The universal functorial Lefschetz invariant

Lück, Wolfgang

Fundamenta Mathematicae, Tome 159 (1999), p. 167-215 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW-complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and $L^2$-torsion of mapping tori. We examine its behaviour under fibrations.

Détail

Publié le : 1999-01-01

Zbl 0938.57017

EUDML-ID : urn:eudml:doc:212398

Bibliographie

[00000] [1] Almkvist, G.: The Grothendieck ring of the category of endomorphisms, J. Algebra 28 (1974), 375-388 | Zbl 0281.18012

[00001] [2] Brown, R. F.: The Lefschetz Fixed Point Theorem, Scott and Foresman, 1971. | Zbl 0216.19601

[00002] [3] Burghelea, D., Friedlander, L. and Kappeler, T.: Torsion for manifolds with boundary and glueing formulas, preprint, 1996. | Zbl 0959.58041

[00003] [4] D. Burghelea, Friedlander, L., Kappeler, T. and McDonald, P.: Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal. 6 (1996), 751-859 | Zbl 0874.57025

[00004] [5] Carey, A. L. and Mathai, V.: $L^2$-acyclicity and $L^2$-torsion invariants, Amer. Math. Soc., Contemp. Math. 105 (1990), 141-155

[00005] [6] Cohen, M. M.: A Course in Simple Homotopy Theory, Grad. Texts in Math. 10, Springer, 1973.

[00006] [7] Deseyve, M.: Verallgemeinerte Lefschetz Zahlen, Diplomarbeit, Mainz, 1994.

[00007] [8] Dodziuk, J. and Mathai, V.: Approximating $L^2$-invariants of amenable covering spaces: A combinatorial approach, preprint, 1996. | Zbl 0936.57018

[00008] [9] Dold, A.: The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281-2937 | Zbl 0284.55007

[00009] [10] Dold, A.: The fixed point transfer of fibre preserving maps, Math. Z. 148 (1976), 215-244 | Zbl 0329.55007

[00010] [11] Fel'shtyn, A. L. and Hill, R.: Dynamical zeta functions, Nielsen theory and Reidemeister torsion, in: Nielsen Theory and Dynamical Systems, C. K. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc. (1993), 43-68 | Zbl 0793.58028

[00011] [12] Fel'shtyn, A. L. and Hill, R.: The Reidemeister zeta function with applications to Nielsen theory and connections to Reidemeister torsion, K-Theory 8 (1994), 367-393 | Zbl 0814.58033

[00012] [13] Fried, D.: Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), 539-563 | Zbl 0603.58020

[00013] [14] Geoghegan, R. and Nicas, A.: Lefschetz trace formulae, zeta functions and torsion in dynamics, in: Nielsen Theory and Dynamical Systems, C. K. McCord (ed.), Contemp. Math. 152, Amer. Math. Soc. (1993), 141-157 | Zbl 0807.19004

[00014] [15] Geoghegan, R. and Nicas, A.: Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces, in: Nielsen Theory and Dynamical Systems, Amer. J. Math. 116 (1994), 397-446 | Zbl 0812.55001

[00015] [16] Grayson, D.: The K-theory of endomorphisms, J. Algebra 48 (1977), 439-446 | Zbl 0413.18010

[00016] [17] Jiang, B.: Estimation of the Nielsen numbers, Chinese Math. 5 (1964), 330-339

[00017] [18] Jiang, B.: Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., 1983. | Zbl 0512.55003

[00018] [19] Jiang, B.: Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151-185. | Zbl 0855.55001

[00019] [20] Jiang, B. and Wang, S.: Lefschetz numbers and Nielsen numbers for homeomorphisms on aspherical manifolds, in: Topology Hawaii (Honolulu, 1990), K. H. Dovermann (ed.), World Sci. (1992), 119-136 | Zbl 1039.55502

[00020] [21] Jiang, B. and Wang, S.: Twisted topological invariants associated with representations, in: Topics in Knot Theory (Erzurum, 1992), M. E. Bozhüyük (ed.) (1993), 211-227

[00021] [22] Laitinen, E. and Lück, W.: Equivariant Lefschetz classes, Osaka J. Math. 26 (1989), 491-525 | Zbl 0701.55002

[00022] [23] Lott, J.: Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510 | Zbl 0770.58040

[00023] [24] Lott, J. and Lück, W.: $L^2$-topological invariants of 3-manifolds, Invent. Math. 120 (1995), 15-60 | Zbl 0876.57050

[00024] [25] Lück, W.: The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), 367-384 | Zbl 0626.57011

[00025] [26] Lück, W.: The transfer maps induced in the algebraic $K_0$- and $K_1$-groups by a fibration I, Math. Scand. 59 (1986), 93-121 | Zbl 0589.57020

[00026] [27] Lück, W.: The transfer maps induced in the algebraic $K_0$- and $K_1$-groups by a fibration II, J. Pure Appl. Algebra 45 (1987), 143-169 | Zbl 0657.57010

[00027] [28] Lück, W.: Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer, 1989. | Zbl 0679.57022

[00028] [29] Lück, W.: $L^2$-torsion and 3-manifolds, in: Low-Dimensional Topology (Knoxville, TN, 1992), K. Johannson (ed.), Conf. Proc. Lecture Notes Geom. Topology III, Internat. Press (1994), 75-107

[00029] [30] Lück, W.: $L^2$-Betti numbers of mapping tori and groups, Topology 33 (1994), 203-214 | Zbl 0847.55004

[00030] [31] Lück, W.: $L^2$-invariants of regular coverings of compact manifolds and CW-complexes, in: Handbook of Geometry, R. J. Davermann and R. B. Sher (eds.), Elsevier, 1998, to appear.

[00031] [32] Lück, W. and Ranicki, A.: Surgery transfer, in: Algebraic Topology and Transformation Groups (Göttingen, 1987), T. tom Dieck (ed.), Lecture Notes in Math. 1361 (1988), 167-246 Springer,

[00032] [33] Lück, W. and Ranicki, A.: Surgery obstructions of fibre bundles, J. Pure Appl. Algebra 81 (1992), 139-189 | Zbl 0755.57013

[00033] [34] Lück, W. and Rothenberg, M.: Reidemeister torsion and the K-theory of von Neumann algebras, K-Theory 5 (1991), 213-264 | Zbl 0748.57007

[00034] [35] Lück, W. and Schick, T.: $L^2$-torsion of hyperbolic manifolds of finite volume, preprint, Münster, 1997. | Zbl 0947.58024

[00035] [36] Lydakis, M. G.: Fixed point problems, equivariant stable homotopy theory, and a trace map for the algebraic K-theory of a point, Topology 34 (1995), 959-999 | Zbl 0857.55010

[00036] [37] Mathai, V.: $L^2$-analytic torsion, J. Funct. Anal. 107 (1992), 369-386 | Zbl 0756.58047

[00037] [38] Milnor, J.: Infinite cyclic coverings, in: Proc. Conf. on the Topology of Manifolds (East Lansing, MI, 1967), Prindle, Weber & Schmidt, Boston, MA (1968), 115-133

[00038] [39] Okonek, C.: Bemerkungen zur K-Theorie äquivarianter Endomorphismen, Arch. Math. (Basel) 40 (1983), 132-138 | Zbl 0592.18012

[00039] [40] Pedersen, E. K. and Taylor, L.: The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), 887-896 | Zbl 0415.55010

[00040] [41] Reidemeister, K.: Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1938), 586-593 | Zbl 0013.36903

[00041] [42] Wecken, F.: Fixpunktklassen II, ibid. 118 (1942), 216-243 | Zbl 67.1094.01