2Questions: 1)Def. of Reg. Neighhoods 2)Zero in Homlgy and Hmtpy.

[Bacle]

Hi, everyone:

A couple of things:

1) I wonder if someone can give me a good explanation/ref. for "regular neighborhood"

of a manifold. The standard Google searches have not been enough to find a clear

def; some hits require reading some 10 pages, others just seem unclear.


2) Does anyone know of example of cycles that are zero in homology but not
in homotopy or viceversa?. I know the result of homology being the Abelianization
of Pi_1 , but I don't know well what this map does at the level of individual
cycles/classes.

Thanks.

 

[Hurkyl]

(I assume you meant the fundamental group pi_1 and not the fundamental groupoid Pi_1)


The map from the fundamental group to the first homology group is induced by the identity map -- it sends (the homotopy class of) any loop to (homology class of) the cycle defined by that loop.

The kernel of abelianization, incidentally, is generated by commutators -- elements of the form ghg^{-1}h^{-1}. (I'm writing the group operation as multiplication) Furthermore, such a commutator is zero if and only if g and h commute.

 

[mathwonk]

very roughly, a closed loop in a manifold is homologous to zero if it is the boundary of some piece of surface, and is homotopic to a point if it is the boundary of some disc. So it is harder to be homotopic to a point than to be homologous to zero. Think of a loop that divides a surface with two handles in half, separating the two handles.

 

[lavinia]

Hi, everyone:

A couple of things:

2) Does anyone know of example of cycles that are zero in homology but not
in homotopy or viceversa?. I know the result of homology being the Abelianization
of Pi_1 , but I don't know well what this map does at the level of individual
cycles/classes.

Thanks.

the plane minus two points has fundamental group the free group on two generators, the two simple loops that circle the two points. draw a commutator and that is an example.

try the Klein bottle. its fundamental group has two generators a and b. each generates an infinite cyclic group and the relation ab = (-b)a