The main aim of this paper is to present the construction of the Morse– Smale complex of a compact smooth manifold M with boundary and to establish the connection to the topology of M . This approach to connecting the analysis of an appropriate function f : M → R — respectively the dynamical system associated to such a function and a Riemannian metric g — with the topology of the manifold was i...

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: ...

Segmentation of 3D data (some time 4D data) is a very challenging problem in applications exploiting Marine GIS data. To tackle this problem, this paper proposes a topological approach based on the Digital Morse theory which is a kind of Discrete Morse theory to high dimension Grid points. The essence of the approach concerns detecting critical points in the High dimension Data, which represent...

In this paper we prove the Cheeger–Müller theorem for $$L^2$$ -analytic torsion form under assumption that there exists a fiberwise Morse function and Novikov–Shubin invariants are positive.

In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlyin...

If a complex analytic function, f , has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of f has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along f have isolated support. We also investigate an index-theo...