Fundamental group to second homology group

1. May 2, 2010

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?

2. May 3, 2010

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?)

3. May 6, 2010

lavinia

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.