[Paper Review] Nielsen coincidence theory in arbitrary codimensions

Ulrich Koschorke|ArXiv.org|Aug 3, 2004

Advanced Differential Geometry Research6 references4 citations

TL;DR

This paper extends Nielsen coincidence theory to arbitrary codimensions by introducing a new invariant, the Nielsen coincidence number $N(f_1,f_2)$, derived from normal bordism in the path space $E(f_1,f_2)$. It proves that $MCC(f_1,f_2) = N(f_1,f_2)$ in three key cases: when $m < 2n - 2$, when $N$ is the circle, and when $M$ and $N$ are spheres under a James-Hopf injectivity condition, establishing $N(f_1,f_2)$ as a sharp lower bound for the minimum number of coincidence components under homotopy.

ABSTRACT

Given two maps f_1, f_2 : M^m \longrightarrow N^n between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC (f_1, f_2) of pathcomponents of the coincidence space of maps f'_1, f'_2 where f'_i is homotopic to f_i, i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N (f_1, f_2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N (f_1, f_2) turns out to coincide with MCC (f_1, f_2): (i) when m &lt; 2n - 2; (ii) when N is the unit circle; and (iii) when M and N are spheres and a certain injectivity condition involving James-Hopf invariants is satisfied. We also exhibit situations where N (f_1, f_2) vanishes, but MCC (f_1, f_2) is strictly positive.

Motivation & Objective

• To generalize Nielsen fixed point and coincidence theory beyond the classical case of equal dimensions ($m = n$) to arbitrary codimensions ($m \neq n$).
• To define a new Nielsen-type invariant $N(f_1,f_2)$ as a lower bound for the minimum number of coincidence components $MCC(f_1,f_2)$ under homotopy.
• To determine when $N(f_1,f_2)$ equals $MCC(f_1,f_2)$, thus identifying it as a complete obstruction to loosening maps.
• To analyze the role of path components in the space $E(f_1,f_2)$, which parametrize Reidemeister classes, and their contribution to the normal bordism class $\widetilde{\omega}(f_1,f_2)$.
• To investigate the failure of $N(f_1,f_2)$ to detect looseness by constructing examples where $N(f_1,f_2) = 0$ but $MCC(f_1,f_2) > 0$.

Proposed method

• Construct the space $E(f_1,f_2)$ of pairs $(x, \theta)$, where $x \in M$ and $\theta$ is a path in $N$ from $f_1(x)$ to $f_2(x)$, to encode Reidemeister classes.
• Define the normal bordism class $\widetilde{\omega}(f_1,f_2) \in \Omega_{m-n}(E(f_1,f_2); \widetilde{\varphi})$, where $\widetilde{\varphi} = \text{pr}^*(f_1^*TN - TM)$, using Hatcher-Quinn theory.
• Decompose $\widetilde{\omega}(f_1,f_2)$ into components indexed by path components $A \in \pi_0(E(f_1,f_2))$, and define $N(f_1,f_2)$ as the number of essential components (nontrivial in bordism).
• Use the generalized Whitney trick via paths in $E(f_1,f_2)$ to construct homotopies $f_i \sim f_i'$ that eliminate coincidences when $N(f_1,f_2) = 0$.
• Apply framed bordism and stable homotopy theory to analyze $\Gamma_k$ maps and their behavior under reflections and sign changes in self-intersection invariants.
• Use Toda’s tables and EHP sequences to verify injectivity and non-injectivity of the $\Gamma_k$ homomorphism in specific stable homotopy groups.

Experimental results

Research questions

• RQ1 When is the Nielsen coincidence number $N(f_1,f_2)$ equal to the minimum coincidence component count $MCC(f_1,f_2)$?
• RQ2 Under what codimension conditions does $N(f_1,f_2)$ serve as a complete obstruction to loosening two maps?
• RQ3 Can $N(f_1,f_2)$ vanish while $MCC(f_1,f_2)$ remains positive, indicating a failure of the Nielsen number to detect non-looseness?
• RQ4 How do the path components of $E(f_1,f_2)$, corresponding to Reidemeister classes, influence the structure of the normal bordism class $\widetilde{\omega}(f_1,f_2)$?
• RQ5 What is the behavior of the $\Gamma_k$ homomorphism in stable homotopy groups, and when is it injective or non-injective?

Key findings

• When $m < 2n - 2$, the Nielsen number $N(f_1,f_2)$ equals $MCC(f_1,f_2)$, so $N(f_1,f_2) = 0$ if and only if the pair $(f_1,f_2)$ is loose.
• When $N$ is the unit circle $S^1$, $N(f_1,f_2) = MCC(f_1,f_2)$ holds, extending the classical Nielsen theory to codimension 1.
• For spheres $M = S^m$, $N = S^n$, if a certain injectivity condition involving James-Hopf invariants is satisfied, then $N(f_1,f_2) = MCC(f_1,f_2)$.
• There exist examples where $N(f_1,f_2) = 0$ but $MCC(f_1,f_2) > 0$, showing that the Nielsen number is not always a complete obstruction in general codimensions.
• The homomorphism $\Gamma_k: \pi_{m}(S^n) \to \pi^S_{m-1-k(n-1)}$ is injective on $\pi_{n+1}(S^n)$ for $n \geq 2$, and on $\pi_{n+2}(S^n)$ and $\pi_5(S^2)$, with explicit verification via Pontryagin-Thom construction and Toda’s tables.
• For $\Gamma: \pi_{24}(S^6) \to \pi^S_{23}$, the map is not injective, as the domain $\mathbb{Z}_{24} \oplus \mathbb{Z}_6 \oplus \mathbb{Z}_2$ has larger order than the relevant torsion part of the target, and $2\Gamma_4 \equiv 0$.