The universal functorial Lefschetz invariant
Fundamenta Mathematicae (1999)
- Volume: 161, Issue: 1-2, page 167-215
- ISSN: 0016-2736
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topLück, Wolfgang. "The universal functorial Lefschetz invariant." Fundamenta Mathematicae 161.1-2 (1999): 167-215. <http://eudml.org/doc/212398>.
@article{Lück1999,
abstract = {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.},
author = {Lück, Wolfgang},
journal = {Fundamenta Mathematicae},
keywords = {universal functorial Lefschetz invariants; Grothendieck group of endomorphisms of modules; transfer maps; CW-complex; Grothendieck group; fibration; -torsion of mapping tori; Lefschetz number; Nielsen number; transfer map},
language = {eng},
number = {1-2},
pages = {167-215},
title = {The universal functorial Lefschetz invariant},
url = {http://eudml.org/doc/212398},
volume = {161},
year = {1999},
}
TY - JOUR
AU - Lück, Wolfgang
TI - The universal functorial Lefschetz invariant
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 1-2
SP - 167
EP - 215
AB - 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.
LA - eng
KW - universal functorial Lefschetz invariants; Grothendieck group of endomorphisms of modules; transfer maps; CW-complex; Grothendieck group; fibration; -torsion of mapping tori; Lefschetz number; Nielsen number; transfer map
UR - http://eudml.org/doc/212398
ER -
References
top- [1] Almkvist, G.: The Grothendieck ring of the category of endomorphisms, J. Algebra 28 (1974), 375-388 Zbl0281.18012
- [2] Brown, R. F.: The Lefschetz Fixed Point Theorem, Scott and Foresman, 1971. Zbl0216.19601
- [3] Burghelea, D., Friedlander, L. and Kappeler, T.: Torsion for manifolds with boundary and glueing formulas, preprint, 1996. Zbl0959.58041
- [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 Zbl0874.57025
- [5] Carey, A. L. and Mathai, V.: -acyclicity and -torsion invariants, Amer. Math. Soc., Contemp. Math. 105 (1990), 141-155
- [6] Cohen, M. M.: A Course in Simple Homotopy Theory, Grad. Texts in Math. 10, Springer, 1973.
- [7] Deseyve, M.: Verallgemeinerte Lefschetz Zahlen, Diplomarbeit, Mainz, 1994.
- [8] Dodziuk, J. and Mathai, V.: Approximating -invariants of amenable covering spaces: A combinatorial approach, preprint, 1996. Zbl0936.57018
- [9] Dold, A.: The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281-2937 Zbl0284.55007
- [10] Dold, A.: The fixed point transfer of fibre preserving maps, Math. Z. 148 (1976), 215-244 Zbl0329.55007
- [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 Zbl0793.58028
- [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 Zbl0814.58033
- [13] Fried, D.: Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), 539-563 Zbl0603.58020
- [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 Zbl0807.19004
- [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 Zbl0812.55001
- [16] Grayson, D.: The K-theory of endomorphisms, J. Algebra 48 (1977), 439-446 Zbl0413.18010
- [17] Jiang, B.: Estimation of the Nielsen numbers, Chinese Math. 5 (1964), 330-339
- [18] Jiang, B.: Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., 1983. Zbl0512.55003
- [19] Jiang, B.: Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151-185. Zbl0855.55001
- [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 Zbl1039.55502
- [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
- [22] Laitinen, E. and Lück, W.: Equivariant Lefschetz classes, Osaka J. Math. 26 (1989), 491-525 Zbl0701.55002
- [23] Lott, J.: Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510 Zbl0770.58040
- [24] Lott, J. and Lück, W.: -topological invariants of 3-manifolds, Invent. Math. 120 (1995), 15-60 Zbl0876.57050
- [25] Lück, W.: The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), 367-384 Zbl0626.57011
- [26] Lück, W.: The transfer maps induced in the algebraic - and -groups by a fibration I, Math. Scand. 59 (1986), 93-121 Zbl0589.57020
- [27] Lück, W.: The transfer maps induced in the algebraic - and -groups by a fibration II, J. Pure Appl. Algebra 45 (1987), 143-169 Zbl0657.57010
- [28] Lück, W.: Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer, 1989. Zbl0679.57022
- [29] Lück, W.: -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
- [30] Lück, W.: -Betti numbers of mapping tori and groups, Topology 33 (1994), 203-214 Zbl0847.55004
- [31] Lück, W.: -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.
- [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,
- [33] Lück, W. and Ranicki, A.: Surgery obstructions of fibre bundles, J. Pure Appl. Algebra 81 (1992), 139-189 Zbl0755.57013
- [34] Lück, W. and Rothenberg, M.: Reidemeister torsion and the K-theory of von Neumann algebras, K-Theory 5 (1991), 213-264 Zbl0748.57007
- [35] Lück, W. and Schick, T.: -torsion of hyperbolic manifolds of finite volume, preprint, Münster, 1997. Zbl0947.58024
- [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 Zbl0857.55010
- [37] Mathai, V.: -analytic torsion, J. Funct. Anal. 107 (1992), 369-386 Zbl0756.58047
- [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
- [39] Okonek, C.: Bemerkungen zur K-Theorie äquivarianter Endomorphismen, Arch. Math. (Basel) 40 (1983), 132-138 Zbl0592.18012
- [40] Pedersen, E. K. and Taylor, L.: The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), 887-896 Zbl0415.55010
- [41] Reidemeister, K.: Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1938), 586-593 Zbl0013.36903
- [42] Wecken, F.: Fixpunktklassen II, ibid. 118 (1942), 216-243 Zbl67.1094.01
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