Abstract
The main result of this paper states that if \(F: T \times I \rightarrow T\) is a homotopy on 2-dimensional torus and \(\pi _{1}(T,x_{0}) = <u,v|[u,v]=1>\), then the one-parameter Lefschetz class L(F) of F is given by
where N(F) is the one-parameter Nielsen number of F, and \(\alpha = [u] \in H_{1}(\pi _{1}(T),\mathbb {Z})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brooks, R.B.S., Brown, R.F., Park, J., Taylor, D.H.: Nielsen numbers of maps of tori. Proc. Am. Math. Soc. 52, 398–400 (1975)
Dimovski, D.: One-parameter fixed point indices. Pac. J. Math. 2, 164 (1994)
Dimovski, D., Geoghegan, R.: One-parameter fixed point theory. Forum Math. 2, 125–154 (1990)
Gonçalves, D.L., Kelly, M.R.: Maps into the torus and minimal coincidence sets for homotopies. Fundam. Math. 172, 99–106 (2002)
Geoghegan, R., Nicas, A.: Parametrized Lefschetz–Nielsen fixed point theory and Hochschild homology traces. Am. J. Math. 116, 397–446 (1994)
Geoghegan, R., Nicas, A.: Trace and torsion in the theory of flows. Topology 33(4), 683–719 (1994)
Schirmer, H.: Fixed point sets of homotopies. Pac. J. Math. 108(1), 191–202 (1983)
Silva, W.L.: Minimal fixed point set of fiber-preserving maps on T-bundles over \(S^{1}\). Topol. Appl. 173, 240–263 (2014)
Whitehead, G.W.: Elements of Homotopy Theory. Springer, Berlin (1918)
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.
Rights and permissions
About this article
Cite this article
Silva, W.L. One-parameter Lefschetz class of homotopies on torus. J. Fixed Point Theory Appl. 22, 26 (2020). https://doi.org/10.1007/s11784-020-0762-3
Published:
DOI: https://doi.org/10.1007/s11784-020-0762-3