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By the work of Li, a compact co-Kähler manifold M is a mapping torus K ϕ , where K is a Kähler manifold and ϕ is a Hermitian isometry. We show here that there is always a finite cyclic cover M of the form M ∼ = K × S 1 , where ∼ = is equivariant diffeomorphism with respect to an action of S 1 on M and the action of S 1 on K × S 1 by translation on the second factor. Furthermore, the covering transformations act diagonally on S 1 , K and are translations on the S 1 factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle. MSC classification [2010]: Primary 53C25; Secondary 53B35, 53C55, 53D05.Key words and phrases: co-Kähler manifolds, mapping tori.Let (M 2n+1 , J, ξ, η, g) be an almost contact metric manifold given by the conditionsThe authors of [CDM] use the term cosymplectic for Li's co-Kähler because they view these manifolds as odd-dimensional versions of symplectic manifoldseven as far as being a convenient setting for time-dependent mechanics [DT]. Li's characterization, however, makes clear the true underlying Kähler structure, so we have chosen to follow his terminology.

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Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homo-topical obstruction is described which detects when an action is Hamiltonian. This new entity, the AA-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. Furthermore , new results in symplectic geometry also arise from this homotopical approach.

Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

In this article we study the higher topological complexity TCr(X) in the case when X is an aspherical space, X = K(π, 1) and r ≥ 2. We give a characterisation of TCr(K(π, 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [8], joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TCr(π) in terms of cohomological dimension of subgroups of π × π × · · · × π (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of the Higman's groups. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [17] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC-generating function ∞ r=1 TCr+1(X)x r encoding the values of the higher topological complexity TCr(X) for all values of r. We show that in many examples (including the case when X = K(H, 1) with H being a RAA group) the TC-generating function is a rational function of the form P (x)(1−x) 2 where P (x) is an integer polynomial with P (1) = cat(X).