Equivariant fixed point index and fixed point transfer in nonzero dimensions

Prieto, Carlos; Ulrich, Hanno

doi:10.1090/S0002-9947-1991-1062875-4

Equivariant fixed point index and fixed point transfer in nonzero dimensions
HTML articles powered by AMS MathViewer

by Carlos Prieto and Hanno Ulrich

Trans. Amer. Math. Soc. 328 (1991), 731-745

DOI: https://doi.org/10.1090/S0002-9947-1991-1062875-4

PDF | Request permission

Abstract:

Dold’s fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant dimension." Those invariants change the dimension correspondingly. A formula for the index of a situation over a space with trivial group action is exhibited. For the transfer, a generalization of Dold’s Lefschetz-Hopf trace formula is proved.

References

J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107–133. MR 436137
Mónica Clapp, Duality and transfer for parametrized spectra, Arch. Math. (Basel) 37 (1981), no. 5, 462–472. MR 643290, DOI 10.1007/BF01234383
A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602
Albrecht Dold, The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281–297. MR 380776, DOI 10.1007/BF01389731
Albrecht Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976), no. 3, 215–244. MR 433440, DOI 10.1007/BF01214520
Albrecht Dold and Dieter Puppe, Duality, trace, and transfer, Proceedings of the International Conference on Geometric Topology (Warsaw, 1978) PWN, Warsaw, 1980, pp. 81–102. MR 656721
J. Ize, I. Massabò, and A. Vignoli, Degree theory for equivariant maps. I, Trans. Amer. Math. Soc. 315 (1989), no. 2, 433–510. MR 935940, DOI 10.1090/S0002-9947-1989-0935940-8
Czes Kosniowski, Equivariant cohomology and stable cohomotopy, Math. Ann. 210 (1974), 83–104. MR 413081, DOI 10.1007/BF01360033
G. Lewis, J. P. May, and J. McClure, Ordinary $RO(G)$-graded cohomology, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 208–212. MR 598689, DOI 10.1090/S0273-0979-1981-14886-2
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
Carlos Prieto, Coincidence index for fiber-preserving maps: an approach to stable cohomotopy, Manuscripta Math. 47 (1984), no. 1-3, 233–249. MR 744321, DOI 10.1007/BF01174595

graded stable cohomotopy over

and

graded

equivariant cohomotopy

A fixed point theoretical approach to the Segal conjecture

58

Una fórmula de Lefschetz-Hopf para el índice de coincidencia equivariante parametrizado

5

Fixpunkttransfer für stetige Familien von

Räumen

Existenz und axiomatische Characterisierung

Hanno Ulrich, Fixed point theory of parametrized equivariant maps, Lecture Notes in Mathematics, vol. 1343, Springer-Verlag, Berlin, 1988. MR 967325, DOI 10.1007/BFb0079799
Stefan Waner, Equivariant homotopy theory and Milnor’s theorem, Trans. Amer. Math. Soc. 258 (1980), no. 2, 351–368. MR 558178, DOI 10.1090/S0002-9947-1980-0558178-7