An $L^2$-Cheeger Müller theorem on compact manifolds with boundary

Benjamin Waßermann

doi:10.5802/ambp.400

L^{2}

-Cheeger Müller theorem on compact manifolds with boundary

Benjamin Waßermann ¹

¹ Karlsruher Institut für Technologie Fakultät für Mathematik Institut für Algebra und Geometrie Englerstr. 2 Mathebau (20.30) 76131 Karlsruhe Germany

Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 1, pp. 71-116.

Résumé

We generalize a Cheeger–Müller type theorem for flat, unitary bundles on infinite covering spaces over manifolds with boundary, proven by Burghelea, Friedlander and Kappeller. Employing recent anomaly results by Brüning, Ma and Zhang, we prove an analogous statement for a general flat bundle that is only required to have a unimodular restriction to the boundary.

Publié le : 2022-01-21

DOI : 10.5802/ambp.400

Affiliations des auteurs :

Benjamin Waßermann ¹

¹ Karlsruher Institut für Technologie Fakultät für Mathematik Institut für Algebra und Geometrie Englerstr. 2 Mathebau (20.30) 76131 Karlsruhe Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AMBP_2021__28_1_71_0,
     author = {Benjamin Wa{\ss}ermann},
     title = {An $L^2${-Cheeger} {M\"uller} theorem on compact manifolds with boundary},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {71--116},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {28},
     number = {1},
     year = {2021},
     doi = {10.5802/ambp.400},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.400/}
}

TY  - JOUR
AU  - Benjamin Waßermann
TI  - An $L^2$-Cheeger Müller theorem on compact manifolds with boundary
JO  - Annales mathématiques Blaise Pascal
PY  - 2021
SP  - 71
EP  - 116
VL  - 28
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.400/
DO  - 10.5802/ambp.400
LA  - en
ID  - AMBP_2021__28_1_71_0
ER  -

%0 Journal Article
%A Benjamin Waßermann
%T An $L^2$-Cheeger Müller theorem on compact manifolds with boundary
%J Annales mathématiques Blaise Pascal
%D 2021
%P 71-116
%V 28
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.400/
%R 10.5802/ambp.400
%G en
%F AMBP_2021__28_1_71_0

Benjamin Waßermann. An $L^2$-Cheeger Müller theorem on compact manifolds with boundary. Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 1, pp. 71-116. doi : 10.5802/ambp.400. https://ambp.centre-mersenne.org/articles/10.5802/ambp.400/

Bibliographie
Cité par

[1] Miklos Abert; Nicolas Bergeron; Ian Biringer; Tsachik Gelander; Nikolay Nikolov; Jean Raimbault; Iddo Samet On the growth of $L^{2}$ -invariants for sequences of lattices in Lie groups, Ann. Math., Volume 185 (2017) no. 3, pp. 711-790 | MR | Zbl

[2] Michael F. Atiyah Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsai, 1974) (Astérisque), Volume 32-33, Société Mathématique de France, 1976, pp. 43-72 | Numdam | Zbl

[3] Augustin Banyaga; David Hurtubise Lectures on Morse homology, Kluwer Texts in the Mathematical Sciences, 29, Kluwer Academic Publishers, 2004, x+324 pages | DOI

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

[5] Jean-Michel Bismut; Henri Gillet; Christophe Soulé Complex immersions and Arakelov geometry, The Grothendieck Festschrift (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 249-331 | MR | Zbl

[6] Jean-Michel Bismut; Weiping Zhang An extension of a theorem by Cheeger and Müller. With an appendix by Francois Laudenbach., Astérisque, 205, Société Mathématique de France, 1992, 235 pages | Numdam

[7] Maxim Braverman; Alan Carey; Michael Farber; Varghese Mathai $L^{2}$ -torsion without the determinant class condition and extended $L^{2}$ cohomology, Commun. Contemp. Math., Volume 7 (2005) no. 4, pp. 421-462 | DOI | MR | Zbl

[8] Jochen Brüning; Xiaonan Ma An anomaly formula for Ray-Singer metrics on manifolds with boundary, C. R. Math. Acad. Sci. Paris, Volume 335 (2002) no. 7, pp. 603-608 | DOI | MR | Zbl

[9] Jochen Brüning; Xiaonan Ma On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 1-2, pp. 1085-1117 | DOI | MR | Zbl

[10] Dan Burghelea; Leonid Friedlander; Thomas Kappeler Torsions for manifolds with boundary and glueing formulas, Math. Nachr., Volume 208 (1999), pp. 31-91 | DOI | MR | Zbl

[11] Dan Burghelea; Leonid Friedlander; Thomas Kappeler Relative Torsion, Commun. Contemp. Math., Volume 3 (2001) no. 1, pp. 15-85 | DOI | MR | Zbl

[12] Dan Burghelea; Leonid Friedlander; Thomas Kappeler; Patrick Macdonald Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal., Volume 6 (1996) no. 5, pp. 751-859 | DOI | MR | Zbl

[13] Alan L. Carey; Varghese Mathai $L^{2}$ -torsion invariants, J. Funct. Anal., Volume 110 (1992) no. 2, pp. 377-409 | DOI | MR | Zbl

[14] Thomas A. Chapman Topological invariance of the Whitehead torsion, Am. J. Math., Volume 96 (1974), pp. 488-497 | DOI | MR | Zbl

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

[16] Jozeg Dodziuk de Rham–Hodge theory for $L^{2}$ -cohomology of infinite coverings, Topology, Volume 16 (1977) no. 2, pp. 157-165 | DOI | MR | Zbl

[17] Mikhael Gromov; Mikhail A. Shubin Von Neumann spectra near zero, Geom. Funct. Anal., Volume 1 (1991) no. 4, pp. 375-404 | DOI | MR | Zbl

[18] Andrew Hassell Analytic surgery and analytic torsion, Commun. Anal. Geom., Volume 6 (1998) no. 2, pp. 255-289 | DOI | MR | Zbl

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

[20] Wolfgang Lück $L^{2}$ -invariants: Theory and applications to geometry and $K$ -theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages

[21] Wolfgang Lück; Thomas Schick $L^{2}$ -torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 518-567 | DOI | MR | Zbl

[22] Xiaonan Ma; Weiping Zhang An anomaly formula for $L^{2}$ -analytic torsions on manifolds with boundary, Analysis, geometry and topology of elliptic operators, World Scientific, 2006, pp. 235-262 | Zbl

[23] Varghese Mathai $L^{2}$ -analytic torsion, J. Funct. Anal., Volume 107 (1992) no. 2, pp. 369-386 | DOI | Zbl

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

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

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

[27] Werner Müller; Jonathan Pfaff 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

[28] Werner Müller; Frédéric Rochon Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps (2019) (https://arxiv.org/abs/1903.06199)

[29] Lizhen Qin On moduli spaces and CW Structures Arising from Morse Theory On Hilbert Manifolds, J. Topol. Anal., Volume 2 (2010) no. 4, pp. 469-526 | MR | Zbl

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

[31] Thomas Schick Analysis and Geometry of Boundary Manifolds of Bounded Geometry (1998) (https://arxiv.org/abs/math/9810107)

[32] Mikhail A. Shubin De Rham Theorem for extended $L^{2}$ -cohomology, Voronezh winter mathematical schools. Dedicated to Selim Krein (Translations), Volume 2, American Mathematical Society, 1998, pp. 217-231 | MR | Zbl

[33] S. M. Vishik Analytic torsion of boundary value problems, Dokl. Akad. Nauk SSSR, Volume 300 (1987) no. 6, pp. 1293-1298 | Zbl

[34] Benjamin Waßermann The $L^{2}$ -Cheeger–Müller Theorem for Representations of Hyperbolic Lattices, Ph. D. Thesis, Karlsruhe Institute for Technology (2020)

[35] Weiping Zhang An extended Cheeger–Müller theorem for covering spaces, Topology, Volume 44 (2005) no. 6, pp. 1093-1131 | DOI | Zbl

Cité par Sources :