Generalizations of the Coincidence Value Property

TL;DR

This paper explores generalized conditions for the existence of coincidence points of commuting continuous maps on Hausdorff spaces, linking these properties to group actions, fiber bundles, and sheaf theory, with specific results for spheres and disks.

Contribution

It introduces generalized notions of the coincidence value property and connects these to topological structures like group actions and sheaf theory, expanding understanding of coincidence phenomena.

Findings

Established conditions for coincidence points on spheres and disks.

Linked coincidence properties to group actions and fiber bundles.

Developed a sheaf-theoretic approach to analyze coincidence values.

Abstract

For $f,g:X\longrightarrow X$ continuous and commuting maps of a Hausdorff space, we investigate various conditions on $X$ and on the pair $(f,g)$ which provide existence of a coincidence value. We introduce generalized notions of the coincidence value property and use this added flexibility to determine how various coincidence properties of $X$ are related to group actions on $X$ and to coincidence properties of associated fibre bundles and adjunction spaces. We also present a sheaf theoretical approach to obtaining information concerning coincidence values through construction of an "almost constant" presheaf. In particular, we prove several partial results concerning the special cases where $X$ is either a low dimensional dimensional sphere or the closed unit disk.