How to show 2-tori is diffeomorphic to S^3

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SUMMARY

The discussion centers on the mathematical relationship between 2-tori and S^3, specifically addressing the embedding of 2-tori into S^3. It is established that a torus is not diffeomorphic to S^3; rather, the focus is on how to embed 2-tori into S^3 using the mapping from R^2 to R^4 defined by (x,y) -> (1/2^.5)(cos x, sin x, cos y, sin y). Additionally, the discussion introduces a smooth function F: C^2\{0} to C, defined as F(z1,z2) = z1^p + z2^q, where p and q are relatively prime integers greater than or equal to 2, and explores the diffeomorphism of S^3 intersect F^(-1)(0) with 2-tori.

PREREQUISITES
  • Understanding of differential topology concepts
  • Familiarity with embedding theorems in topology
  • Knowledge of smooth functions and their properties
  • Basic understanding of R^n and its mappings
NEXT STEPS
  • Study the embedding of manifolds, particularly in the context of S^3
  • Learn about the properties of smooth functions in differential topology
  • Investigate the implications of the Whitney embedding theorem
  • Explore the concept of diffeomorphism in higher-dimensional topology
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to understand the relationship between different manifolds and their embeddings.

robforsub
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Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3
 
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robforsub said:
Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3

A torus is not diffeomorphic to S^3.
 
My mistake, it should be how 2-tori is embedded into S^3
 
robforsub said:
My mistake, it should be how 2-tori is embedded into S^3

map R^2 into R^4 by (x,y) -> (1/2^.5)(cos x, sin x, cos y, sin y)

The image is a torus in S^3
 
What if there is a smooth function F:C^2\{0} to C, defined as F(z1,z2)=z1^p+z2^q with p and q>=2 and they are relatively prime, then how to show that S^3 intersect F^(-1)(0) is diffeomorphic to 2-tori?
 

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