Bounds for fixed points on products of hyperbolic surfaces

TL;DR

This paper establishes finite bounds on fixed point indices, Lefschetz numbers, and Nielsen numbers for self-homeomorphisms of products of hyperbolic surfaces, advancing understanding in topological fixed point theory.

Contribution

It provides the first finite bounds for fixed point indices, Lefschetz, and Nielsen numbers on products of hyperbolic surfaces, answering a special case of Jiang's question.

Findings

Bound $ ext{B}$ for fixed point index is finite.

Bounds for Lefschetz number $L(f)$ are established.

Bounds for Nielsen number $N(f)$ are provided.

Abstract

For the product $S_1\times S_2$ of any two connected compact hyperbolic surfaces $S_1$ and $S_2$, we give a finite bound $\mathcal{B}$ such that for any self-homeomorphism $f$ of $S_1\times S_2$ and any fixed point class $F$ of $f$, the index $|ind(f, F)|\leq \mathcal{B}$, which is an affirmative answer for a special case of a question asked by Boju Jiang. Moreover, we also give bounds for the Lefschetz number $L(f)$ and the Nielsen number $N(f)$ of the homeomorphism $f$.