Helga Schirmer

Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary

Abstract Nielsen coincidence theory is extended to manifolds with boundary. For X and Y compact connected oriented n -manifolds with boundary, and for maps ƒ : X → Y and g : ( X , ∂ X ) → ( Y , ∂ Y ), a coincidence index—which is a local version of Nakaokas Lefschetz coincidence number—and a Nielsen coincidence number are defined and their properties explored. As an application, coincidence-producing maps g are characterized if Y is acyclic over the rationals and many new examples of coincidence-producing maps are constructed.

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANRs. A Nielsen number of the complement N(f; X, A) and a Nielsen number of the boundary n(f; X, A) are defined. N(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least n(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly N(f; X - A) fixed points on C1(X−A) and n(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).

Nielsen theory of roots of maps of pairs

Abstract A relative root Nielsen number N rel (ƒ; c) is introduced which is a homotopy invariant lower bound for the number of roots at c for a map of pairs of spaces ƒ : (X, A) → (Y, B) and c ϵ Y . Conditions are given which ensure that N rel (ƒ; c) is a sharp lower bound. The standard method for the computation of the root Nielsen number N(ƒ; c) from the homomorphism of the fundamental group induced by ƒ is extended to obtain formulae for N rel (ƒ; c) . Many of the results depend on the location of c .

Nielsen type numbers for periodic points on nonconnected spaces

Abstract Two homotopy invariant Nielsen type numbers exist for periodic points of a self-map ƒ: X → X of a compact connected ANR X, namely NP n (ƒ) , the Nielsen type number of period n, and NΦ n (ƒ) , the Nielsen type number of the nth iterate. The first is a lower bound for the number of periodic points of least period n, and the second a lower bound for the number of periodic points of all periods m ⩽ n. Both these Nielsen type numbers are extended here to the case where the ANR X is no longer connected. Calculations and many examples are given.

Nielsen type numbers for periodic points on pairs of spaces

Abstract Two homotopy invariant Nielsen type numbers exist for periodic points of a self-map ƒ: X → X of a compact connected ANR X , namely NP n (ƒ) , the Nielsen type number of period n , and NΦ n (ƒ) , the Nielsen type number of the n th iterate. The first is a lower bound for the number of periodic points of least period n , and the second a lower bound for the number of periodic points of all periods m ⩽ n . Both these Nielsen type numbers are extended here to the case of maps ƒ: (X, A) → (X, A) of pairs of compact ANRs. Calculations and many examples are given.

Fixed point sets of deformations of pairs of spaces

Abstract Let (X,A) be a pair of compact polyhedra which satisfies conditions similar to those needed in order to realize the Nielsen number of the identity map in the cases where A = o or A = X. Deformations of (X,A) with minimal fixed point sets are constructed, and are used to construct deformations of (X,A) with prescribed fixed point sets. The size and location of these fixed point sets depends only on the Euler characteristics of the components of X and A. As an application it follows that if M is an even-dimensional compact connected triangulable manifold with boundary Bd M, then there is no difference between possible fixed point sets of deformations of M and of deformations of (M, Bd M).

Fixed point sets of homeomorphisms of compact surfaces

Every closed and non-empty subset of a compact surfaceS can be the fixed point set of a homeomorphism, andS also admits fixed point free homeomorphisms if it does not have the fixed point property. A partial extension to higher dimensions states that every closed and non-empty subset of a compactn-manifold can be the fixed point set of a surjective self-map.

On fixed point sets of homeomorphisms of then-ball

Conditions are investigated under which a subsetA can be the fixed point set of a homeomorphism ofBn. If eitherA ∩ ∂Bn ≠ Ø andn arbitrary orA ∩ ∂Bn=Ø andn even it is necessary and sufficient thatA is non-empty and closed. IfA ∩ ∂Bn=Ø andn odd, conditions which are either necessary or sufficient (but not both) are given.

The absolute degree and the Nielsen root number of compositions and Cartesian products of maps

Abstract Brouwers homological degree has the multiplicative property for the composition of maps. That is, if f :X→Y and g :Y→Z are maps between closed oriented manifolds X,Y,Z of the same dimension, then | deg (g∘f)|=| deg (f)|| deg (g)| . Hopfs absolute degree is defined for maps between all n -manifolds, whether orientable or not, and is equal to the absolute value of the Brouwer degree if the manifolds are orientable. It is shown that the absolute degree does not always have the multiplicative property for compositions, but that it does have this property for orientable maps, i.e., for maps that do not map any orientation-reversing loop to a contractible one. If at least one of f and g is not an orientable map, the absolute degree of the composition g∘f can still be calculated from the absolute degrees of f and g if additional information about these two maps and a “correction term” κ(f,g) that depends on the homomorphisms of the fundamental groups induced by f and g are included. Although the Nielsen root number is closely related to the absolute degree, the multiplicative property for compositions can fail to hold for it even if the manifolds are orientable, but it does hold after the insertion of the correction term κ(f,g) . Other interpretations of this correction term are presented. Given maps f i :X i →Y i between n i -manifolds, for i=1,2 , the Brouwer degree of their Cartesian product f 1 ×f 2 :X 1 ×X 2 →Y 1 ×Y 2 has the multiplicative property | deg (f 1 ×f 2 )|=| deg (f 1 )|| deg (f 2 )| . The results obtained concerning the multiplicative property for the composition of maps are used to investigate the multiplicative property for the Cartesian product of maps. We include an appendix on maps of aspherical spaces: Building on previous results of Brooks and Odenthal we show that if f :X→Y is a map of connected compact infrasolvmanifolds of the same dimension, then the Nielsen root number and absolute degree of f are equal.

The absolute degree and Nielsen root number of a fibre-preserving map

Abstract Let p :M→X and q :N→Y be locally trivial bundles, with fibres F and G , respectively, where all the spaces are connected closed manifolds, but neither the manifolds nor the bundles need be orientable. Assume further that dim X =dim Y and dim F =dim G so dim M =dim N . A fibre-preserving map f :M→N induces a map f :X→Y of the base and maps f x :F x →G f (x) of the fibres. The purpose of this paper is to relate the Nielsen root number NR ( f ) of f with that of f and f x and to do the same for the absolute degree A (f) . If both f and f x are orientable maps or if f is both orientable and root-essential (that is, if NR ( f )>0), then the multiplicative property A (f)= A ( f )· A (f x ) is shown to be valid. Applying this property to selfmaps of compact solvmanifolds produces a computational result for the absolute degree of such a map. If f is a root-essential nonorientable map, the multiplicative property for the absolute degree must be modified in a way that includes a factor κ ( f ) that describes a relationship between the root class structure of f x and that of f . The Nielsen root numbers of any root-essential fibre-preserving map are found to satisfy κ(f)·NR(f)=NR( f )·NR(f x ) . A fibre-preserving version FG (f) of the classical geometric degree G (f) is defined and it is shown that FG (f)= G (f) if and only if the absolute degree has the multiplicative property. Letting MR [ f ] denote the minimum number of points in the pre-image of a given point of N among all maps homotopic to f and FMR [ f ] the same with regard to fibre-preserving maps and homotopies, every fibre-preserving map satisfies FMR[f]=MR[f x ]·MR[ f ] . Moreover, if none of the manifolds M , X and F are two-dimensional, then FMR [ f ]= MR [ f ] if and only if NR(f)=NR( f )·NR(f x ) . A new bundle and pairing, the fibrewise orientation bundle and the orientation bundle pairing, are introduced in order to relate the orientation bundles of base, fibre, and total space of a locally trivial bundle.