What Are n-Dimensional Holes in Topology?

[WWGD]

TL;DR

Trying to find a Formal Definition of the term.

Is it reasonable to define the n-dimensional holes of a topological space X as the non-zero Homology/Homotopy classes of X?

Can we read these as obstructions to continuously shrinking a simple closed curve * to a point within the space?

*I understand this is what we mean by a cycle.

 

[Orodruin]

1. Almost, but not quite. You can generate the homotopy group by homotopy classes that surround a hole each. However, if there are several holes you may have non-zero homotopy classes that surround several holes.

2. No, that is only the first homotopy group. The second homotopy group would correspond to looking at the obstruction to shrink something homeomorphic to a two-dimensional sphere to a point and so on.

 

[WWGD]

1. Almost, but not quite. You can generate the homotopy group by homotopy classes that surround a hole each. However, if there are several holes you may have non-zero homotopy classes that surround several holes.

2. No, that is only the first homotopy group. The second homotopy group would correspond to looking at the obstruction to shrink something homeomorphic to a two-dimensional sphere to a point and so on.

Thank you, you're right, I should adapt 'holes' to the right dimension. Curves, I guess n-cycles instead of just curves.

 

[lavinia]

@WWGD I am not. sure what you mean by a hole. Do you have an intuitive idea?

Here is why I ask.

The third homotopy group of the 2 sphere is $Z$. So one might be tempted to say that the 2 sphere has a 3 dimensional hole. But it is a 2 dimensional manifold. Hmm...

The second homotopy group of the two dimensional torus is zero. So no hole there. But its second homology is $Z$ so maybe one needs homology as well. But then underlying this there must be some non- formal idea of what a hole is.

I suppose one might say that if a closed n-manifold surrounds a compact (n+1) dimensional volume, it creates a hole in some sense. But some n-manifolds do not surround a volume for instance the projective plane is not the boundary of a 3 manifold. Still, one might abandon homotopy and homology and instead say that a closed n manifold forms a n dimensional hole if it is a boundary of another manifold. In that case the projective plane does not form a two dimensional hole.

The Klein bottle is a boundary so by this idea it would form a 2 dimensional hole. On the other hand, its second homotopy group is zero and its second integer homology group is zero. One might say that its second integer cohomology group is $Z_2$ but then what is meant by a hole whose double is not a hole?

More generally what is meant by a hole that is detected by a torsion homology or homotopy class?

Another confusing case to me is a space with a non-abelian fundamental group. What kind of hole is made by a closed loop that is the boundary of a two dimensional singular chain? An example of historical interest is the Poincare homology 3 sphere. Its fundamental group has order 120 and is a simple group. So every homotopy class is a homology boundary.

 

[WWGD]

@WWGD I am not. sure what you mean by a hole. Do you have an intuitive idea?

Here is why I ask.

The third homotopy group of the 2 sphere is $Z$. So one might be tempted to say that the 2 sphere has a 3 dimensional hole. But it is a 2 dimensional manifold. Hmm...

The second homotopy group of the two dimensional torus is zero. So no hole there. But its second homology is $Z$ so maybe one needs homology as well. But then underlying this there must be some non- formal idea of what a hole is.

I suppose one might say that if a closed n-manifold surrounds a compact (n+1) dimensional volume, it creates a hole in some sense. But some n-manifolds do not surround a volume for instance the projective plane is not the boundary of a 3 manifold. Still, one might abandon homotopy and homology and instead say that a closed n manifold forms a n dimensional hole if it is a boundary of another manifold. In that case the projective plane does not form a two dimensional hole.

The Klein bottle is a boundary so by this idea it would form a 2 dimensional hole. On the other hand, its second homotopy group is zero and its second integer homology group is zero. One might say that its second integer cohomology group is $Z_2$ but then what is meant by a hole whose double is not a hole?

More generally what is meant by a hole that is detected by a torsion homology or homotopy class?

Another confusing case to me is a space with a non-abelian fundamental group. What kind of hole is made by a closed loop that is the boundary of a two dimensional singular chain? An example of historical interest is the Poincare homology 3 sphere. Its fundamental group has order 120 and is a simple group. So every homotopy class is a homology boundary.

Sorry for the delay in Replying. I had had a discussion similarly where someone kept using the term but never pinned it down. I guess 'Holes' are singularities of some sort and homotopy, homology may detect different types of 'singularities' . Sort of exploring the meaning of the term. Homology-wise, I thought of it as just any non-zero element, i.e., a cycle that does not bound. EDIT: But it may become even more complicated in that holes may exist for some coefficient rings but not for others. That seems puzzling to me.

 

[lavinia]

@WWGD Maybe the term comes from two dimensional orientable surfaces. These are all classified by the number of torus like holes. So the sphere has no holes, the torus one, two tori stuck together two and so forth.