Fundamental group to second homology group

In summary, the conversation discusses the relationship between an embedded loop and a torus in a smooth compact 3 manifold. The torus is the boundary of a tubular neighborhood of the loop. The question is raised whether a non-null homotopic loop implies a non-null homologous torus. The answer is that it does not, as the torus is clearly bound by the definition given. The conversation also touches on how an element of the fundamental group can determine a homology class, and how the torus may initially seem to work but ultimately does not.
  • #1
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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  • #2
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?)
 
  • #3
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
 

1. What is the fundamental group?

The fundamental group is a mathematical concept in topology that measures the number of "holes" or "loops" in a topological space. It is denoted by π1(X) and is a group, meaning it satisfies certain algebraic properties.

2. What is the second homology group?

The second homology group, denoted by H2(X), is another mathematical concept in topology that measures the number of "two-dimensional holes" in a topological space. It is also a group and is related to the fundamental group through the Hurewicz homomorphism.

3. How is the fundamental group related to the second homology group?

The fundamental group is the first homotopy group in a series of homotopy groups, while the second homology group is the first homology group in a series of homology groups. The two groups are related through the Hurewicz homomorphism, which maps elements of the fundamental group to elements of the second homology group.

4. Why is the fundamental group important?

The fundamental group is important in topology because it helps classify and distinguish topological spaces. It is also a useful tool in solving problems and making calculations in algebraic topology.

5. How is the fundamental group used in real-world applications?

The fundamental group has various applications in fields such as physics, engineering, and computer science. It is used to study and understand the behavior of physical systems, create efficient algorithms, and analyze data in areas such as machine learning and image processing.

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