Lefschetz periodic point free self-maps of compact manifolds

Let f be a self-map of a compact connected manifold M. We characterize Lefschetz periodic point free continuous self-maps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous self-maps on connected compact manifolds, Topology Appl. 158 (16) (2011) 2165–2169].     

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: CPn the n-dimensional complex projective space, HPn the n-dimensional quaternion projective space, Sn the n-dimensional sphere and Sp×Sq the product space of the p-dimensional with the q-dimensional spheres.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p³, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.     

A combinatorial Lefschetz fixed point formula

We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdⁿK →K is expanding directions preserving.     

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H ⁰(E) and H ⁰(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π ₁ (G/H) = 0.     

On Lefschetz series

Let R = ⊕ _i=0^∞ R ⁱ be a connected graded commutative algebra over the field ℚ of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f) on the ℚ vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f). The main tool we used is the graded algebra version of Gr?bner basis theory. At the end of this paper, some examples and applications are given.     

Least number of periodic points of self-maps of Lie groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f :M → M of a compact smooth manifold of dimension at least 3:N Fn(f) = min{#Fix(gn); g ～f; g is continuous} and N J Dn(f) = min{#Fix(gn); g ～ f; g is smooth}.In general,N J Dn(f) may be much greater than N Fn(f).If M is a torus,then the invariants are equal.We show that for a self-map of a nonabelian compact Lie group,with free fundamental group,the equality holds  all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.     

Local contribution to the Lefschetz fixed point formula

Summary For a class of self-correspondencesC calledweakly hyperbolic, we give a computable formula for the contribution of a fixed point component to the Lefschetz number ofC. The formula applies to Lefschetz numbers of cohomology with coefficients in a constructible complex of sheaves (such as intersection homology).Oblatum 9-XI-1990 & 29-IV-1992In memory of J.L. VerdierPartially supported by NSF grant # DMS8802638 and DMS9001941Partially supported by NSF grant # DMS8803083 and DMS9106522     

On the growth of the number of periodic points for smooth self-maps of a compact manifold

Let be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension . We show that in the homotopy class of there is a map with less then periodic points, up to any given fixed period .

     

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

Let M be a smooth compact and simply-connected manifold with simply-connected boundary \(\partial M\), r be a fixed odd natural number. We consider f, a \(C^1\) self-map of M, preserving \(\partial M\). Under the assumption that the dimension of M is at least 4, we define an invariant \(D_r(f;M,\partial M)\) that is equal to the minimal number of r-periodic points for all maps preserving \(\partial M\) and \(C^1\)-homotopic to f. As an application, we give necessary and sufficient conditions for a reduction of a set of r-periodic points to one point in the \(C^1\)-homotopy class.     

On automorphisms and embeddings of free periodic groups

The paper gives a construction of a free monoid of rank 2 in the group of automorphisms of free periodic groups B(m, n) of any odd period n ≥ 665 and any rank m > 1.Moreover, it is proved that if the period is any prime numbern > 1003 and the group B(m, n) is nested in some n-periodic group G as a normal subgroup, then B(m, n) is a direct factor in G.     

A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps

Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.