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**11 - 12**of**12**### The boundary-Wecken classification of surfaces

, 2004

"... Let X be a compact 2-manifold with nonempty boundary ∂X and let f: (X, ∂X) → (X, ∂X) be a boundary-preserving map. Denote by MF∂[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N∂(f) is th ..."

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Let X be a compact 2-manifold with nonempty boundary ∂X and let f: (X, ∂X) → (X, ∂X) be a boundary-preserving map. Denote by MF∂[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N∂(f) is the sum of the number of essential fixed point classes of the restriction ¯ f: ∂X → ∂X and the number of essential fixed point classes of f that do not contain essential fixed point classes of ¯f. We prove that if X is the Möbius band with one (open) disc removed, then MF∂[f] − N∂(f) ≤ 1 for all maps f: (X, ∂X) → (X, ∂X). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Möbius band, then X is boundary-Wecken, that is, MF∂[f] = N∂(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Möbius band with one disc removed, then X is not boundary-Wecken, but MF∂[f]−N∂(f) ≤ 1. All other surfaces are totally non-boundary-Wecken, that is, given an integer k ≥ 1, there is a map fk: (X, ∂X) → (X, ∂X) such that MF∂[fk]− N∂(fk) ≥ k.

### OBSTRUCTION THEORY AND MINIMAL NUMBER OF COINCIDENCES FOR MAPS FROM A COMPLEX INTO A MANIFOLD

"... Abstract. The Nielsen coincidence theory is well understood for a pair of maps between n-dimensional compact manifolds for n greater than or equal to three. We consider coincidence theory of a pair (f, g): K → N n,where K is a finite simplicial complex of the same dimension as the manifold N n. We c ..."

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Abstract. The Nielsen coincidence theory is well understood for a pair of maps between n-dimensional compact manifolds for n greater than or equal to three. We consider coincidence theory of a pair (f, g): K → N n,where K is a finite simplicial complex of the same dimension as the manifold N n. We construct an algorithm to find the minimal number of coincidences in the homotopy class of the pair based on the obstruction to deform the pair to coincidence free. Some particular cases are analyzed including the one where the target is simply connected. 1.