Higher Lefschetz Traces and Spherical Euler Characteristics

Geoghegan, Ross; Nicas, Andrew; Oprea, John

doi:10.1090/S0002-9947-96-01615-7

Higher Lefschetz Traces and Spherical Euler Characteristics
HTML articles powered by AMS MathViewer

by Ross Geoghegan, Andrew Nicas and John Oprea PDF

Trans. Amer. Math. Soc. 348 (1996), 2039-2062 Request permission

Abstract:

Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler.

References

J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107–133. MR 436137
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Albrecht Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965), 1–8. MR 193634, DOI 10.1016/0040-9383(65)90044-3
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, Lectures on algebraic topology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin-New York, 1980. MR 606196
R. Geoghegan and A. Nicas, Higher Euler characteristics (I), L’Enseign. Math. 41 (1995), 3–62.
D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840–856. MR 189027, DOI 10.2307/2373248
Daniel Henry Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729–756. MR 275424, DOI 10.2307/2373349
Daniel Henry Gottlieb, Applications of bundle map theory, Trans. Amer. Math. Soc. 171 (1972), 23–50. MR 309111, DOI 10.1090/S0002-9947-1972-0309111-X
Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
R. J. Knill, On the homology of a fixed point set, Bull. Amer. Math. Soc. 77 (1971), 184–190. MR 300274, DOI 10.1090/S0002-9904-1971-12674-5
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Gregory Lupton and John Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261–288. MR 1282893, DOI 10.1090/S0002-9947-1995-1282893-4
Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508