[Paper Review] Maps on graphs can be deformed to be coincidence-free

P. Christopher Staecker|arXiv (Cornell University)|Jun 27, 2010

Computational Geometry and Mesh Generation3 references4 citations

TL;DR

This paper proves that any two continuous maps between connected graphs can be made coincidence-free via homotopy if the codomain is not homeomorphic to a circle. The key method uses a 'road traffic' analogy to homotopically reroute maps around coincidence points using forks in the graph structure, showing that nontrivial coincidence invariants like the Nielsen number or Reidemeister trace must vanish in this setting, invalidating earlier claims in related work.

ABSTRACT

We give a construction to remove coincidence points of continuous maps on graphs (1-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidence index, Nielsen coincidence number, or coincidence Reidemeister trace in this setting, and the results of our previous paper "A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles" are invalid.

Motivation & Objective

• To resolve a fundamental flaw in prior work claiming to compute nontrivial coincidence invariants for maps on bouquets of circles.
• To establish conditions under which coincidence points of maps on graphs can be removed via homotopy.
• To demonstrate that the coincidence index, Nielsen number, and Reidemeister trace must all vanish when the codomain is not a circle.
• To correct the flawed use of the coincidence index in earlier literature, particularly in [3], by showing its homotopy instability.
• To clarify the distinction between fixed point theory and coincidence theory in the context of graphs.

Proposed method

• Uses a homotopy-based strategy inspired by a road traffic analogy, where two maps are viewed as cars on a network of single-lane roads.
• Applies a local homotopy transformation near each coincidence point in the interior of an edge, redirecting the maps via a trivalent vertex (a 'fork') to avoid collision.
• Employs a topological construction where maps retreat to a common vertex, traverse different edges to pass each other, and resume their paths with reversed order.
• Relies on the assumption that the codomain graph is not a manifold (specifically not homeomorphic to a circle), ensuring the existence of a trivalent vertex for the rerouting.
• Uses a lemma to reduce the problem to coincidence points in the interior of edges, enabling repeated application of the local homotopy construction.
• Applies the construction iteratively to remove all coincidence points, proving that any pair of maps can be made coincidence-free under the stated condition.

Experimental results

Research questions

• RQ1 Can any pair of continuous maps between graphs be made coincidence-free through homotopy when the codomain is not a circle?
• RQ2 Why does the coincidence index fail to be well-behaved in the setting of graphs, particularly for bouquets of circles?
• RQ3 What is the correct behavior of the Nielsen number and Reidemeister trace for maps on graphs when the codomain is not a circle?
• RQ4 How does the failure of the coincidence index in this setting invalidate the results of prior work, such as the formula for the coincidence Reidemeister trace in [3]?
• RQ5 In what cases can nontrivial coincidence invariants still exist for maps on graphs?

Key findings

• Any two continuous maps from a connected graph X to a connected graph Y can be made coincidence-free via homotopy if Y is not homeomorphic to the circle.
• The coincidence index, Nielsen number, and Reidemeister trace must all vanish for maps on graphs when the codomain is not a circle, as all coincidences can be removed by homotopy.
• The construction in [3] for computing the coincidence Reidemeister trace is fundamentally flawed because it relies on an ill-behaved coincidence index that changes unpredictably when the coincidence value crosses the wedge point.
• The error in [3] arises specifically from defining the coincidence index near non-wedge points as if the space were a differentiable manifold, which fails under homotopy when the image passes through the wedge point.
• When the codomain is the circle, nontrivial coincidence invariants can still exist; for example, MC(f,g) = |deg(f) - deg(g)| for maps f,g: S¹ → S¹.
• The result shows that coincidence theory on graphs is not a generalization of fixed point theory, as fixed points of self-maps on graphs may be essential (non-removable), while coincidences with the identity can always be removed.