Abstract
For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi-Bracci-Izawa-Suwa (2016) in the framework of Čech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particular we define the local homology class and give some explicit expressions. These all together lead to a generalization of the classical Lefschetz coincidence point formula.
Similar content being viewed by others
References
Biasi, C., Libardi, A.K.M., Monis, T.F.M.: The Lefschetz coincidence class of \(p\) maps. Forum Math. 27, 1717–1728 (2015). https://doi.org/10.1515/forum-2013-0038
Bisi, C., Bracci, F., Izawa, T., Suwa, T.: Localized intersection of currents and the Lefschetz coincidence point theorem. Annali di Mat. Pura ed Applicata 195, 601–621 (2016). https://doi.org/10.1007/s10231-015-0480-4
Brasselet, J.-P.: Définition combinatoire des homomorphismes de Poincaré, Alexander et Thom pour une pseudo-variété, Astérisque 82–83, Soc. Math. France, pp. 71–91 (1981)
Brouwer, L.E.J.: Über eineindeutige, stetige Transformationen von Flächen in sich. Math. Ann. 69, 176–180 (1910)
Fulton, W.: Intersection Theory. Springer, New York (1984)
Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology, vol. I. Academic Press, New York (1972)
Lefschetz, S.: Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28, 1–49 (1926)
Lefschetz, S.: On the fixed point formula. Ann. Math. 38, 819–822 (1937)
Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle IV. J. Math. Pures et Appl. 2, 151–217 (1886)
Saveliev, P.: Lefschetz coincidence theory for maps between spaces of different dimensions. Topol. Appl. 116, 137–152 (2001)
Steenrod, N.E.: The work and influence of Professor Lefschetz in algebraic topology, Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefschetz, Princeton Univ. Press, Princeton, pp. 24–43 (1957)
Suwa, T.: Indices of Vector Fields and Residues of Singular Holomorphic Foliations. Hermann, Paris (1998)
Suwa, T.: Residue theoretical approach to intersection theory, Real and Complex Singularities. Contemp. Math. Am. Math. Soc. 459, 207–261 (2008)
Suwa, T.: Complex Analytic Geometry (tentative title), in preparation
Vick, J.W.: Homology Theory: An Introduction to Algebraic Topology. Gaduate Texts in Mathematics. Springer, New York (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
J.-P. Brasselet: Partially supported by CNRS, Aix-Marseille University and FAPESP Grant No. 2015/06697-9. T. Suwa: Partially supported by the JSPS Grant nos. 24540060 and 16K05116.
Rights and permissions
About this article
Cite this article
Brasselet, JP., Suwa, T. Local and global coincidence homology classes. J. Fixed Point Theory Appl. 23, 24 (2021). https://doi.org/10.1007/s11784-021-00857-1
Accepted:
Published:
DOI: https://doi.org/10.1007/s11784-021-00857-1
Keywords
- Alexander duality
- Thom class
- Intersection product with map
- Coincidence homology class
- Lefschetz coincidence point formula