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.
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.
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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
@unpublished{AIF_0__0_0_A112_0, author = {Ludwig, Ursula}, title = {A {Morse-Bott} type complex and the {Bismut{\textendash}Zhang} torsion for intersection cohomology},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}, }
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] On the Hodge theory of stratified spaces, Hodge theory and -analysis (Advanced Lectures in Mathematics), Volume 39, International Press, 2017, pp. 1-78 | MR | Zbl
[2] 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] Resolvent, heat kernel, and torsion under degeneration to fibered cusps, Memoirs of the American Mathematical Society, 1314, American Mathematical Society, 2021 | DOI | Zbl
[4] A Cheeger–Müller theorem for manifolds with wedge singularities, Anal. PDE, Volume 15 (2022) no. 3, pp. 567-642 | DOI | Zbl
[5] Witten’s perturbation on strata, Asian J. Math., Volume 21 (2017) no. 1, pp. 47-125 | DOI | MR | Zbl
[6] Morse–Bott theory and equivariant cohomology, The Floer memorial volume (Progress in Mathematics), Volume 133, Birkhäuser, 1995, pp. 123-183 | DOI | MR | Zbl
[7] Morse-Bott homology, Trans. Am. Math. Soc., Volume 362 (2010) no. 8, pp. 3997-4043 | DOI | MR | Zbl
[8] 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] 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] 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] Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom., Volume 9 (1991) no. 3, pp. 211-243 | DOI | MR | Zbl
[12] 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] On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 3-4, pp. 1085-1117 | DOI | MR | Zbl
[14] 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] A torsion Jacquet–Langlands correspondence, Astérisque, 409, Société Mathématique de France, 2019, x+226 pages | DOI | Zbl
[16] Analytic torsion and the heat equation, Ann. Math., Volume 109 (1979) no. 2, pp. 259-322 | DOI | MR | Zbl
[17] 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] -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] Intersection -torsion and analytic torsion for pseudomanifolds, Math. Z., Volume 194 (1987) no. 2, pp. 193-216 | DOI | MR | Zbl
[20] -duality for stratified pseudomanifolds, Geom. Topol., Volume 13 (2009) no. 1, pp. 49-86 | DOI | MR | Zbl
[21] Opérateurs pseudodifférentiels sur les variétés à coins fibrés, Ann. Inst. Fourier, Volume 65 (2015) no. 4, pp. 1799-1880 | DOI | Zbl
[22] Witten’s complex and infinite-dimensional Morse theory, J. Differ. Geom., Volume 30 (1989) no. 1, pp. 207-221 | DOI | MR | Zbl
[23] Über die Torsion einer Überdeckung, J. Reine Angew. Math., Volume 173 (1935), pp. 245-254 | DOI | MR | Zbl
[24] The Arnold–Givental conjecture and moment Floer homology, Int. Math. Res. Not. (2004) no. 42, pp. 2179-2269 | DOI | MR | Zbl
[25] Reidemeister torsion, spectral sequences, and Brieskorn spheres, J. Reine Angew. Math., Volume 429 (1992), pp. 75-89 | DOI | MR | Zbl
[26] Polynomial invariants of non-locally flat knots, Ph. D. Thesis, New York University (2001)
[27] 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] Intersection homology of stratified fibrations and neighborhoods, Adv. Math., Volume 215 (2007) no. 1, pp. 24-65 | DOI | Zbl
[29] Floer homology of connected sum of homology -spheres, Topology, Volume 35 (1996) no. 1, pp. 89-136 | DOI | MR | Zbl
[30] Intersection homology theory, Topology, Volume 19 (1980) no. 2, pp. 135-162 | DOI | MR | Zbl
[31] Intersection homology. II, Invent. Math., Volume 72 (1983) no. 1, pp. 77-129 | DOI | MR | Zbl
[32] 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] Simplicial Intersection Homology, Appendix to R. MacPherson & K. Vilonen, “Elementary construction of perverse sheaves”, Invent. Math. 84, p. 403-435, 1986
[34] Linking pairings on singular spaces, Comment. Math. Helv., Volume 58 (1983), pp. 96-110 | DOI | Zbl
[35] 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] 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] Intersection torsion and analytic torsion of spaces with conical singularities (2020) (https://arxiv.org/abs/2001.07801)
[38] 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] Floer homology of families. I, Algebr. Geom. Topol., Volume 8 (2008) no. 1, pp. 435-492 | DOI | MR | Zbl
[40] An introduction to intersection homology theory, Chapman & Hall/CRC, 2006, xiv+229 pages | DOI | MR | Zbl
[41] Gradient flows of Morse–Bott functions, Math. Ann., Volume 318 (2000) no. 4, pp. 731-759 | DOI | MR | Zbl
[42] On the Thom–Smale complex, Appendix to [9], 1992
[43] Determinats of regular singular Sturm–Liouville operators, Math. Nachr., Volume 194 (1998), pp. 139-170 | DOI | Zbl
[44] A gluing formula for the analytic torsion on singular spaces, Anal. PDE, Volume 6 (2013) no. 1, pp. 221-256 | DOI | MR | Zbl
[45] Analytic torsion for group actions, J. Differ. Geom., Volume 34 (1991) no. 2, pp. 431-481 | DOI | MR | Zbl
[46] 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] Torsion and fibrations, J. Reine Angew. Math., Volume 498 (1998), pp. 1-33 | DOI | MR | Zbl
[48] Morse–Smale–Witten complex for gradient-like vector fields on stratified spaces, Singularity theory, World Scientific, 2007, pp. 683-713 | DOI | MR | Zbl
[49] Comparison between two complexes on a singular space, J. Reine Angew. Math., Volume 724 (2017), pp. 1-52 | DOI | MR | Zbl
[50] A complex in Morse theory computing intersection homology, Ann. Inst. Fourier, Volume 67 (2017) no. 1, pp. 197-236 | DOI | Numdam | MR | Zbl
[51] 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] 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] Notes on Topological Stability, Mimeographed Notes, Harvard, 1970
[54] Notes on topological stability, Bull. Am. Math. Soc., Volume 49 (2012) no. 4, pp. 475-506 | DOI | Zbl
[55] Analytic torsion of arithmetic quotients of the symmetric space , Geom. Funct. Anal., Volume 27 (2017) no. 6, pp. 1378-1449 | DOI | MR | Zbl
[56] Approximation of -analytic torsion for arithmetic quotients of the symmetric space , J. Inst. Math. Jussieu, Volume 19 (2020) no. 2, pp. 307-350 | DOI | MR | Zbl
[57] Analytic torsion on manifolds with edges, Adv. Math., Volume 231 (2012) no. 2, pp. 1000-1040 | DOI | MR | Zbl
[58] User’s guide to spectral sequences, Mathematics Lecture Series, 12, Publish or Perish Inc., 1985, xiv+423 pages | MR | Zbl
[59] Whitehead torsion, Bull. Am. Math. Soc., Volume 72 (1966), pp. 358-426 | DOI | MR | Zbl
[60] Analytic torsion and -torsion of Riemannian manifolds, Adv. Math., Volume 28 (1978) no. 3, pp. 233-305 | DOI | MR | Zbl
[61] Analytic torsion and -torsion for unimodular representations, J. Am. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | DOI | MR | Zbl
[62] Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal., Volume 263 (2012) no. 9, pp. 2615-2675 | DOI | MR | Zbl
[63] 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] Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps, Geom. Funct. Anal., Volume 30 (2020) no. 3, pp. 910-954 | DOI | Zbl
[65] 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] 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] 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] 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] 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] 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] The geometry of topological stability, London Mathematical Society Monographs. New Series, 9, Clarendon Press, 1995 (Oxford Science Publications) | DOI | Zbl
[72] Asymptotics of analytic torsion for hyperbolic three-manifolds, Comment. Math. Helv., Volume 94 (2019) no. 3, pp. 459-531 | DOI | MR | Zbl
[73] -torsion and the Laplacian on Riemannian manifolds, Adv. Math., Volume 7 (1971), pp. 145-210 | DOI | MR | Zbl
[74] Homotopieringe und Linsenräume, Abh. Math. Semin. Univ. Hamb., Volume 11 (1935) no. 1, pp. 102-109 | DOI | MR | Zbl
[75] 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] Laplacien hypoelliptique, torsion analytique, et théorème de Cheeger–Müller, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 2817-2999 | DOI | MR | Zbl
[77] 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] On gradient dynamical systems, Ann. Math., Volume 74 (1961), pp. 199-206 | DOI | MR | Zbl
[79] 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] Ensembles et morphismes stratifiés, Bull. Am. Math. Soc., Volume 75 (1969), pp. 240-284 | DOI | MR | Zbl
[81] Stratified mappings – structure and triangulability, Lecture Notes in Mathematics, 1102, Springer, 1984, ix+160 pages | DOI | MR | Zbl
[82] Analytic torsion of a bounded generalized cone, Commun. Math. Phys., Volume 290 (2009) no. 3, pp. 813-860 | DOI | MR | Zbl
[83] 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] Supersymmetry and Morse theory, J. Differ. Geom., Volume 17 (1982) no. 4, pp. 661-692 | DOI | MR | Zbl
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