Hilsum & G. Skandalis, Invariance par homotopie de la signature à coefficients dans un fibré presque plat, J. reine angew. Math. 423 (1992), 73–99. Zbl0731.55013MR1142484
  26. [26] B. Hughes & S. Weinberger, Surgery and stratified spaces, in Surveys on surgery theory, Vol. 2, Ann. of Math. Stud. 149, Princeton Univ. Press, 2001, 319–352. Zbl0982.57009MR1818777
  27. [27] E. Hunsicker & R. Mazzeo, Harmonic forms on manifolds with edges, Int. Math. Res. Not.2005 (2005), 3229–3272. Zbl1089.58007MR2186793
  28. [28] M. Karoubi, K -theory, Grundl. Math. Wiss. 226, Springer, 1978. Zbl0382.55002
  29. [29] G. Kasparov, Equivariant K K -theory and the Novikov conjecture, Invent. Math.91 (1988), 147–201. Zbl0647.46053MR918241
  30. [30] G. Kasparov, Novikov’s conjecture on higher signatures: the operator K -theory approach, in Representation theory of groups and algebras, Contemp. Math. 145, Amer. Math. Soc., 1993, 79–99. Zbl0788.19005MR1216182
  31. [31] F. Kirwan & J. Woolf, An introduction to intersection homology theory, second éd., Chapman & Hall/CRC, Boca Raton, FL, 2006. Zbl1106.55001MR2207421
  32. [32] Y. A. Kordyukov, L p -theory of elliptic differential operators on manifolds of bounded geometry, Acta Appl. Math.23 (1991), 223–260. Zbl0743.58030MR1120831
  33. [33] E. C. Lance, Hilbert C * -modules, London Mathematical Society Lecture Note Series 210, Cambridge Univ. Press, 1995. Zbl0822.46080MR1325694
  34. [34] E. Leichtnam, J. Lott & P. Piazza, On the homotopy invariance of higher signatures for manifolds with boundary, J. Differential Geom.54 (2000), 561–633. Zbl1032.58012MR1823315
  35. [35] E. Leichtnam & P. Piazza, The b -pseudodifferential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem, Mém. Soc. Math. Fr. 68 (1997). Zbl0942.58003MR1488084
  36. [36] E. Leichtnam & P. Piazza, Dirac index classes and the noncommutative spectral flow, J. Funct. Anal.200 (2003), 348–400. Zbl1030.58018MR1979016
  37. [37] E. Leichtnam & P. Piazza, Elliptic operators and higher signatures, Ann. Inst. Fourier (Grenoble) 54 (2004), 1197–1277. Zbl1069.58014MR2127848
  38. [38] E. Leichtnam & P. Piazza, Cut-and-paste on foliated bundles, in Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math. 366, Amer. Math. Soc., 2005, 151–192. Zbl1074.58010MR2114488
  39. [39] M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner Texte zur Mathematik 136, 1997. Zbl1156.58302MR1449639
  40. [40] J. N. Mather, Stratifications and mappings, in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, 1973, 195–232. Zbl0286.58003MR368064
  41. [41] R. Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), 1615–1664. Zbl0745.58045MR1133743
  42. [42] G. A. Meladze & M. A. Shubin, Algebras of pseudodifferential operators on unimodular Lie groups, Dokl. Akad. Nauk SSSR279 (1984), 542–545. Zbl0617.35138MR771745
  43. [43] R. B. Melrose, Pseudodifferential operators, corners and singular limits, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, 1991, 217–234. Zbl0743.58033MR1159214
  44. [44] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Math. 4, A K Peters Ltd., 1993. Zbl0796.58050MR1348401
  45. [45] A. S. Miščenko & A. T. Fomenko, The index of elliptic operators over C * -algebras, Izv. Akad. Nauk SSSR Ser. Mat.43 (1979), 831–859. Zbl0416.46052MR548506
  46. [46] A. Mishchenko, Homotopy invariants of non-simply connected manifolds. I. Rational invariants, Math. USSR, Izvestija 4 (1970), 509–519. Zbl0232.55015
  47. [47] H. Moscovici & F. Wu, Straight Chern character for Witt spaces, in Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, Amer. Math. Soc., 1997, 103–113. Zbl0891.19002MR1478705
  48. [48] E. Pedersen, J. Roe & S. Weinberger, On the homotopy invariance of the boundedly controlled analytic signature of a manifold over an open cone, London Math. Society Lecture Notes Series 227, 1993. Zbl0959.58036
  49. [49] M. J. Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Math. 1768, Springer, 2001. Zbl0988.58003MR1869601
  50. [50] P. Piazza & T. Schick, Bordism, rho-invariants and the Baum-Connes conjecture, J. Noncommut. Geom.1 (2007), 27–111. Zbl1158.58012MR2294190
  51. [51] J. Rosenberg, Analytic Novikov for topologists, in Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 226, Cambridge Univ. Press, 1995, 338–372. Zbl0955.57027MR1388305
  52. [52] J. Rosenberg & S. Weinberger, Higher G -signatures for Lipschitz manifolds, K -Theory 7 (1993), 101–132. Zbl0791.58004MR1235284
  53. [53] J. Rosenberg & S. Weinberger, The signature operator at 2, Topology45 (2006), 47–63. Zbl1103.58012MR2170494
  54. [54] B.-W. Schulze, The iterative structure of corner operators, preprint arXiv:0905.0977. 
  55. [55] P. H. Siegel, Witt spaces: a geometric cycle theory for K O -homology at odd primes, Amer. J. Math.105 (1983), 1067–1105. Zbl0547.57019MR714770
  56. [56] Y. P. Solovyov & E. V. Troitsky, C * -algebras and elliptic operators in differential topology, Translations of Mathematical Monographs 192, Amer. Math. Soc., 2001. Zbl0958.46038MR1787114
  57. [57] D. P. Sullivan, Geometric topology: localization, periodicity and Galois symmetry, K -Monographs in Math. 8, Springer, 2005. Zbl1078.55001MR2162361
  58. [58] R. Thom, Les classes caractéristiques de Pontrjagin des variétés triangulées, in Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, 54–67. Zbl0088.39201MR102071
  59. [59] A. Verona, Stratified mappings—structure and triangulability, Lecture Notes in Math. 1102, Springer, 1984. Zbl0543.57002MR771120
  60. [60] S. Weinberger, Homotopy invariance of η -invariants, Proc. Nat. Acad. Sci. U.S.A.85 (1988), 5362–5363. Zbl0659.57016MR952817
  61. [61] S. Weinberger, Higher ρ -invariants, in Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Contemp. Math. 231, Amer. Math. Soc., 1999, 315–320. Zbl0946.57037MR1707352

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.