Corpus ID: 237491999

Combinatorial models for topological Reeb spaces

@inproceedings{Trygsland2021CombinatorialMF,
  title={Combinatorial models for topological Reeb spaces},
  author={Paul Trygsland},
  year={2021}
}
There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on stratified spaces are introduced, including both smooth and combinatorial examples. As a consequence of the simplicial approach taken, the theory comes with a spectral sequence for computing (generalized) homology. We also model the homotopy type of Reeb… 

Figures from this paper

References

SHOWING 1-10 OF 26 REFERENCES
Categorified Reeb Graphs
TLDR
A natural construction for smoothing a Reeb graph to reduce its topological complexity is obtained and an ‘interleaving’ distance is defined which is stable under the perturbation of a function.
Floer's infinite dimensional Morse theory and homotopy theory
This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology. Its objectives are as follows: (A) to describe some of our ideas concerning what
On the classifying spaces of discrete monoids
This is proved by constructing directly, for each connected simplicial complex X, a monoid M such that BM = X. The monoid M has one generator for each simplex (+ outside of a maximal tree in X. This
Discrete Morse theory and classifying spaces
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of
The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem
Let G be a group or monoid which is presented by means of a complete rewriting system. Then one can use the resulting normal forms to collapse the classifying space of G down to a quotient complex
Simplicial Homotopy Theory
TLDR
Simplicial sets, model categories, and cosimplicial spaces: applications for homotopy coherence, results and constructions, and more.
Reeb graphs for shape analysis and applications
TLDR
An overview of the mathematical properties of Reeb graphs is provided and its history in the Computer Graphics context is reconstructed, with an eye towards directions of future research.
The Fundamental Group of the Hawaiian earring is not Free
  • B. D. Smit
  • Mathematics, Computer Science
    Int. J. Algebra Comput.
  • 1992
TLDR
A short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of the Hawaiian earring, which is a countably infinite union of circles that are all tangent to a single line at the same point.
Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition
TLDR
The method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data, and is not dependent on any particular clustering algorithm, i.e. any clustering algorithms may be used with Mapper.
Classifying spaces and spectral sequences
© Publications mathematiques de l’I.H.E.S., 1968, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www.
...
1
2
3
...