We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M . If the K-action is Hamiltonian, we obtain results on the cohomology and the K-equivariant cohomology of M .

We compute an invariant counting gradient flow lines (including closed orbits) in S-valued Morse theory, and relate it to Reidemeister torsion for manifolds with χ = 0, b1 > 0. Here we extend the results in [6] following a different approach. However, this paper is written in a self-contained manner and may be read independently of [6]. The motivation of this work is twofold: on the one hand, i...

SupposeM is a finite simplicial complex and that for 0 = t0, t1, . . . , tr = 1 we have a discrete Morse function Fti : M → R. In this paper, we study the births and deaths of critical cells for the functions Fti and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the triangulation of M is the same for each ti, and then generalize to the c...

Let ϕ t be a continuous flow on a metric space X and I be an isolated invariant set

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar c...

Let f be a Morse function on a closed manifold M , and v be a Riemannian gradient of f satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function f associates to these data the Morse complex M * (f, v). In the present paper we introduce a new class of vector fields (f-gradie...

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.

How to put a metric on a given simplicial complex? One way is to declare all edges to have unit length, and to regard all triangles as equilateral triangles in the Euclidean plane. This yields the equilateral flat metric, also known as regular metric. Many other options are possible; for example, one can assign different lengths to the various edges. The metric is called acute (resp. non-obtuse...