Simply connected curve

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In summary, the conversation discusses the difference between simply connected and multiply connected regions in complex functions. A simply connected region means that every closed curve can be shrunk to a point, while a multiply connected region has curves that cannot be shrunk to a point. The example of an apple being simply connected and the number eight being multiply connected is given. The conversation also delves into the concept of branches of the logarithm and explains why a circle is not simply connected. The idea of null homotopic loops and the covering space argument is mentioned as another way to prove the circle is not simply connected.
  • #1
saravanan13
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Dear Friends

In the complex functions, I completely understand the simply connected region but not the multiply connected region?
An apple is a simply connected region but No. 8 is multiply connected. How?
 
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  • #2
saravanan13 said:
Dear Friends

In the complex functions, I completely understand the simply connected region but not the multiply connected region?
An apple is a simply connected region but No. 8 is multiply connected. How?

Simply connected means that every closed curve can be shrunk to a point. On the figure eight there are infinitely many curves that can not be shrunk to a point. Also on a circle.

A disk minus a point is not simply connected. For instance look at a branch of the logarithm in the unit disk minus the origin.
 
  • #3
lavinia said:
Simply connected means that every closed curve can be shrunk to a point. On the figure eight there are infinitely many curves that can not be shrunk to a point. Also on a circle.

A disk minus a point is not simply connected. For instance look at a branch of the logarithm in the unit disk minus the origin.

Thank
I could not follow your Second statement. Especially " branch of logarithm". Why can't a circle be simply connected?
 
  • #4
saravanan13 said:
Thank
I could not follow your Second statement. Especially " branch of logarithm". Why can't a circle be simply connected?

The angle function on a circle is defined only locally but its exterior derivative is globally defined. Therefore it is a closed 1 form that is not the exterior derivative of a function.

The integral of the derivative of the complex logarithm around a circle centered at the origin is the same as the integral of the angle function.

Another way to look at this is - suppose the curve that loops around the circle once were null homotopic. Then there would be a map from a disk to the circle that was equal to this curve on the boundary of the disk. Stokes Theorem then tells you that the integral of the exterior derivative of the angle function over this loop must be zero. But the intergral is not zero. It is 2pi.

I suggest that you look at the covering space argument that also proves that the circle is not simply connected. This avoids homology and uses purely topological arguments.

Intuitively, a null homotopic loop on the circle would have to retrace its path and return to its end point in the opposite direction that it came in. The loop that goes around once does not do this.
 
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  • #5


Hello,

I can help explain the concept of simply connected and multiply connected regions in complex functions. In mathematics, a simply connected curve is a curve that does not contain any holes or "islands." This means that any closed loop drawn on this curve can be continuously shrunk to a single point without leaving the curve. On the other hand, a multiply connected curve contains at least one hole or "island" and therefore cannot be continuously shrunk to a single point without leaving the curve.

To better understand this concept, let's use the examples you mentioned. An apple can be seen as a simply connected region because it does not have any holes or "islands" in its shape. However, the number 8 can be seen as a multiply connected region because it has two holes or "islands" in its shape. If you were to draw a closed loop around each of the holes in the number 8, you would not be able to shrink them to a single point without leaving the curve.

I hope this explanation helps clarify the difference between simply connected and multiply connected regions in complex functions. Let me know if you have any further questions.

Best regards,
 

1. What is a simply connected curve?

A simply connected curve is a type of curve in mathematics that does not intersect itself and is connected in a single piece. This means that any two points on the curve can be connected by a continuous path without leaving the curve.

2. How is a simply connected curve different from a non-simply connected curve?

A non-simply connected curve is a curve that can be separated into two or more pieces by removing a point or a set of points. This means that there are multiple paths between certain points on the curve, and it is not considered a single connected piece like a simply connected curve.

3. What is the significance of a simply connected curve in topology?

In topology, a simply connected curve is important because it is the simplest type of curve and it serves as a building block for more complex shapes. It also helps in defining and understanding other topological concepts such as connectedness and homotopy.

4. Can a simply connected curve be infinite in length?

Yes, a simply connected curve can be infinite in length. The key characteristic of a simply connected curve is that it is connected in a single piece and does not intersect itself, regardless of its length.

5. What are some real-world examples of simply connected curves?

Some examples of simply connected curves in real-world applications include a straight line, a circle, an ellipse, and a parabola. These curves are considered simply connected because they do not intersect themselves and can be continuously traced without leaving the curve.

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