On the homotopy invariance of torsion for covering spaces
Authors:
Varghese Mathai and Melvin Rothenberg
Journal:
Proc. Amer. Math. Soc. 126 (1998), 887897
MSC (1991):
Primary 58G11, 58G18, 58G25
MathSciNet review:
1469424
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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]
Huzihiro
Araki, Misoo
Bae Smith, and Larry
Smith, On the homotopical significance of the type of von Neumann
algebra factors, Comm. Math. Phys. 22 (1971),
71–88. MR
0288587 (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), no. 5, 751–859. MR 1415762
(97i:58177), http://dx.doi.org/10.1007/BF02246786
 [BFK]
D. Burghelea, L. Friedlander and T. Kappeler, Torsion for manifolds with boundary and glueing formulae, preprint 1996.
 [CM]
Alan
L. Carey and Varghese
Mathai, 𝐿²torsion invariants, J. Funct. Anal.
110 (1992), no. 2, 377–409. MR 1194991
(94a:58211), http://dx.doi.org/10.1016/00221236(92)90036I
 [CFM]
A. Carey, M. Farber and V. Mathai, Determinant Lines, Von Neumann Algebras and torsion, J. Reine Angew. Math. 484 (1997), 153181. CMP 97:09
 [Di]
Jacques
Dixmier, von Neumann algebras, NorthHolland Mathematical
Library, vol. 27, NorthHolland Publishing Co., AmsterdamNew York,
1981. With a preface by E. C. Lance; Translated from the second French
edition by F. Jellett. MR 641217
(83a:46004)
 [Do]
Jozef
Dodziuk, de RhamHodge theory for 𝐿²cohomology of
infinite coverings, Topology 16 (1977), no. 2,
157–165. MR 0445560
(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. 151167.
 [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
𝐾theory, J. Amer. Math. Soc.
6 (1993), no. 2,
249–297. MR 1179537
(93h:57032), http://dx.doi.org/10.1090/S08940347199311795370
 [FK]
Bent
Fuglede and Richard
V. Kadison, Determinant theory in finite factors, Ann. of
Math. (2) 55 (1952), 520–530. MR 0052696
(14,660a)
 [L]
John
Lott, Heat kernels on covering spaces and topological
invariants, J. Differential Geom. 35 (1992),
no. 2, 471–510. MR 1158345
(93b:58140)
 [Lu]
W.
Lück, Approximating 𝐿²invariants by their
finitedimensional analogues, Geom. Funct. Anal. 4
(1994), no. 4, 455–481. MR 1280122
(95g:58234), http://dx.doi.org/10.1007/BF01896404
 [Lu1]
Wolfgang
Lück, 𝐿²torsion and 3manifolds,
Lowdimensional topology (Knoxville, TN, 1992) Conf. Proc. Lecture Notes
Geom. Topology, III, Int. Press, Cambridge, MA, 1994,
pp. 75–107. MR 1316175
(96g:57019)
 [LuR]
Wolfgang
Lück and Mel
Rothenberg, Reidemeister torsion and the 𝐾theory of von
Neumann algebras, 𝐾Theory 5 (1991),
no. 3, 213–264. MR 1162441
(93g:57025), http://dx.doi.org/10.1007/BF00533588
 [M]
J.
Brüning and R.
Seeley, The expansion of the resolvent near a singular stratum of
conical type, J. Funct. Anal. 95 (1991), no. 2,
255–290. MR 1092127
(93g:58146), http://dx.doi.org/10.1016/00221236(91)900309
 [Mi]
J.
Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 0196736
(33 #4922), http://dx.doi.org/10.1090/S000299041966114842
 [RS]
D.
B. Ray and I.
M. Singer, 𝑅torsion and the Laplacian on Riemannian
manifolds, Advances in Math. 7 (1971), 145–210.
MR
0295381 (45 #4447)
 [ASS]
 H. Araki, MS.B. Smith and L. Smith, On the homotopical significance of the type of von Neumann algebra factors, Commun. Math. Phys. 22 (1971), 7188. 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), 751858. 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), 377409. MR 94a:58211
 [CFM]
 A. Carey, M. Farber and V. Mathai, Determinant Lines, Von Neumann Algebras and torsion, J. Reine Angew. Math. 484 (1997), 153181. CMP 97:09
 [Di]
 J. Dixmier, Von Neumann algebras, NorthHolland, Amsterdam (1981). MR 83a:46004
 [Do]
 J. Dodziuk, De RhamHodge theory for cohomology of infinite coverings, Topology 16 (1977), 157165. 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. 151167.
 [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 Ktheory, JAMS 6 (1993), 249298. MR 93h:57032
 [FK]
 B. Fuglede and R.V. Kadison, Determinant theory in finite factors, Annals of Math. 55 (1952), 520530. MR 14:660a
 [L]
 J. Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 (1992), 471510. MR 93b:58140
 [Lu]
 W. Lück, Approximating invariants by their finite dimensional analogues, Geom. and Func. Analysis 4 (1994), 455481. MR 95g:58234
 [Lu1]
 W. Lück, Torsion and 3manifolds, LowDimensional Topology (Knoxville, TN, 1992; K. Johannson, editor), Conf. Proc. and Lecture Notes Geom. Topology, vol. III, Internat. Press, Cambridge, MA, 1994, pp. 75107. MR 96g:57019
 [LuR]
 W. Lück and M. Rothenberg, Reidemeister torsion and the Ktheory of von Neumann algebras, KTheory 5 (1991), 213264. MR 93g:57025
 [M]
 V. Mathai, analytic torsion, J. Func. Anal. 107 (1992), 369386. MR 93g:58146
 [Mi]
 J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc., 72, (1966), 358426. MR 33:4922
 [RS]
 D. Ray and I. M. Singer, Rtorsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145210. 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:
http://dx.doi.org/10.1090/S000299399804595X
PII:
S 00029939(98)04595X
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
