The universal functorial Lefschetz invariant

Wolfgang Lück

Fundamenta Mathematicae (1999)

  • Volume: 161, Issue: 1-2, page 167-215
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Lück, 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. [1] Almkvist, G.: The Grothendieck ring of the category of endomorphisms, J. Algebra 28 (1974), 375-388 Zbl0281.18012
  2. [2] Brown, R. F.: The Lefschetz Fixed Point Theorem, Scott and Foresman, 1971. Zbl0216.19601
  3. [3] Burghelea, D., Friedlander, L. and Kappeler, T.: Torsion for manifolds with boundary and glueing formulas, preprint, 1996. Zbl0959.58041
  4. [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. [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. [6] Cohen, M. M.: A Course in Simple Homotopy Theory, Grad. Texts in Math. 10, Springer, 1973. 
  7. [7] Deseyve, M.: Verallgemeinerte Lefschetz Zahlen, Diplomarbeit, Mainz, 1994. 
  8. [8] Dodziuk, J. and Mathai, V.: Approximating L 2 -invariants of amenable covering spaces: A combinatorial approach, preprint, 1996. Zbl0936.57018
  9. [9] Dold, A.: The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281-2937 Zbl0284.55007
  10. [10] Dold, A.: The fixed point transfer of fibre preserving maps, Math. Z. 148 (1976), 215-244 Zbl0329.55007
  11. [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. [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. [13] Fried, D.: Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), 539-563 Zbl0603.58020
  14. [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. [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. [16] Grayson, D.: The K-theory of endomorphisms, J. Algebra 48 (1977), 439-446 Zbl0413.18010
  17. [17] Jiang, B.: Estimation of the Nielsen numbers, Chinese Math. 5 (1964), 330-339 
  18. [18] Jiang, B.: Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., 1983. Zbl0512.55003
  19. [19] Jiang, B.: Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), 151-185. Zbl0855.55001
  20. [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. [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. [22] Laitinen, E. and Lück, W.: Equivariant Lefschetz classes, Osaka J. Math. 26 (1989), 491-525 Zbl0701.55002
  23. [23] Lott, J.: Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), 471-510 Zbl0770.58040
  24. [24] Lott, J. and Lück, W.: L 2 -topological invariants of 3-manifolds, Invent. Math. 120 (1995), 15-60 Zbl0876.57050
  25. [25] Lück, W.: The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), 367-384 Zbl0626.57011
  26. [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 Zbl0589.57020
  27. [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 Zbl0657.57010
  28. [28] Lück, W.: Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer, 1989. Zbl0679.57022
  29. [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. [30] Lück, W.: L 2 -Betti numbers of mapping tori and groups, Topology 33 (1994), 203-214 Zbl0847.55004
  31. [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. [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. [33] Lück, W. and Ranicki, A.: Surgery obstructions of fibre bundles, J. Pure Appl. Algebra 81 (1992), 139-189 Zbl0755.57013
  34. [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. [35] Lück, W. and Schick, T.: L 2 -torsion of hyperbolic manifolds of finite volume, preprint, Münster, 1997. Zbl0947.58024
  36. [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. [37] Mathai, V.: L 2 -analytic torsion, J. Funct. Anal. 107 (1992), 369-386 Zbl0756.58047
  38. [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. [39] Okonek, C.: Bemerkungen zur K-Theorie äquivarianter Endomorphismen, Arch. Math. (Basel) 40 (1983), 132-138 Zbl0592.18012
  40. [40] Pedersen, E. K. and Taylor, L.: The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), 887-896 Zbl0415.55010
  41. [41] Reidemeister, K.: Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1938), 586-593 Zbl0013.36903
  42. [42] Wecken, F.: Fixpunktklassen II, ibid. 118 (1942), 216-243 Zbl67.1094.01

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.