Examining Collapsibility in Arc Complexes

#Collapsibility of Arc Complexes

We show that the arc complex of a shape with a marked point inside is a strongly collapsible structure. For a Mobius strip with a few marked points on its edge, the arc complex is simpler but not strongly collapsible.

#What are We Studying?

In this work, we look into the shapes made up of points and lines, specifically focusing on their properties and how they can be simplified. The study of these properties is called topology.

#What is Collapsibility?

Collapsibility is a way to see if a shape can be simplified in a specific manner. A shape is collapsible if you can keep removing certain parts of it until you're left with a single point. This concept helps us understand how shapes can be changed while still keeping their essential qualities.

#Important Contributions

• A shape called the Dunce hat and Bing's house serve as examples of complex shapes that, while they can’t be simplified in the way we described, are still interesting to study.
• When a shape is collapsible, another related shape called its barycentric subdivision will also be collapsible.
• It has been shown that two shapes linked together will be collapsible if at least one of them is.

#Collapsible Shapes

An important focus of our study is on collapsible shapes, specifically how they relate to other shapes.

#Strong Collapses

There is a stronger way to simplify shapes, called a strong collapse. It means removing parts in a way that certain conditions about the shape are still met.

#What are Arc Complexes?

Arc complexes are made by connecting certain kinds of paths on surfaces. For surfaces with boundaries and marked points, these paths help define the structure and relationships of different points on the surface.

#The Basic Shapes

1. Polygons: These are flat shapes with straight sides. When we take a polygon and mark points on its edges, we have a way to create an arc complex.

2. Crowns: These are shapes like the surface of a donut with points marked on it. Depending on how we mark points, we can create different arc complexes.

3. Möbius Strip: This is a surface that has only one side and one edge, which leads to interesting properties when marked points are included.

#Inner Arc Complex

The inner arc complex focuses on the paths that connect the interior points of the shape to its boundary points. For non-orientable crowns, this inner arc complex has interesting features.

#Full Arc Complex

The full arc complex is created by considering all possible paths on the surface. It helps us understand how complex these shapes can be while also looking at their collapsibility.

#Results and Proofs

We outline our main results on the collapsibility of different arc complexes. For each type of shape, we define their properties and show step-by-step how to simplify them.

#The Arc Complex of a Crown

For a crown, the full arc complex can be strongly simplified. By analyzing how the arcs divide the surface, we can confirm its strong collapsibility.

#Inner Arc Complex of a Non-Orientable Crown

We apply a similar approach to the inner arc complex of a non-orientable crown. By making a few changes, we show that it can be simplified effectively.

#Full Arc Complex of Non-Orientable Crown

For the full arc complex of a non-orientable crown, we demonstrate that it can be transformed into a simpler structure by going through similar steps.

#Integral Strips

An integral strip is a specific shape where arcs connect points in a meaningful way. We show that this also has the property of being strongly collapsible.

#Conclusion

The study of arc complexes offers insights into the nature of shapes and their properties. Through the examination of various surfaces and their marked points, we can illustrate the ways in which they can be simplified while maintaining their fundamental characteristics.

#Final Thoughts

This exploration into the topology of arc complexes and their collapsibility leads to a deeper understanding of shapes. By uncovering the relationships among them, we provide a clear path for future research in the field.