Fundamental group to second homology group

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SUMMARY

The discussion centers on the relationship between the fundamental group and the second homology group in the context of a smooth compact 3-manifold containing an embedded loop. It is established that if the loop is not null homotopic, it does not necessarily imply that the torus, which serves as the boundary of a tubular neighborhood of the loop, is not null homologous. The participants clarify that the torus, defined as the boundary of an open set in the manifold, can indeed be null homologous despite the loop's properties. The conversation emphasizes the distinction between homotopy and homology classes in algebraic topology.

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  • Understanding of fundamental groups in algebraic topology
  • Knowledge of homology groups, specifically second homology groups
  • Familiarity with smooth compact 3-manifolds
  • Concept of null homotopy and null homology
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  • Study the relationship between fundamental groups and homology classes in algebraic topology
  • Explore the properties of smooth compact 3-manifolds and their boundaries
  • Learn about the implications of null homotopy versus null homology
  • Investigate examples of toroidal structures in algebraic topology
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Mathematicians, particularly those specializing in algebraic topology, graduate students studying topology, and researchers exploring the properties of manifolds and their homological characteristics.

lavinia
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In a smooth compact 3 manifold there is an embedded loop - a diffeomorph of the circle

Consider a torus that is the boundary of a tubular neighborhood of this loop.

If the loop is not null homotopic does that imply that the torus is not null homologous?
 
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Why would it? Doesn't the torus clearly bound? (I.e. you've defined it as the boundary of an open set in your 3-manifold, right?)
 
zhentil said:
Why would it? Doesn't the torus clearly bound? (I.e. you've defined it as the boundary of an open set in your 3-manifold, right?)

yes. Stupid question.

I am trying to understand how an element of the fundamental group can determine a homology class - but this element is null homologous though not null homotopic. The homology class would be 2 dimensional. For a moment I thought the torus might work - but that thought is as yopu pointed out - empty.

Thanks for your reply though
 

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