# Random Simple-Homotopy Theory

@article{Benedetti2021RandomST, title={Random Simple-Homotopy Theory}, author={Bruno Benedetti and Crystal Lai and Davide Lofano and Frank H. Lutz}, journal={ArXiv}, year={2021}, volume={abs/2107.09862} }

We implement an algorithm RSHT (Random Simple-Homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions. For triangulated d-manifolds with d ≤ 6, we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex… Expand

#### Figures and Tables from this paper

#### References

SHOWING 1-10 OF 45 REFERENCES

Simplicial Manifolds, Bistellar Flips and a 16-Vertex Triangulation of the Poincaré Homology 3-Sphere

- Mathematics, Computer Science
- Exp. Math.
- 2000

A computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices is presented and it is shown that if a d-manifold, with d ≥ 5, admits any triangulation on n vertices, it admits a noncombinatorial triangu lation on n + 12 vertices. Expand

Random Discrete Morse Theory and a New Library of Triangulations

- Mathematics, Computer Science
- Exp. Math.
- 2014

The existing libraries of examples in computational topology are “too easy” for testing algorithms based on discrete Morse theory and a new library containing more complicated (and thus more meaningful) test examples is proposed. Expand

Extremal Examples of Collapsible Complexes and Random Discrete Morse Theory

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2017

It is proved that after repeated barycentric subdivisions, the discrete Morse vectors found by randomized algorithms have, on average, an exponential number of critical cells asymptotically almost surely. Expand

A potential smooth counterexample in dimension 4 to the Poincare conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture

- Mathematics
- 1985

WE DESCRIBE a homotopy 4-sphere X4, built with the usual zero and 4-handle and two lhandles and two 2-handles (Figure 28). Of course X4 is homeomorphic to S4 [Freedman] but considerable effort has… Expand

The worst way to collapse a simplex

- Mathematics
- Israel Journal of Mathematics
- 2021

In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose… Expand

A USER'S GUIDE TO DISCRETE MORSE THEORY

- Mathematics
- 2002

A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us… Expand

The dunce hat in a minimal non-extendably collapsible 3-ball

- Mathematics
- 2009

We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space. We show there is a simplicial 3-ball with 8 vertices that is collapsible, but also… Expand

The 9-vertex complex projective plane

- Mathematics
- 1983

In the early days of topology, most of the objects of interest were defined in terms of triangulations, describing a topological space as a union of finitely many vertices, edges, triangles, and… Expand

Knots in Collapsible and Non-Collapsible Balls

- Mathematics, Computer Science
- Electron. J. Comb.
- 2013

The first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible, and the first explicit triangulation of a 3-sphere S_{18, 125} (with only 18 vertices) that isNot locally constructible is presented. Expand

Algebraic K-theory of group rings and the cyclotomic trace map

- Mathematics
- 2017

We prove that the Farrell-Jones assembly map for connective alge- braic K-theory is rationally injective, under mild homological niteness condi- tions on the group and assuming that a weak version of… Expand