Lefschetz Coincidence Number Research Articles

Nielsen coincidence theory on infra-solvmanifolds of Sol

We derive averaging formulas for the Lefschetz coincidence numbers, the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type (R). As an application, we compare our formula for the Nielsen coincidence numbers with a result of Jezierski (1992) for pairs of maps on some infra-solvmanifolds of Sol. For all pairs of self-maps of a nonorientable infra-solvmanifold of Sol, we determine the sets of all possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers.

Lefschetz coincidence numbers of solvmanifolds with Mostow conditions

For any two continuous maps $f,g$ between two solvmanifolds of same dimension satisfying the Mostow condition, we give a technique of computation of the Lefschetz coincidence number of $f,g$. This result is an extension of the result of Ha, Lee and Penninckx for completely solvable case.

Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds

AbstractWe prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension. In order to obtain these formulas, we use the averaging formulas for the Lefschetz coincidence number and for the Nielsen coincidence number and we develop an averaging formula for the Reidemeister coincidence number. We also give a simple proof of the averaging formula for the Lefschetz coincidence number.Mathematics Subject Classification 2000: 55M20; 57S30.

Anosov theorem for coincidences on special solvmanifolds of type $(\mathrm {R})$

Suppose that S' and S' are simply connected solvable Lie groups of type (R) with the same dimension. We show that the Lefschetz coincidence numbers of maps f,g: Γ\S → Γ'\S' between special solvmanifolds Γ\S → Γ'\S' can be computed algebraically as follows: where F * , G * are the matrices, with respect to any preferred bases, of morphisms of Lie algebras induced by f and g. This generalizes a recent result by S. W. Kim and J. B. Lee to special solvmanifolds of type (R). Moreover, we can drop the dimension match condition imposed in the latter result.

Lefschetz numbers for maps of fiber spaces

Expressions for the Lefschetz numbers of maps of fiber spaces in terms of those of maps of the fibers and the base are obtained. A similar problem is considered for Lefschetz coincidence numbers.

The coincidence lefschetz number for self-maps of lie groups

Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.

Maps into the torus and minimal coincidence sets for homotopies

LetX;Y be manifolds of the same dimension. Given continuous mappings fi;gi : X ! Y , i = 0; 1, we consider the 1-parameter coincidence problem of nding homotopies ft;gt, 0 t 1, such that the number of coincidence points for the pair ft;gt is independent of t. When Y is the torus and f0;g0 are coincidence free we produce coincidence free pairs f1;g1 such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the Lefschetz coincidence number.

Coincidences of maps into homogeneous spaces

Let f,g:X→M be maps between two closed connected orientable n-manifolds where M=G/K is the homogeneous space of left cosets of a compact connected Lie group G by a finite subgroup K. In this note, we obtain a simple formula for the Lefschetz coincidence number L(f,g) in terms of topological degree, generalizing some previously known formulas for fixed points. Our approach, by means of Nielsen root theory, also allows us to give a simpler and more geometric proof of the fact that all coincidence classes of f and g have coincidence index of the same sign.

Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, II

McCord (1991) claimed that Nielsen coincidence numbers and Lefschetz coincidence numbers are related by the inequality N(f, g) ⩾ ¦L(f, g)¦ for all maps f, g: S 1 → S 2 between compact orientable solvmanifolds of the same dimension. It was further claimed that N(f, g) = ¦L(f, g)¦ when S 2 is a nilmanifold. A mistake in that paper has been discovered. In this paper, that mistake is partially repaired. A new proof of the equality N(f, g) = ¦L(f, g)¦ for nilmanifolds is given, and a variety of conditions for maps on orientable solvmanifolds are established which imply the inequality N(f, g) ⩾ ¦L(f, g)¦. However, it still remains open whether N(f,g) ⩾ ¦L(f,g)¦ for all maps between orientable solvmanifolds.

Some applications of coincidence theory

As a first application we compute the Lefschetz coincidence number of maps between manifolds $$\mathcal{M}$$ whose rational singular cohomologyH* $$(\mathcal{M};\mathbb{Q})$$ has a simple system of generators. For the second let $$\mathcal{M}$$ be anH-manifold with multiplicationm. Define for $$x \in \mathcal{M}$$ ,m 2 (x,x)=m(x,x) andm k (x)=m(x,m k−1 (x)), for allk>2. All roots of equationm k (x)=m 8 (x),k>s such thatm k (x)=m 3 (x) butm i (x)≠m j (x) for allk>i>j,s≥j≥0 andk−s does not dividei−j, split into a finite number of equivalence classes. We compute precisely the numbers of classes such that their members satisfy the above property.

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

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 Nakaoka's 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.

Estimating Nielsen numbers on infrasolvmanifolds

A well-known lower bound for the number of fixed points of a selfmap /: X —» X is the Nielsen number N(f). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number L(f), on the other hand, is readily computable, but does not give a lower bound for the number of fixed points. In this paper, we investigate conditions on the space X which guarantee either N(f) = \L(f) or N(f) > \L(f) . By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) > \L(f) for all self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable subgroup of finite index). Moreover, for infranilmanifolds, there is a Lefschetz number formula which computes N(f). 1. Estimating Nielsen numbers. Consider a continuous self-map /: X -> X. Let Fix(/) denote the fixed point set {x e X\f(x) = x} . One of the fundamental problems of fixed point theory is to estimate (preferably from below) the cardinality of this set. The Nielsen number N(f) provides such an estimate: it is an integer homotopy invariant which provides a lower bound on the number of fixed points of g, for all maps g homotopic to /. This estimate is sharp for all compact manifolds save surfaces of negative Euler characteristic. Its one drawback is that it is very difficult to compute N(f) from its definition, so that other means must be sought. At least, since the Nielsen number provides a lower bound for the original topological object |Fix(/)|, it would be useful to find lower bounds for N(f). We will refer to the search for lower bounds to N(f) as the problem of estimating N(f) while the search for other algebraic-topological means of finding the exact value of N(f) will be referred to as the problem of computing N(f). The Lefschetz number L(f) is a (reasonably) computable invariant, but in general, there is no relation between L(f) and either N(f) or |Fix(/)|. One approach to computing the Nielsen number is to find conditions on either the space X or the map / which allow N(f) and L(f) to be related. The Jiang condition, for example, is a condition on the map / which, when satisfied, computes N(f) from L(f)

Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds

Suppose M 1, M 2 are compact, connected orientable manifolds of the same dimension. Then for all pairs of maps ƒ, g: M 1→ M 2, the Nielsen coincidence number N(ƒ, g) and the Lefschetz coincidence number L(ƒ, g) are measures of the number of coincidences of ƒ and g: points x∈ M 1 with f( x)= g( x). A manifold is a nilmanifold (solvmanifold) if it is a homogenous space of a nilpotent (solvable) Lie group. If M 1 and M 2 are compact connected orientable solvmanifolds, then N( ƒ, g)⩾| L( ƒ, g)| for all ƒ and g, with equality for all ƒ and g if M 2 is a nilmanifold.

On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem

Let f, g : M1 → M2 be maps where M1 and M2 are connected triangulable oriented n-manifolds so that the set of coincidences Cf,g = {x ∈ M1 | f(x) = g(x)} is compact in M1. We define a Nielsen equivalence relation on Cf,g and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if M2 = M2/K where M2 is a connected simply connected topological group and K is a discrete subgroup then all the Nielsen coincidence classes of f and g have the same coincidence index. In particular, when M1 and M2 are compact, f and g are deformable to be coincidence free if the Lefschetz coincidence number L(f, g) vanishes. 0. Introduction. In topological fixed point theory, the Lefschetz fixed point theorem states that the vanishing of the Lefschetz number L(f) of a self-map f : X → X on a compact manifold is necessary to deforming f to be fixed point free. The converse, however, is false in general. If X satisfies the so-called Jiang condition, L(f) = 0 is also sufficient. The essence of the Jiang condition is that all fixed point classes have the same index. Thus the Nielsen number N(f) divides L(f) which is easier to compute. A theorem of Anosov asserts that |L(f)| = N(f) for any self-map f on a nilmanifold. This result is recently generalized to coincidences independently by Jezierski [7], McCord [10], Brooks and Wong [3]. For nilmanifolds, coincidence classes have the same index. The technique used in [3] is a local version of the root theory developed by R. Brooks [1] (see also [9,V]). It is shown in [3] that if φ : N →M is a map from an oriented n-manifold N to an oriented triangulable n-manifold M such that the set of roots Rφ = {x ∈ N | φ(x) = e} is compact, then all the root classes have the same index. The purpose of this note is to use this fact about roots to show that (local) coincidence and hence (local) fixed point 1991 Mathematics Subject Classification: Primary 55M20.