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

Waßermann, Benjamin

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

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.

[1] Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo 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] Atiyah, Michael F. 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] Banyaga, Augustin; Hurtubise, David Lectures on Morse homology, Kluwer Texts in the Mathematical Sciences, 29, Kluwer Academic Publishers, 2004, x+324 pages | DOI

[4] 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

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

[6] Bismut, Jean-Michel; Zhang, Weiping 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] Braverman, Maxim; Carey, Alan; Farber, Michael; Mathai, Varghese $L^2$-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] Brüning, Jochen; Ma, Xiaonan 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] Brüning, Jochen; Ma, Xiaonan On the gluing formula for the analytic torsion, Math. Z., Volume 273 (2013) no. 1-2, pp. 1085-1117 | DOI | MR | Zbl

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

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

[12] Burghelea, Dan; Friedlander, Leonid; Kappeler, Thomas; Macdonald, Patrick 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] Carey, Alan L.; Mathai, Varghese $L^2$-torsion invariants, J. Funct. Anal., Volume 110 (1992) no. 2, pp. 377-409 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[22] Ma, Xiaonan; Zhang, Weiping 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] Mathai, Varghese $L^2$-analytic torsion, J. Funct. Anal., Volume 107 (1992) no. 2, pp. 369-386 | DOI | Zbl

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

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

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

[27] 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

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

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

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

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

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

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