Homotopy and Homology version of Cauchy's Theorem

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SUMMARY

The discussion focuses on the necessity of assumptions in Cauchy's Theorem, specifically its Homotopy and Homology versions. The user provides examples demonstrating that if the function is not holomorphic in the interior of the path or if the path is not a 1-cycle, the integral does not yield the expected result. The user struggles with expressing paths as cycles and understanding the homotopy aspect, indicating a gap in knowledge regarding these concepts. The provided links to the theorem's statements and proofs serve as foundational resources for further exploration.

PREREQUISITES
  • Understanding of complex analysis, specifically Cauchy's Theorem.
  • Familiarity with homology and homotopy concepts in topology.
  • Knowledge of holomorphic functions and their properties.
  • Ability to work with 1-cycles and integration in complex domains.
NEXT STEPS
  • Study the proofs of Cauchy's Theorem in both Homotopy and Homology contexts.
  • Learn about the construction and properties of 1-cycles in complex analysis.
  • Explore the relationship between holomorphic functions and their integrals over closed paths.
  • Investigate advanced topics in topology, focusing on homotopy theory and its applications in complex analysis.
USEFUL FOR

Students of complex analysis, mathematicians interested in topology, and anyone seeking to deepen their understanding of Cauchy's Theorem and its implications in both homological and homotopical contexts.

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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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