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

 

[blue_raver22]

Hello,

I would like to provide some clarification and explanation for the two questions posed:

1) A regular neighborhood of a manifold refers to a small open set surrounding a point on the manifold that is homeomorphic to a Euclidean space. In other words, it is a neighborhood that is "nice" or "well-behaved" in a topological sense, meaning it is easy to work with and has simple properties. This concept is often used in differential topology and algebraic topology.

2) In terms of cycles that are zero in homology but not in homotopy, one example is the figure-eight space, which has a non-trivial cycle in homotopy but is zero in homology. This is because the figure-eight space is not simply connected, so its fundamental group is non-trivial. However, its first homology group is zero, meaning there are no non-trivial cycles. The reverse can also occur, where a cycle is non-trivial in homology but trivial in homotopy. An example of this is the infinite cylinder, which has a non-trivial cycle in homology but is simply connected, so its fundamental group is trivial.

I hope this helps to clarify these concepts. Let me know if you have any further questions.