A Morse-Bott type complex and the Bismut–Zhang torsion for intersection cohomology
[Un complexe de Morse–Bott et la torsion de Bismut–Zhang en cohomologie d’intersection]
Ludwig, Ursula1
1 University of Münster, Mathematics Münster, Einsteinstrasse 62, 48149 Münster (Germany)
Annales de l'Institut Fourier, Online first, 68 p.

Dans la première partie de cet article on construit, en utilisant la théorie de Morse, pour une pseudovariété stratifiée et une perversité au sens de la théorie de Goresky et MacPherson un complexe cohomologique. Ce complexe calcule la cohomologie d’intersection de la pseudovariété. Dans la deuxième partie on utilise ce complexe ainsi que le Laplacien de Witten pour définir la torsion de Bismut–Zhang, qui, conjecturellement, va servir dans un théorème de Cheeger–Müller pour des pseudovariétés à singularités coniques.

In the first part of this article we establish, for a compact pseudomanifold and a given perversity in the sense of Goresky and MacPherson a Morse theoretical cochain complex, which computes the intersection cohomology of the space. In the second part we use this cochain complex as well as the model Witten Laplacian to define the Bismut–Zhang torsion of a pseudomanifold. Conjecturally the Bismut–Zhang torsion will serve as the “topological” side in a Cheeger–Müller theorem for spaces with iterated conical singularities.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3648
Classification : 58J52, 37B30, 55N33
Keywords: Topological and analytic torsion, Morse–Bott complex, intersection cohomology.
Mot clés : Torsion analytic et topologique, complexe de Morse–Bott, cohomologie d’intersection.

Ludwig, Ursula 1

1 University of Münster, Mathematics Münster, Einsteinstrasse 62, 48149 Münster (Germany) 
@unpublished{AIF_0__0_0_A112_0,
     author = {Ludwig, Ursula},
     title = {A {Morse-Bott} type complex and the {Bismut{\textendash}Zhang} torsion for intersection cohomology},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3648},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Ludwig, Ursula
TI  - A Morse-Bott type complex and the Bismut–Zhang torsion for intersection cohomology
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3648
LA  - en
ID  - AIF_0__0_0_A112_0
ER  - 
%0 Unpublished Work
%A Ludwig, Ursula
%T A Morse-Bott type complex and the Bismut–Zhang torsion for intersection cohomology
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3648
%G en
%F AIF_0__0_0_A112_0
Ludwig, Ursula. A Morse-Bott type complex and the Bismut–Zhang torsion for intersection cohomology. Annales de l'Institut Fourier, Online first, 68 p.

[1] Albin, Pierre On the Hodge theory of stratified spaces, Hodge theory and L 2 -analysis (Advanced Lectures in Mathematics), Volume 39, International Press, 2017, pp. 1-78 | MR | Zbl

[2] Albin, Pierre; Rochon, Frédéric; Sher, David Analytic torsion and R-torsion of Witt representations on manifolds with cusps, Duke Math. J., Volume 167 (2018) no. 10, pp. 1883-1950 | DOI | MR | Zbl

[3] Albin, Pierre; Rochon, Frédéric; Sher, David Resolvent, heat kernel, and torsion under degeneration to fibered cusps, Memoirs of the American Mathematical Society, 1314, American Mathematical Society, 2021 | DOI | Zbl

[4] Albin, Pierre; Rochon, Frédéric; Sher, David A Cheeger–Müller theorem for manifolds with wedge singularities, Anal. PDE, Volume 15 (2022) no. 3, pp. 567-642 | DOI | Zbl

[5] Álvarez López, Jesús A.; Calaza, Manuel Witten’s perturbation on strata, Asian J. Math., Volume 21 (2017) no. 1, pp. 47-125 | DOI | MR | Zbl

[6] Austin, David M.; Braam, Peter J. Morse–Bott theory and equivariant cohomology, The Floer memorial volume (Progress in Mathematics), Volume 133, Birkhäuser, 1995, pp. 123-183 | DOI | MR | Zbl

[7] Banyaga, Augustin; Hurtubise, David E. Morse-Bott homology, Trans. Am. Math. Soc., Volume 362 (2010) no. 8, pp. 3997-4043 | DOI | MR | Zbl

[8] Bergeron, Nicolas; Venkatesh, Akshay The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Volume 12 (2013) no. 2, pp. 391-447 | DOI | MR | Zbl

[9] Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller, Astérisque, 205, Société Mathématique de France, 1992, 235 pages (with an appendix by François Laudenbach) | MR

[10] Bismut, Jean-Michel; Zhang, Weiping Milnor and Ray–Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 136-212 | DOI | MR | Zbl

[11] Brasselet, Jean Paul; Hector, Gilbert; Saralegi, Martin Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom., Volume 9 (1991) no. 3, pp. 211-243 | DOI | MR | Zbl

[12] Brüning, Jochen; Ma, Xiaonan An anomaly formula for Ray–Singer metrics on manifolds with boundary, Geom. Funct. Anal., Volume 16 (2006) no. 4, pp. 767-837 | DOI | MR | Zbl

[13] Brüning, Jochen; Ma, Xiaonan On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 3-4, pp. 1085-1117 | DOI | MR | Zbl

[14] Brylinski, Jean-Luc Equivariant intersection cohomology, Kazhdan-Lusztig theory and related topics. Proceedings of an AMS special session, held May 19-20, 1989 at the University of Chicago, Lake Shore Campus, Chicago, IL, USA (Contemporary Mathematics), Volume 139, American Mathematical Society, 1992, pp. 5-32 | DOI | MR | Zbl

[15] Calegari, Frank; Venkatesh, Akshay A torsion Jacquet–Langlands correspondence, Astérisque, 409, Société Mathématique de France, 2019, x+226 pages | DOI | Zbl

[16] Cheeger, Jeff Analytic torsion and the heat equation, Ann. Math., Volume 109 (1979) no. 2, pp. 259-322 | DOI | MR | Zbl

[17] Cheeger, Jeff On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proceedings of Symposia in Pure Mathematics), Volume XXXVI, American Mathematical Society, 1980, pp. 91-146 | Zbl

[18] Cheeger, Jeff; Goresky, Mark; MacPherson, Robert L 2 -cohomology and intersection homology of singular algebraic varieties, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 303-340 | Zbl

[19] Dar, Aparna Intersection R-torsion and analytic torsion for pseudomanifolds, Math. Z., Volume 194 (1987) no. 2, pp. 193-216 | DOI | MR | Zbl

[20] Debord, Claire; Lescure, Jean-Marie K-duality for stratified pseudomanifolds, Geom. Topol., Volume 13 (2009) no. 1, pp. 49-86 | DOI | MR | Zbl

[21] Debord, Claire; Lescure, Jean-Marie; Rochon, Frédéric Opérateurs pseudodifférentiels sur les variétés à coins fibrés, Ann. Inst. Fourier, Volume 65 (2015) no. 4, pp. 1799-1880 | DOI | Zbl

[22] Floer, Andreas Witten’s complex and infinite-dimensional Morse theory, J. Differ. Geom., Volume 30 (1989) no. 1, pp. 207-221 | DOI | MR | Zbl

[23] Franz, Wolfgang Über die Torsion einer Überdeckung, J. Reine Angew. Math., Volume 173 (1935), pp. 245-254 | DOI | MR | Zbl

[24] Frauenfelder, Urs The Arnold–Givental conjecture and moment Floer homology, Int. Math. Res. Not. (2004) no. 42, pp. 2179-2269 | DOI | MR | Zbl

[25] Freed, Daniel S. Reidemeister torsion, spectral sequences, and Brieskorn spheres, J. Reine Angew. Math., Volume 429 (1992), pp. 75-89 | DOI | MR | Zbl

[26] Friedman, Greg Polynomial invariants of non-locally flat knots, Ph. D. Thesis, New York University (2001)

[27] Friedman, Greg Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata, Topology Appl., Volume 134 (2003) no. 2, pp. 69-109 | DOI | Zbl

[28] Friedman, Greg Intersection homology of stratified fibrations and neighborhoods, Adv. Math., Volume 215 (2007) no. 1, pp. 24-65 | DOI | Zbl

[29] Fukaya, Kenji Floer homology of connected sum of homology 3-spheres, Topology, Volume 35 (1996) no. 1, pp. 89-136 | DOI | MR | Zbl

[30] Goresky, Mark; MacPherson, Robert Intersection homology theory, Topology, Volume 19 (1980) no. 2, pp. 135-162 | DOI | MR | Zbl

[31] Goresky, Mark; MacPherson, Robert Intersection homology. II, Invent. Math., Volume 72 (1983) no. 1, pp. 77-129 | DOI | MR | Zbl

[32] Goresky, Mark; MacPherson, Robert Stratified Morse theory, Singularities, Part 1 (Arcata, Calif., 1981) (Proceedings of Symposia in Pure Mathematics), Volume 40, American Mathematical Society, 1983, pp. 517-533 | MR | Zbl

[33] Goresky, Mark; MacPherson, Robert Simplicial Intersection Homology, Appendix to R. MacPherson & K. Vilonen, “Elementary construction of perverse sheaves”, Invent. Math. 84, p. 403-435, 1986

[34] Goresky, Mark; Siegel, Paul Linking pairings on singular spaces, Comment. Math. Helv., Volume 58 (1983), pp. 96-110 | DOI | Zbl

[35] Hartmann, Luiz; Spreafico, Mauro The analytic torsion of a cone over a sphere, J. Math. Pures Appl., Volume 93 (2010) no. 4, pp. 408-435 | DOI | MR | Zbl

[36] Hartmann, Luiz; Spreafico, Mauro The analytic torsion of a cone over an odd dimensional manifold, J. Geom. Phys., Volume 61 (2011) no. 3, pp. 624-657 | DOI | MR | Zbl

[37] Hartmann, Luiz; Spreafico, Mauro Intersection torsion and analytic torsion of spaces with conical singularities (2020) (https://arxiv.org/abs/2001.07801)

[38] Helffer, Bernard; Sjöstrand, Johannes Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Commun. Partial Differ. Equations, Volume 10 (1985) no. 3, pp. 245-340 | DOI | MR | Zbl

[39] Hutchings, Michael Floer homology of families. I, Algebr. Geom. Topol., Volume 8 (2008) no. 1, pp. 435-492 | DOI | MR | Zbl

[40] Kirwan, Frances; Woolf, Jonathan An introduction to intersection homology theory, Chapman & Hall/CRC, 2006, xiv+229 pages | DOI | MR | Zbl

[41] Latschev, Janko Gradient flows of Morse–Bott functions, Math. Ann., Volume 318 (2000) no. 4, pp. 731-759 | DOI | MR | Zbl

[42] Laudenbach, François On the Thom–Smale complex, Appendix to [9], 1992

[43] Lesch, Matthias Determinats of regular singular Sturm–Liouville operators, Math. Nachr., Volume 194 (1998), pp. 139-170 | DOI | Zbl

[44] Lesch, Matthias A gluing formula for the analytic torsion on singular spaces, Anal. PDE, Volume 6 (2013) no. 1, pp. 221-256 | DOI | MR | Zbl

[45] Lott, John; Rothenberg, Mel Analytic torsion for group actions, J. Differ. Geom., Volume 34 (1991) no. 2, pp. 431-481 | DOI | MR | Zbl

[46] Lück, Wolfgang Analytic and topological torsion for manifolds with boundary and symmetry, J. Differ. Geom., Volume 37 (1993) no. 2, pp. 263-322 | DOI | MR | Zbl

[47] Lück, Wolfgang; Schick, Thomas; Thielmann, Thomas Torsion and fibrations, J. Reine Angew. Math., Volume 498 (1998), pp. 1-33 | DOI | MR | Zbl

[48] Ludwig, Ursula Morse–Smale–Witten complex for gradient-like vector fields on stratified spaces, Singularity theory, World Scientific, 2007, pp. 683-713 | DOI | MR | Zbl

[49] Ludwig, Ursula Comparison between two complexes on a singular space, J. Reine Angew. Math., Volume 724 (2017), pp. 1-52 | DOI | MR | Zbl

[50] Ludwig, Ursula A complex in Morse theory computing intersection homology, Ann. Inst. Fourier, Volume 67 (2017) no. 1, pp. 197-236 | DOI | Numdam | MR | Zbl

[51] Ludwig, Ursula An extension of a theorem by Cheeger and Müller to spaces with isolated conical singularities, Duke Math. J., Volume 169 (2020) no. 13, pp. 2501-2570 | DOI | MR | Zbl

[52] Ludwig, Ursula An index formula for the intersection Euler characteristic of an infinite cone, Math. Z., Volume 296 (2020) no. 1-2, pp. 99-126 | DOI | MR | Zbl

[53] Mather, John N. Notes on Topological Stability, Mimeographed Notes, Harvard, 1970

[54] Mather, John N. Notes on topological stability, Bull. Am. Math. Soc., Volume 49 (2012) no. 4, pp. 475-506 | DOI | Zbl

[55] Matz, Jasmin; Müller, Werner Analytic torsion of arithmetic quotients of the symmetric space SL(n,ℝ)/SO(n), Geom. Funct. Anal., Volume 27 (2017) no. 6, pp. 1378-1449 | DOI | MR | Zbl

[56] Matz, Jasmin; Müller, Werner Approximation of L 2 -analytic torsion for arithmetic quotients of the symmetric space SL(n,ℝ)/SO(n), J. Inst. Math. Jussieu, Volume 19 (2020) no. 2, pp. 307-350 | DOI | MR | Zbl

[57] Mazzeo, Rafe; Vertman, Boris Analytic torsion on manifolds with edges, Adv. Math., Volume 231 (2012) no. 2, pp. 1000-1040 | DOI | MR | Zbl

[58] McCleary, John User’s guide to spectral sequences, Mathematics Lecture Series, 12, Publish or Perish Inc., 1985, xiv+423 pages | MR | Zbl

[59] Milnor, John W. Whitehead torsion, Bull. Am. Math. Soc., Volume 72 (1966), pp. 358-426 | DOI | MR | Zbl

[60] Müller, Werner Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305 | DOI | MR | Zbl

[61] Müller, Werner Analytic torsion and R-torsion for unimodular representations, J. Am. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | DOI | MR | Zbl

[62] Müller, Werner; Pfaff, Jonathan Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal., Volume 263 (2012) no. 9, pp. 2615-2675 | DOI | MR | Zbl

[63] Müller, Werner; Pfaff, Jonathan The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume, J. Funct. Anal., Volume 267 (2014) no. 8, pp. 2731-2786 | DOI | MR | Zbl

[64] Müller, Werner; Rochon, Frédéric Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps, Geom. Funct. Anal., Volume 30 (2020) no. 3, pp. 910-954 | DOI | Zbl

[65] Müller, Werner; Rochon, Frédéric Exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds, Math. Z., Volume 298 (2021) no. 1-2, pp. 79-106 | DOI | Zbl

[66] Müller, Werner; Vertman, Boris The metric anomaly of analytic torsion on manifolds with conical singularities, Commun. Partial Differ. Equations, Volume 39 (2014) no. 1, pp. 146-191 | DOI | MR | Zbl

[67] Oancea, Alexandru La suite spectrale de Leray-Serre en homologie de Floer des variétés symplectiques compactes à bord de type contact, Ph. D. Thesis, Université Paris Sud (2003)

[68] Pfaff, Jonathan Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z., Volume 277 (2014) no. 3-4, pp. 953-974 | DOI | MR | Zbl

[69] Pfaff, Jonathan Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups, Ann. Global Anal. Geom., Volume 45 (2014) no. 4, pp. 267-285 | DOI | MR | Zbl

[70] Pfaff, Jonathan A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, J. Inst. Math. Jussieu, Volume 16 (2017) no. 4, pp. 673-743 | DOI | MR | Zbl

[71] du Plessis, Andrew; Wall, Terry The geometry of topological stability, London Mathematical Society Monographs. New Series, 9, Clarendon Press, 1995 (Oxford Science Publications) | DOI | Zbl

[72] Raimbault, Jean Asymptotics of analytic torsion for hyperbolic three-manifolds, Comment. Math. Helv., Volume 94 (2019) no. 3, pp. 459-531 | DOI | MR | Zbl

[73] Ray, Daniel B.; Singer, Isadore M. R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210 | DOI | MR | Zbl

[74] Reidemeister, Kurt Homotopieringe und Linsenräume, Abh. Math. Semin. Univ. Hamb., Volume 11 (1935) no. 1, pp. 102-109 | DOI | MR | Zbl

[75] Schwartz, Marie-Hélène Classes caractéristiques definies par une stratification d’une variété analytique complexe, C. R. Acad. Sci. Paris, Volume 260 (1965), p. 3262-3264, 3535–3537 | Zbl

[76] Shen, Shu Laplacien hypoelliptique, torsion analytique, et théorème de Cheeger–Müller, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 2817-2999 | DOI | MR | Zbl

[77] Siegel, Paul Witt spaces: A geometric cycle theory for KO-homology at odd primes, Am. J. Math., Volume 105 (1983), pp. 1067-1105 hdl.handle.net/1721.1/91309 | DOI | Zbl

[78] Smale, Stephen On gradient dynamical systems, Ann. Math., Volume 74 (1961), pp. 199-206 | DOI | MR | Zbl

[79] Thom, René Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris, Volume 228 (1949), pp. 973-975 | MR | Zbl

[80] Thom, René Ensembles et morphismes stratifiés, Bull. Am. Math. Soc., Volume 75 (1969), pp. 240-284 | DOI | MR | Zbl

[81] Verona, Andrei Stratified mappings – structure and triangulability, Lecture Notes in Mathematics, 1102, Springer, 1984, ix+160 pages | DOI | MR | Zbl

[82] Vertman, Boris Analytic torsion of a bounded generalized cone, Commun. Math. Phys., Volume 290 (2009) no. 3, pp. 813-860 | DOI | MR | Zbl

[83] Vishik, S. M. Generalized Ray–Singer conjecture. I. A manifold with a smooth boundary, Commun. Math. Phys., Volume 167 (1995) no. 1, pp. 1-102 | DOI | MR | Zbl

[84] Witten, Edward Supersymmetry and Morse theory, J. Differ. Geom., Volume 17 (1982) no. 4, pp. 661-692 | DOI | MR | Zbl

Cité par Sources :