## Illinois Journal of Mathematics

### Analytic torsion on manifolds under locally compact group actions

Guangxiang Su

#### Abstract

For a complete Riemannian manifold without boundary which a unimodular locally compact group properly cocompact acts on it, under some conditions, we define and study the analytic torsion on it by using the $G$-trace defined in ($L^{2}$-index formula for proper cocompact group actions, preprint). For a fiber bundle $\pi:M\to B$, if there is a unimodular locally compact group acts fiberwisely properly and cocompact on it, we define the torsion form for it, and show that the zero degree part of the torsion form is the analytic torsion. This can be viewed as an extension of the $L^{2}$-analytic torsion.

#### Article information

Source
Illinois J. Math., Volume 57, Number 1 (2013), 171-193.

Dates
First available in Project Euclid: 23 June 2014

https://projecteuclid.org/euclid.ijm/1403534491

Digital Object Identifier
doi:10.1215/ijm/1403534491

Mathematical Reviews number (MathSciNet)
MR3224566

Zentralblatt MATH identifier
1300.58011

Subjects
Primary: 58J52: Determinants and determinant bundles, analytic torsion

#### Citation

Su, Guangxiang. Analytic torsion on manifolds under locally compact group actions. Illinois J. Math. 57 (2013), no. 1, 171--193. doi:10.1215/ijm/1403534491. https://projecteuclid.org/euclid.ijm/1403534491

#### References

• A. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32 (1976), 43–72.
• N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Springer, Berlin, 2003.
• J. M. Bismut and J. Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), 291–363.
• M. Braverman, A. Carey, M. Farber and V. Mathai, $L^{2}$ torsion without the determinant class condition and extended $L^{2}$ cohomology, Commun. Contemp. Math. 7 (2005), 421–462.
• D. Burghelea, L. Friedlander, T. Kappeler and P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal. 6 (1996), 751–859.
• J. M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992).
• J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259–332.
• A. Carey and V. Mathai, $L^{2}$-torsion invariants, J. Funct. Anal. 110 (1992), 377–409.
• G. Dong and M. Rothenberg, Analytic torsion forms for noncompact fiber bundles, MPIM preprint, 1997.
• J. Heitsch and C. Lazarov, Spectral asymptotics of foliated manifolds, Illinois J. Math. 38 (1994), 653–678.
• J. Heitsch and C. Lazarov, Riemann–Roch–Grothendieck and torsion for foliations, J. Geom. Anal. 12 (2002), 437–468.
• J. Lott, Heat kernels on covering spaces andtopological invariants, J. Differential Geom. 35 (1992), 471–510.
• V. Mathai, $L^{2}$-analytic torsion, J. Funct. Anal. 107 (1992), 369–386.
• J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. (N.S.) 72 (1996), 358–426.
• W. Müller, Analytic torsion and the R-torsion of Riemannian manifolds, Adv. Math. 28 (1978), 233–305.
• W. Müller, Analytic torsion and the R-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721–753.
• D. Quillen, Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl. 19 (1985), 31–34.
• D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210.
• H. Wang, $L^{2}$-index formula for proper cocompact group actions, preprint, \arxivurlarXiv:1106.4542v3.
• E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661–692.
• W. Zhang, An extended Cheeger–Müller theorem for covering spaces, Topology 44 (2005), 1093–1131.