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1999 | 161 | 1-2 | 167-215
Tytuł artykułu

The universal functorial Lefschetz invariant

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
161
Numer
1-2
Strony
167-215
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-10-16
poprawiono
1998-09-18
Twórcy
  • Fachbereich Mathematik und Informatik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, lueck@math.uni-muenster.de
Bibliografia
  • [1] Almkvist, G.: The Grothendieck ring of the category of endomorphisms, J. Algebra 28 (1974), 375-388
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  • [3] Burghelea, D., Friedlander, L. and Kappeler, T.: Torsion for manifolds with boundary and glueing formulas, preprint, 1996.
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  • [5] Carey, A. L. and Mathai, V.: $L^2$-acyclicity and $L^2$-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 $L^2$-invariants of amenable covering spaces: A combinatorial approach, preprint, 1996.
  • [9] Dold, A.: The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281-2937
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  • [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
  • [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
  • [13] Fried, D.: Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), 539-563
  • [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
  • [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
  • [16] Grayson, D.: The K-theory of endomorphisms, J. Algebra 48 (1977), 439-446
  • [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.
  • [19] Jiang, B.: Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151-185.
  • [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
  • [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
  • [23] Lott, J.: Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510
  • [24] Lott, J. and Lück, W.: $L^2$-topological invariants of 3-manifolds, Invent. Math. 120 (1995), 15-60
  • [25] Lück, W.: The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), 367-384
  • [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
  • [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
  • [28] Lück, W.: Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer, 1989.
  • [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
  • [30] Lück, W.: $L^2$-Betti numbers of mapping tori and groups, Topology 33 (1994), 203-214
  • [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.
  • [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
  • [34] Lück, W. and Rothenberg, M.: Reidemeister torsion and the K-theory of von Neumann algebras, K-Theory 5 (1991), 213-264
  • [35] Lück, W. and Schick, T.: $L^2$-torsion of hyperbolic manifolds of finite volume, preprint, Münster, 1997.
  • [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
  • [37] Mathai, V.: $L^2$-analytic torsion, J. Funct. Anal. 107 (1992), 369-386
  • [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
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  • [41] Reidemeister, K.: Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1938), 586-593
  • [42] Wecken, F.: Fixpunktklassen II, ibid. 118 (1942), 216-243
Typ dokumentu
Bibliografia
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