Complements of Curves in Closed Surfaces: Homeomorphic?

In summary, the conversation discusses a result regarding non-isotopic curves in a compact, oriented surface S. The initial statement is incorrect, as it is not true when removing a disk and a meridian. There is a corrected result mentioned in Farb and Margalit's Primer on Mapping Class Groups, which states that there is an orientation-preserving homeomorphism of S taking one simple-closed curve to another if and only if the corresponding cut surfaces are homeomorphic.
  • #1
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Hi, let ## \alpha, \gamma ## be non-isotopic curves in a compact, oriented surface S. There is a result to the effect that ## S-\alpha## is homeo. to ## S- \gamma ## . This is not true as stated; we can , e.g., remove a disk (trivial class) in a copy of S and then remove a meridian ( a generator of ## H^1 (T^2) ##). Does anyone know the corrected result? This is not true either if we only remove essential curves. It seems like if ##\alpha, \gamma## actually _were_ isotopic , then this would be true. Or maybe if they were homologous?
Thanks.
 
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  • #2
I seem to remember coming across the result you are looking for in Farb and Margalit's Primer on Mapping Class Groups, in chapter 1, if I am not mistaken. I think maybe they call it the change of coordinates principle.
 
  • #3
http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf [Broken]
 
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  • #4
Thanks; in case anyone else is interested, here it is, towards the bottom of p.39:

There is an orientation-preserving homeo. of a surface S taking one s.c.c * ## \gamma## to ## \beta## iff the
corresponding cut surfaces ##S -\gamma ## , ##S- \beta ## are homeomorphic.

*s.c.c: Simple-Closed Curve.
 
  • #5


Hi there,

Thank you for sharing your thoughts on this topic. I can provide some clarification on the statement made in the content. The concept of homeomorphism is a fundamental concept in topology, which deals with the properties of geometric objects that are preserved under continuous transformations. In this case, we are dealing with closed surfaces, which are topological spaces that are compact, connected, and have no boundary.

The statement that ##S-\alpha## is homeomorphic to ##S-\gamma## is not necessarily true. As you have pointed out, there are cases where this may not hold, such as when we remove a disk and a meridian. In fact, the corrected result would depend on the specific properties of the curves ##\alpha## and ##\gamma##. If they are homologous, meaning they have the same winding number around a given point, then the statement would be true. However, if they are not homologous, then the statement may not hold.

Furthermore, the statement also assumes that the curves ##\alpha## and ##\gamma## are non-isotopic, which means they cannot be continuously deformed into each other without intersecting. This is an important distinction because if the curves were isotopic, then they would have the same topological properties and the statement would hold.

In summary, the corrected result would depend on the specific properties of the curves and whether they are homologous or isotopic. It is important to carefully consider these factors when making statements about homeomorphism in closed surfaces. I hope this helps clarify the topic. Let me know if you have any further questions.
 

1. What is the definition of a complement of a curve in a closed surface?

The complement of a curve in a closed surface is the set of all points in the surface that are not on the curve. In other words, it is the region surrounding the curve.

2. How is a complement of a curve in a closed surface determined?

The complement of a curve in a closed surface is determined by first defining the curve and then identifying all the points in the surface that are not on the curve. This can be done by visualizing the curve and the surface, or by using mathematical equations to define the curve and its boundary.

3. Are all complements of curves in closed surfaces homeomorphic?

No, not all complements of curves in closed surfaces are homeomorphic. Homeomorphism is a topological property that requires the two spaces to be continuously deformable into one another. While some complements of curves in closed surfaces may be homeomorphic, others may not be due to differences in their topological properties.

4. What is the significance of studying complements of curves in closed surfaces?

Studying complements of curves in closed surfaces is important in understanding the topological properties of these surfaces. It also allows for the exploration of different types of curves and their interactions with the surface. This can have applications in various fields such as mathematics, physics, and engineering.

5. Can complements of curves in closed surfaces be used to solve real-world problems?

Yes, complements of curves in closed surfaces can be applied to real-world problems. For example, in computer graphics and animation, understanding the complement of a curve in a closed surface can help in creating realistic 3D models. It can also be used in the study of fluid dynamics and the behavior of fluids around objects with curves and edges.

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