On the homotopy invariance of torsion

for covering spaces

Authors:
Varghese Mathai and Melvin Rothenberg

Journal:
Proc. Amer. Math. Soc. **126** (1998), 887-897

MSC (1991):
Primary 58G11, 58G18, 58G25

DOI:
https://doi.org/10.1090/S0002-9939-98-04595-X

MathSciNet review:
1469424

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the homotopy invariance of torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the cohomology of the covering space vanishes, the homotopy invariance was established by Lück. We also give some applications of our results.

**[ASS]**H. Araki, M-S.B. Smith and L. Smith,*On the homotopical significance of the type of von Neumann algebra factors*, Commun. Math. Phys.**22**(1971), 71-88. MR**44:5783****[BFKM]**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), 751-858. MR**97i:58177****[BFK]**D. Burghelea, L. Friedlander and T. Kappeler,*Torsion for manifolds with boundary and glueing formulae*, preprint 1996.**[CM]**A. Carey and V. Mathai,*Torsion Invariants*, Journal of Functional Analysis**110**(1992), 377-409. MR**94a:58211****[CFM]**A. Carey, M. Farber and V. Mathai,*Determinant Lines, Von Neumann Algebras and torsion*, J. Reine Angew. Math.**484**(1997), 153-181. CMP**97:09****[Di]**J. Dixmier,*Von Neumann algebras*, North-Holland, Amsterdam (1981). MR**83a:46004****[Do]**J. Dodziuk,*De Rham-Hodge theory for cohomology of infinite coverings*, Topology**16**(1977), 157-165. MR**56:3898****[DM]**J. Dodziuk and V. Mathai,*Approximating invariants of amenable covering spaces: A heat kernel approach*, Lipa's Legacy (Proc. Bers Colloq., 1995; J. Dodziuk and L. Keen, editors), Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 151-167.**[DM2]**J. Dodziuk and V. Mathai,*Approximating invariants for amenable covering spaces: A combinatorial approach*, to appear in Journal of Functional Analysis.**[FJ]**F.T. Farrell and L.E. Jones,*Isomorphism conjectures in algebraic K-theory*, JAMS**6**(1993), 249-298. MR**93h:57032****[FK]**B. Fuglede and R.V. Kadison,*Determinant theory in finite factors*, Annals of Math.**55**(1952), 520-530. MR**14:660a****[L]**J. Lott,*Heat kernels on covering spaces and topological invariants*, J. Diff. Geom.**35**(1992), 471-510. MR**93b:58140****[Lu]**W. Lück,*Approximating invariants by their finite dimensional analogues*, Geom. and Func. Analysis**4**(1994), 455-481. MR**95g:58234****[Lu1]**W. Lück,*-Torsion and 3-manifolds*, Low-Dimensional Topology (Knoxville, TN, 1992; K. Johannson, editor), Conf. Proc. and Lecture Notes Geom. Topology, vol. III, Internat. Press, Cambridge, MA, 1994, pp. 75-107. MR**96g:57019****[LuR]**W. Lück and M. Rothenberg,*Reidemeister torsion and the K-theory of von Neumann algebras*, K-Theory**5**(1991), 213-264. MR**93g:57025****[M]**V. Mathai,*analytic torsion*, J. Func. Anal.**107**(1992), 369-386. MR**93g:58146****[Mi]**J. Milnor,*Whitehead torsion*, Bull. Amer. Math. Soc.,**72**, (1966), 358-426. MR**33:4922****[RS]**D. Ray and I. M. Singer,*R-torsion and the Laplacian on Riemannian manifolds*, Advances in Math.**7**(1971), 145-210. MR**45:4447**

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Additional Information

**Varghese Mathai**

Affiliation:
Department of Mathematics, University of Adelaide, Adelaide 5005, Australia

Email:
vmathai@maths.adelaide.edu.au

**Melvin Rothenberg**

Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Email:
mel@math.uchicago.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04595-X

Keywords:
$L^2$ torsion,
invariants,
amenable groups,
residually finite groups,
Whitehead groups,
homotopy invariance

Received by editor(s):
May 16, 1996

Additional Notes:
The second author was supported in part by NSF Grant DMS 9423300

Communicated by:
Jozef Dodziuk

Article copyright:
© Copyright 1998
American Mathematical Society