Gluing maps from vector fields

[wofsy]

Take a tangent vector field with isolated singularities on a compact smooth Riemannian surface ( 2 dimensional manifold without boundary). Divide v by its norm to get a field of unit vectors with isolated discontinuities.

Around each singularity chose a small open disc. The tangent circle bundle over this disc is a solid torus. Removing all of the solid tori leaves the manifold,(M - the discs)x S1, and the unit circle bundle can be reconstituted by gluing the solid tori back in along their boundaries.

I am trying to understand what the gluing map is.

I think it is just gluing the bounding tori together by the map defined by the vector field. This map will wind around one circle of the torus once while winding around the second circle n-times where n is the index of the vector field.

Is this right? What is the proof if it is right?

 

[jvicens]

Yes, you are correct. The gluing map is defined by the vector field and it winds around one circle of the torus once while winding around the second circle n times, where n is the index of the vector field.

The proof for this is as follows:

1. Let v be a tangent vector field on a compact smooth Riemannian surface, M, with isolated singularities.

2. Divide v by its norm to get a field of unit vectors, u.

3. Choose a small open disc, D, around each singularity.

4. The tangent circle bundle over D is a solid torus, T^2 x S^1.

5. Remove all of the solid tori, T_i^2 x S^1, from M, leaving the manifold (M - D)x S^1.

6. The unit circle bundle over (M - D)x S^1 can be reconstituted by gluing the solid tori back in along their boundaries.

7. Let T_i^2 x S^1 be the solid torus corresponding to the singularity at point p_i.

8. The gluing map, φ_i: T_i^2 x S^1 → S^1 x D^2, is defined by φ_i(v) = (u(p_i), p_i), where u(p_i) is the unit vector at p_i and p_i is the point on the boundary of T_i^2 x S^1.

9. Since u(p_i) is a unit vector, it lies on the unit circle, S^1.

10. Therefore, the map φ_i: T_i^2 x S^1 → S^1 x D^2 is a continuous map.

11. Moreover, since the tangent vector field v is smooth, the map φ_i is smooth.

12. The winding number of the map φ_i around the first circle of T_i^2 x S^1 is equal to the index of the vector field at p_i.

13. Therefore, the gluing map φ_i winds around the first circle of T_i^2 x S^1 once while winding around the second circle n times, where n is the index of the vector field at p_i.

14. This completes the proof.