On the multiplicativity of the Euler characteristic

Klein, John; Malkiewich, Cary; Ramzi, Maxime

doi:10.1090/proc/16498

On the multiplicativity of the Euler characteristic
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by John R. Klein, Cary Malkiewich and Maxime Ramzi;

Proc. Amer. Math. Soc. 151 (2023), 4997-5006

DOI: https://doi.org/10.1090/proc/16498

Published electronically: July 21, 2023

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Abstract:

We give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on $\pi _0$.

References

J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107–133. MR 436137
A. J. Berrick and Lars Hesselholt, Topological Hochschild homology and the Bass trace conjecture, J. Reine Angew. Math. 704 (2015), 169–185. MR 3365777, DOI 10.1515/crelle-2013-0051
Shachar Carmeli, Bastiaan Cnossen, Maxime Ramzi, and Lior Yanovski, Characters and transfer maps via categorified traces, Preprint, arXiv:2210.17364, 2022.
Michael Crabb and Ioan James, Fibrewise homotopy theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1998. MR 1646248, DOI 10.1007/978-1-4471-1265-5
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
A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
John R. Klein, Multiplicativity of the Euler characteristic for fibrations, MathOverflow, https://mathoverflow.net/q/416927.
John R. Klein and Cary Malkiewich, The transfer is functorial, Adv. Math. 338 (2018), 1119–1140. MR 3861724, DOI 10.1016/j.aim.2018.09.017
John R. Klein and Cary Malkiewich, Corrigendum to “The transfer is functorial” [Adv. Math. 338 (2018) 1119–1140]. part B, Adv. Math. 410 (2022), no. part B, Paper No. 108752, 2. MR 4500638, DOI 10.1016/j.aim.2022.108752
V. J. Lal, Wall obstruction of a fibration, Invent. Math. 6 (1968), 67–77. MR 231380, DOI 10.1007/BF01389834
P. A. Linnell, Decomposition of augmentation ideals and relation modules, Proc. London Math. Soc. (3) 47 (1983), no. 1, 83–127. MR 698929, DOI 10.1112/plms/s3-47.1.83
John A. Lind and Cary Malkiewich, The transfer map of free loop spaces, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2503–2552. MR 3896088, DOI 10.1090/tran/7497
Cary Malkiewich, Parametrized spectra, a low-tech approach, Preprint, arXiv:1906.04773, 2019.
J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73. MR 1867203, DOI 10.1006/aima.2001.1995
J. P. May and J. Sigurdsson, Parametrized homotopy theory, Mathematical Surveys and Monographs, vol. 132, American Mathematical Society, Providence, RI, 2006. MR 2271789, DOI 10.1090/surv/132
Kate Ponto and Michael Shulman, The linearity of traces in monoidal categories and bicategories, Theory Appl. Categ. 31 (2016), Paper No. 23, 594–689. MR 3518981
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 210112
Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 308195, DOI 10.1007/BFb0059150
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/gsm/145