Why Is an Apple Simply Connected but the Number 8 Multiply Connected?

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Discussion Overview

The discussion revolves around the concepts of simply connected and multiply connected regions in complex functions, using the analogy of an apple as a simply connected region and the number 8 as a multiply connected region. Participants explore the definitions and implications of these terms, seeking to clarify the differences between the two types of connectivity.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that simply connected regions allow every closed curve to be shrunk to a point, while multiply connected regions, like the figure eight, contain curves that cannot be shrunk.
  • There is a claim that a disk minus a point is not simply connected, with a reference to the branch of the logarithm in the unit disk minus the origin.
  • One participant expresses confusion about the statement regarding the circle and its lack of simple connectivity, prompting further explanation.
  • Another participant explains that the angle function on a circle is locally defined but has a globally defined exterior derivative, leading to a closed 1-form that is not the exterior derivative of any function.
  • It is noted that the integral of the derivative of the complex logarithm around a circle is not zero, which is used to argue against the circle being simply connected.
  • A suggestion is made to consider covering space arguments to demonstrate that the circle is not simply connected, emphasizing a topological perspective over homological methods.
  • Intuitive reasoning is provided, stating that a loop on the circle cannot be null homotopic as it does not retrace its path in the opposite direction.

Areas of Agreement / Disagreement

Participants express differing views on the connectivity of the circle, with some asserting it is not simply connected while others seek clarification on this point. The discussion remains unresolved regarding the implications of these concepts.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the definitions of connectivity and the specific mathematical constructs referenced, such as the branch of the logarithm and the angle function.

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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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.
 
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?
 
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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