Homotopy Actions, Cyclic Maps and their Duals

An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1.Comment: 16 pages, LaTeX 2

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.Comment: 9 page

Duan's fixed point theorem: Proof and generalization

Let X be an H-space of the homotopy type of a connected, finite CW-complex, f:X→X any map and pk:X→X the kth power map. Duan proved that pkf:X→X has a fixed point if k≥2. We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces X whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a θ-structure μθ:X→X as defined by Hemmi-Morisugi-Ooshima. The conclusion is that μθf and fμθ each has a fixed point

Nearly trivial homotopy classes between finite complexes

AbstractWe construct examples of essential maps of finite complexes f:X→Y which are trivial of order ⩾n. This latter condition implies that for any space K with cone length ⩽n, the induced map f∗=0:[K,X]→[K,Y]. The main result establishes a connection between the skeleta of the infinite dimensional domains of essential phantom maps and the finite dimensional domains of maps which are trivial of order ⩾n. In particular, there are essential maps f:Σ2i(CPt/S2)→M(Z/ps,2l+3) which are trivial of order ⩾n

Cell polarization in budding and fission yeasts

Polarization is a fundamental cellular property, which is essential for the function of numerous cell types. Over the past three to four decades, research using the best-established yeast systems in cell biological research, Saccharomyces cerevisiae (or budding yeast) and Schizosaccharomyces pombe (or fission yeast), has brought to light fundamental principles governing the establishment and maintenance of a polarized, asymmetric state. These two organisms, though both ascomycetes, are evolutionarily very distant and exhibit distinct shapes and modes of growth. In this review, we compare and contrast the two systems. We first highlight common cell polarization pathways, detailing the contribution of Rho GTPases, the cytoskeleton, membrane trafficking, lipids, and protein scaffolds. We then contrast the major differences between the two organisms, describing their distinct strategies in growth site selection and growth zone dimensions and compartmentalization, which may be the basis for their distinct shape