Abstract
We study Nielsen coincidence theory for maps between manifolds of same dimension regardless of orientation. We use the definition of semi-index of a class, review the definition of defective classes, and study the occurrence of defective root classes. We prove a semi-index product formula for lifting maps and give conditions for the defective coincidence classes to be the only essential classes.
Highlights
In [2, 6] the Nielsen coincidence theory was extended to maps between nonorientable topological manifolds
If we define h = ( f,g) : M → N × N as usual, we may assume that h is in a transverse position, that is, the coincidence set Coin( f, g) = {x ∈ M | f (x) = g(x)} is finite and for each coincidence point x there is a chart Rn × Rn = U ⊂ N × N such that (U, ( f, g)(M) ∩ U, ΔN ∩ U) corresponds to (Rn × Rn, Rn × 0, 0 × Rn)
The semi-index is homotopy invariant, it is well defined for all continuous maps, and if U ⊂ M is an open subset such that Coin( f,g) ∩ U is compact, we can extend this definition to that of the semi-index of a pair on the subset U, which is denoted by | ind |( f, g; U)
Summary
In [2, 6] the Nielsen coincidence theory was extended to maps between nonorientable topological manifolds. We say that two coincidence points x, y ∈ Coin( f ,g) are Nielsen related if there is a path γ : [0,1] → M with γ(0) = x, γ(1) = y such that f γ is homotopic to gγ relative to the endpoints. The semi-index is homotopy invariant, it is well defined for all continuous maps, and if U ⊂ M is an open subset such that Coin( f ,g) ∩ U is compact, we can extend this definition to that of the semi-index of a pair on the subset U, which is denoted by | ind |( f , g; U).
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