Obstruction theory for coincidences of multiple maps

TL;DR

This paper explores the obstruction to deforming multiple maps into coincidence-free maps, extending Lefschetz-type theorems and providing examples where coincidence cannot be eliminated through homotopy.

Contribution

It investigates the converse of the Lefschetz coincidence theorem for multiple maps and constructs examples illustrating the limitations of deforming maps to avoid coincidences.

Findings

Non-vanishing Lefschetz class implies coincidence existence

Constructed example of maps on symplectic 4-manifold with unavoidable coincidences

Demonstrated that some pairs of maps cannot be homotoped to be coincidence free

Abstract

Let $f_1,..., f_k:X\to N$ be maps from a complex $X$ to a compact manifold $N$, $k\ge 2$. In previous works \cite{BLM,MS}, a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class $L(f_1,...,f_k)$ implies the existence of a coincidence $x\in X$ such that $f_1(x)=...=f_k(x)$. In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps $f_1,...,f_k$ to be coincidence free. We construct an example of two maps $f_1,f_2:M\to T$ from a sympletic $4$-manifold $M$ to the $2$-torus $T$ such that $f_1$ and $f_2$ cannot be homotopic to coincidence free maps but for {\it any} $f:M\to T$, the maps $f_1,f_2,f$ are deformable to be coincidence free.