- #1
Diophantus
- 70
- 0
I am trying to show that the connected sum of two topological surfaces does not depend on the open discs removed.
Any hints?
Any hints?
You want to show that for every such surface, any two discs on the surface, and any homeomorphism between their boundaries, there exists a homeomorphism from the surface to itself which restricts to the given homeomorphism on the boundaries of the discs (do you see why this is what you want to show?).
The easiest way I can think to do this is to use a theorem that states every surface is homeomorphic to some polygon with certain edges identified.
It'll still be difficult though, as you'll need to account for several different cases, eg, when the disc lies on an edge.
A topological surface is a mathematical concept that describes a two-dimensional object that is continuous and can be smoothly deformed without tearing or cutting.
A topological surface is different from a geometric surface because it is not limited by the concept of Euclidean space and can have non-Euclidean properties, such as being able to bend or stretch without changing its essential shape.
Some examples of topological surfaces include a sphere, a torus, a Möbius strip, and a Klein bottle. These surfaces can have different properties, such as being orientable or non-orientable, closed or open, and compact or non-compact.
Topological surfaces are used in a variety of scientific fields, such as physics, biology, chemistry, and computer science. They are used to model and understand complex systems, such as the structure of proteins, the behavior of fluids, and the dynamics of networks.
Some current research topics related to topological surfaces include the study of topological phases of matter, the development of new topological materials, and the application of topological methods in computer vision and machine learning. Additionally, there is ongoing research on the classification and characterization of topological surfaces and their connections to other areas of mathematics, such as knot theory and algebraic topology.