Homotopy and Homology version of Cauchy's Theorem

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In summary, the conversation discusses the topic of Cauchy's theorem and the possibility of removing assumptions from its statement. The conversation also includes examples involving the homomorphism and homology versions of the theorem, and the difficulties in understanding and applying these versions. The conversation concludes with a request for assistance on the topic.
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Maybe_Memorie
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Homework Statement



"Give examples showing no assumption in your statement of Cauchy's theorem can be removed. Justify your answer."


Homework Equations



I'm quite poor with LaTeX so I've providing links to the statement and proof of each version of the theorem.

http://www.maths.tcd.ie/~fionnf/cauchy/homotopy.pdf
http://www.maths.tcd.ie/~fionnf/cauchy/homology.pdf

The Attempt at a Solution



This is quite important as it is asked on every Complex Analysis exam my lecturer has set. However there's actually nothing in the notes regarding this. I'll start with the Homology version since the statement is shorter.

All we're assuming is that f is holomorphic in a subset of C and the integration is done on a 1-cycle.

Suppose f=1/z, and the path is a unit circle who's interior contains 0. Then the value of the integral around the path is 2∏i =/= 0, which it should by Cauchy. So f needs to be holomorphic in the interior. However I don't know how to write paths as cycles, so don't know how to finish this example...

The other assumption is that the curve is a 1-cycle. If it isn't a 1-cycle then the path isn't closed. Again, my lack of knowledge of cycles is causing issues here...


For the homotopy version, the holomorphic part is the same as for the last one, but the homotopy part is causing me problems.

I'd appreciate any help.
 
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Can anyone help?
 

FAQ: Homotopy and Homology version of Cauchy's Theorem

1. What is the Homotopy and Homology version of Cauchy's Theorem?

The Homotopy and Homology version of Cauchy's Theorem is a generalization of the classical Cauchy's Theorem in complex analysis. It states that if two continuous functions have the same homotopy and homology groups, then they are homotopic.

2. What is the significance of the Homotopy and Homology version of Cauchy's Theorem?

This theorem is significant because it allows for a more abstract and general proof of Cauchy's Theorem. It also has many applications in various branches of mathematics, such as algebraic topology and differential geometry.

3. How is the Homotopy and Homology version of Cauchy's Theorem used in topology?

In topology, this theorem is used to prove the fundamental theorem of algebra, which states that every non-constant polynomial with complex coefficients has at least one complex root. This is done by showing that a polynomial's homotopy and homology groups are equivalent to those of a constant function.

4. Is there a connection between the Homotopy and Homology version of Cauchy's Theorem and Cauchy's Integral Formula?

Yes, there is a connection between the two. Cauchy's Integral Formula can be proved using the Homotopy and Homology version of Cauchy's Theorem, as it follows from the fact that the homotopy and homology groups of a holomorphic function are trivial.

5. Are there any other versions of Cauchy's Theorem?

Yes, in addition to the Homotopy and Homology version, there are also the simply connected version and the winding number version of Cauchy's Theorem. Each version has its own unique proof and applications in different areas of mathematics.

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