Fundamental group to second homology group

[lavinia]

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?

 

[zhentil]

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?)

 

[lavinia]

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