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The geodesic complexity of a Riemannian manifold is a numerical isometry invariant that is determined by the structure of its cut loci. In this article we study decompositions of cut loci over whose components the tangent cut loci fiber in a convenient way. We establish a new upper bound for geodesic complexity in terms of such decompositions. As an application, we obtain estimates for the geodesic complexity of certain classes of homogeneous manifolds. In particular, we compute the geodesic complexity of complex and quaternionic projective spaces with their standard symmetric metrics., Comment: 22 pages, revised version, to appear in Journal of Applied and Computational Topology

We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the Lusternik--Schnirelmann category of a symplectically aspherical manifold equals its dimension. Symplectically hyperbolic manifolds are symplectically atoroidal, as are symplectically aspherical manifolds whose fundamental group does not contain free abelian subgroups of rank two. Thus we obtain many new calculations of topological complexity, including iterated surface bundles and symplectically aspherical manifolds with hyperbolic fundamental groups. Our result also applies in the greater generality of cohomologically symplectic manifolds., 12 pages, final version, to appear in Math. Z

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths., 13 pages, revised version, to appear in Calc. Var

The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds for the numbers of critical orbits of SO(n)-invariant functions on spaces of n-spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition., 35 pages, revised version, fixed typos and a mistake in Theorem 6.5 pointed out by D. Kotschick. To appear in Algebraic & Geometric Topology

We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length computations. In particular, we show that for large classes of oriented manifolds, e.g. for spin manifolds, the topological complexity of the oriented frame bundle is bounded from below by the dimension of the base manifold., 18 pages, revised version, to appear in Topology and its Applications

As we have mentioned in the introduction, a crucial step towards defining an \(A_\infty \)-algebra structure on the Morse cochain complex of a single Morse function is to consider perturbed Morse trajectories. More precisely, we want to dicuss curves which do not satisfy a negative gradient flow equation, but a perturbed negative gradient flow equation of the form \(\dot{\gamma }(s) + \nabla ^g f\circ \gamma (s) + Z(s,\gamma (s)) = 0 , \)where we pick up the notation from Chap. 1 and where Z is a suitable time-dependent vector field.

This brief chapter is intended to provide the reader with an overview of the construction of Morse (co)homology for finite-dimensional manifolds. We present the main notions and results in a concise way and give references to detailed presentations and proofs whenever appropriate. Moreover, we establish some notation that will be employed throughout this book. Except for notational conventions, a reader familiar with Morse homology might skip this chapter without disadvantages. There are several detailed and recommendable references on Morse homology, see e.g. the textbooks Schwarz, Morse homology, Birkhauser, Basel, 1993, [Sch93], Banyaga et al. Lectures on Morse homology, Kluwer Academic Publishers Group, Dordrecht, 2004, [BH04], Jost, Riemannian geometry and geometric analysis, Springer, Berlin, 2008, [Jos08, Chap. 7], Nicolaescu, An invitation to Morse theory, Springer, New York, 2011, [Nic11] or Audin and Damian, Morse theory and Floer homology, Springer, London, [AD14] as well as the set of lecture notes Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), UC Berkeley, 2002, [Hut02] and the article Weber, Expo Math, 24(2), 127–159, 2006, [Web06].

We continue by taking a closer look at zero- and one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 enable us to show that zero-dimensional moduli spaces are in fact finite sets. This basic observation allows us to define homomorphisms \(C^*(\,f)^{\otimes d} \rightarrow C^*(\, f)\) for every \(d \ge 2\) via counting elements of these zero-dimensional moduli spaces. After constructing these higher order multiplications explicitly, we will study the compactifications of one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 imply that one-dimensional moduli spaces can be compactified to one-dimensional manifolds with boundary, and we will explicitly describe their boundaries. If we impose an additional consistency condition on the chosen perturbations, the boundary spaces will coincide with product of zero-dimensional moduli spaces of perturbed Morse ribbon trees.

Having established \(A_\infty \)-algebra structures on Morse cochain complexes \(C^*(f)\), we consider their duals, namely Morse chain complexes. We will use this brief final chapter to show that these possess the structures of \(A_\infty \)-bimodules over the \(A_\infty \)-algebra \(C^*(f)\).

Having constructed moduli spaces of perturbed Morse ribbon trees in the previous chapter, we want to investigate sequential compactness properties of these moduli spaces. Our starting point is the consideration of certain sequential compactness results for spaces of perturbed Morse trajectories of the three different types we introduced in Chap. 2. We will show that in all three cases, every sequence in the respective moduli space without a convergent subsequence has a subsequence that converges geometrically against a family of trajectories. The notion of geometric convergence will be made precise in Sects. 5.1 and 5.2.

In the remainder of of this book, we will discuss perturbed Morse ribbon trees, which can be interpreted as continuous maps from a tree to the manifold M which edgewise fulfill perturbed negative gradient flow equations. In this chapter, we will make this notion precise in terms of the constructions of Chaps. 2 and 3. Moreover, we will apply the nonlocal transversality result Theorem 3.10 to equip moduli spaces of perturbed Morse ribbon trees with the structures of finite-dimensional manifolds of class \(C^{n+1}\).