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Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With that same model, I should also be able to explain how Betti numbers are related. Incidentally, the dual-fluid model seems to provide an interesting alternative way of viewing Poincaré duality.

Original Question 2012-05-31:

(Please bear with my non-standard terminology; I am not a topologist.)

In $n=3$ space an object such as an $n$-ball can have only type of hole. For the simplest case of one such hole in one $3$-ball, the result is a ring. If the ring is smoothed down and narrowed, the result is a $1$-sphere or circle.

It is easy to generalize "hole" to mean any topologically distinct, simple subtraction of substance from an $n$-ball. By "simple subtraction" I mean this: Define a removal $n$-ball that is initially identical to the first ball. Expand the removal ball slightly by uniformly expanding the $m$-ball defined by the $m$ axes, and contract it slightly by uniformly shrinking the ($n-m$)-ball defined by the remaining subspace of ($n-m$) axes. Intersect the resulting distorted removal ball with the first one. For the first $n$-ball, erase all substance from the locations where it intersects with the removal ball.

This procedure results in three additional hole types for $n=3$ space: voids ($m=0$), splits ($m=2$), and erasures ($m=3$). The original definition of a hole for $n=3$ space becomes $m=1$.

Thus using this generalized hole definition, $n$-space has $(n+1)$ hole types with labels $m=0,1,\ldots,n$. The $m=0$ case is always a void, $m=n-1$ is always a split, and $m=n$ is always an erasure. The space has $(n-2)$ "true holes" labeled $m=1,\ldots,n-2$, with the true holes being those that (a) leave the $n$-ball connected, and (b) provide a path through it. (If the second criterion (b) is omitted, $m=0$ voids could also be called true holes.)

The thinning procedure mentioned earlier simplifies the resulting holes and provides an easy way to observe structure that is shared across all $n$-spaces. For example, an ($n=2,m=0$) void looks like a washer or ring drawn on paper, and it thins into a $1$-sphere. An ($n=3,m=1$) hole looks like a bagel, and also thins into a $1$-sphere.

This pattern produces a very simple rule: Every $n$-space has ($n-2$) true holes, and each of those holes $m$ has the same fundamental topology as an ($n-m-1$)-sphere. Thus our space has one true hole, ($n=3,m=1$), with the fundamental topology of a $1$-sphere or circle, and a void, ($n=3,m=0$), which has the fundamental topology of a $2$-sphere (a hollow shell). For $n=4$ there are two true holes: ($n=4,m=1$) with the form of a $1$-sphere, and ($n=4,m=2$) with the form of a $2$-sphere.

Addendum 2012-06-05: There's no reason for me to be overly abstract about any of this: Bollinger - Generalized Holes for n=0,1,2,3,4

A "1" indicates a "filled n-cell" rather than a vertex, and "0" an empty n-cell. Each solid shown is assumed to exist in empty space, so surrounding zeros are implied.

Notice the (x,y) gap at the center of (4,2), which would allow any sufficiently thin plane-like object in 4D to pass through. At the same time, the object itself retains its structural integrity by forming a closed loop in the (z,t) plane. That loop can be seen easily in the 3x3 units surrounding the empty (x,y) unit.

The (4,2) hole has no analog in 3D, although ironically its fundamental form is the same as a circle or a ring. It's just that in 4D a toroid ends up with two axes of freedom through its hole instead of just one.

Addendum 2012-06-01: Here is a way to visualize why the different $m$ holes have significantly different properties.

In 3D, a ring is a compact object containing a single (genus 1) ($n=3,m=1$) hole. Now the interesting thing about a single hole is that it enables a curious but well-defined relationship between two objects residing in the same $n$-space. In particular, if the hole is of size $m$, you can use it to "point to" any $m$-dimensional location on an object that is indefinitely larger in $m$ dimensions than the hole object. I'll call this the pointer theorem for future reference.

The pointer theorem sounds a bit abstract, but it's really quite easy to visualize. Imagine a ring on a wire, for example. The ring is a compact object with a ($n=3,m=1$) hole in it, and the wire is another 3D object that is indefinitely larger than the ring in exactly one dimension. The pointer theorem therefore applies, so you should be able to "relocate" the ring to any point along the indefinitely long axis of the wire. And it's true! The ring can in fact be moved to "point to" any location on the wire, no matter how longer the wire is. Moreover, the relationship is non-trivial, since you cannot remove the ring from the wire without covering at least some other locations on the wire; the ring is "strung" onto the wire.

While 3D space allows only one type of true hole, notice that a split ($n=3,m=2$), which consists of two "nearby" but separate objects, qualifies as a generalized hole. Does the pointer theorem apply to splits? Yes. The extended object for an $m=2$ hole, regardless of the space size $n$, is a sheet-like object that is indefinitely larger than the hole object in $m=2$ dimensions. Can the two such nearby objects point to any location on such a sheet? Yes. An ordinary sewing machine is a pretty good example, since it has paired parts above and below an indefinitely large sheet of fabric, and those paired parts can be used to stitch any point on a sheet of fabric. The problem in 3D, however, is that unlike the approximation provided by a sewing machine, a real split breaks the compact object apart into two disconnected units.

Now look at the 4D case. For $n=4$ you have have two "true holes," namely ($n=4,m=1$) and ($n=4,m=2$). Unlike the 3D case, either one of these holes can be used to create an continuous (internally connected) compact 4D object with genus 1. The pointer theorem says that an ($n=4,m=1$) hole can be used to point to any location on an indefinitely large rod-like or wire-like $n=4$ object. This makes the ($n=4,m=1$) hole into the 4D equivalent of a ring on a wire. Intriguingly, however, the reduced form of ($n=4,m=1$) is not a $1$-sphere, but a $2$-sphere, since $m-n-1=2$. That makes it topologically equivalent to a hollow ball in 3D space, rather than a ring! What is happening conceptually is that any 4D wire would be orthogonal to our 3D space, and would show up in intersection with our space as a $3$-ball trapped inside the hollow sphere.

But what about that other 4D hole, ($n=4,m=2$)? It is a "sheet-like" hole, just like the one in the earlier 3D split example. The pointer theorem implies that you can "string" a compact object with a genus 1 ($n=4,m=2$) hole onto an indefinitely larger sheet-like 4D object, and then use that compact object to point to any location on the sheet.

What's fascinating conceptually is that a compact genus 1 ($n=4,m=2$) object would remain both intact and "stuck" on such a sheet in very much the same way that a ring gets "stuck" on a long wire in 3D. My own phrase for this is the 4D abacus effect. Just as in 3D an abacus bead can be moved anywhere on a wire but remains trapped on that wire, in 4D a "bead" that has a ($n=4,m=2$) hole can be moved anywhere on a sheet, but remains trapped on that sheet.

Note that this type of genus 1 ($n=4,m=2$) bead is equivalent to a $1$-sphere or ring, just as in the 3D abacus case. If the sheet was rotated into 3D space, this ring would show up as mysteriously connected pieces above and below the sheet. It would look very much like the 3D split in fact, with the addition of a hidden 4D link that keeps the pieces joined.

There are infinitely many different abaci, incidentally. For every $n$-space the corresponding $n$-abaci uses $1$-sphere type beads -- that is, compact genus 1 objects with ($n=n,m=n-2$) holes -- strung onto ($n-2$)-objects. The $5$-abacus is interesting because it attaches a movable ring to any location within a 3D "object." Thus it might provide an interesting physics model for folks who like to model particles as loops attached to and mobile within a 3D space. You can also get a lot more complicated than that, of course, such as by increasing the genus and/or mixing together abaci of different dimensionality.

The main point in all these examples of the pointer theorem is this: Holes with different $m$ cannot be equivalent because they cannot be strung onto the same kinds of objects. In 4D for example, an ($n=4,m=1$) hole can be strung onto a wire-like 4D object, but not onto a sheet-like 4D object. The other true hole type for 4D, ($n=4,m=2$), can be strung onto a sheet-like 4D object, however.

There's another more subtle difference I should mention. While you can in principle use a ($n=4,m=2$) "bead" to point to any location on a wire-like 4D object, you cannot "string" such a bead onto that object. It simply falls off! To see why, imagine trying to capture a thread between two disconnected objects in 3D. The same problem of too many degrees of freedom applies for trying to string a ($n=4,m=2$) bead onto a wire-like 4D object.

So, for any given embedding space $n$, its set of ($n+1$) true and generalized hole types {$m=0,1,\ldots{n}$} are all fundamentally different in terms of the types of interactions they provide between objects in that space.

Finally, an observation: If embedding spaces are not used, the difference between these holes remain, but they get a lot harder to observe and characterize (I think).


So with all of that, here's my simple question: Where do I find all of this in topology?

I'm assuming there must be a similarly simple and regular way to classifying holes in higher dimensional spaces in terms of their sphere analogs, but I've had a lot of trouble finding it the last time I looked. (That was a couple of years ago, so maybe I need to look again.) Most of what I saw dived into the algebraic complexity and special terminology weeds so quickly that I had trouble figuring whether they were really relevant.

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I'm not sure I understand your statement at the beginning ($n=3$). Take a 3 ball and three distinct points on it's surface, then drill into the ball starting from these points with direction to the center. You will end up with a ball with three cylinders removed, the tunnels meeting at the center of the ball. The result is definitely topologically different from your example. Nonetheless the part I removed is homotopically trivial (contractible). For what reason/how do you rule out this kind of hole? –  user20266 May 31 '12 at 18:11
    
Sorry, I wasn't clear enough: Don't drill. Instead, first create a flexible-volume duplicate of the original ball. Next, stretch it outward along the m axes, then inward along the remaining (n-m) axes. Let the distorted n-ball now assume a ghostly character, and co-center it over the original undistorted n-ball. Now give the distorted ball a corrosive character in which it erases every part of the original n-ball with which it occupies the same space. Finally, remove the distorted sphere. The bridge pieces that would result from axial drilling will be erased when using this procedure. –  Terry Bollinger May 31 '12 at 19:36
    
I think Thomas is making the point that there are more holes possible in a 3-ball than are accounted for in your classification. –  Rahul May 31 '12 at 19:37
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Your method of describing and counting holes of different types sounds to me a lot like the computation of the Betti numbers of a topological space. A solid course in Algebraic Topology would be enough for you to decide for yourself if your notion of holes lines up with standard notions in topology. There's no doubt that the algebra is complex and the terminology is special. In my opinion, though, if you are willing to work hard, instead of weeds, you'll find flowers. –  Lee Mosher Jun 2 '12 at 19:36
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A lot of what you describe reminds me of Alexander Duality. This is a piece of topological machinery that (in very casual terms) describes the topology of a space $Y$ that is created by removing some subspace $X$, in terms of the topology of the removed piece. –  NKS Jun 7 '12 at 0:59

1 Answer 1

up vote 5 down vote accepted

I think Alexander Duality is what you are looking for. I gather that you are a non-expert, so I will attempt to describe in fairly informal terms how Alexander duality deals with the questions that you are interested in. Consequently, I'll suppress the inevitable technicalities, since they don't enter into the very geometric situations that you are interested in.

Alexander duality deals with the following situation. Let $S^n$ denote the $n$-dimensional sphere. Note that you can think of the $n$-sphere as being standard $n$-dimensional space $\mathbb{R}^n$ with an extra point "at infinity" added in (take a look at stereographic projection if this is unfamiliar to you). The upshot of this is that working with the $n$-sphere is not too far away from the situation you are interested in. Now take some subspace $X$, like the $m$-balls you are removing. Or take anything else; a solid torus of whatever genus you like, higher-dimensional manifolds, etc. Let $Y$ denote the complement of $X$ inside $S^n$. Then, in very informal terms, Alexander duality asserts that homologically, $Y$ is exactly as complicated as $X$. Somewhat more technically, Alexander duality asserts that for all $q$, there is an isomorphism $$ \tilde H _q(Y) \cong \tilde H^{n-q-1}(X) $$ between the reduced homology of $Y$ and the reduced cohomology of $X$ (for whatever coefficient group we choose). If the meaning of this is somewhat unfamiliar to you, you might be more interested to know that this says that the Betti numbers of $Y$ can be computed from those of $X$. In the range $1 \le q \le n-2$, a consequence of Alexander duality is that $$ B_q(Y) = B_{n-q-1}(X), $$ where $B_k(Z)$ indicates the $k^{th}$ Betti number of a space $Z$. If the piece $X$ that you are removing has $p$ components, this also implies that $B_{n-1}(Y) = p-1$.

As I remarked before, Alexander duality says that the topology of a space obtained by cutting a piece out is, from the point of view of homology (which encapsulates Betti numbers) exactly as complicated as the piece being removed. It is interesting to note that from the point of view of homotopy theory, this is incredibly far from being true. A basic technique in knot theory is to study a knot by studying the topology of its complement in $\mathbb R^3$ (or $S^3$). Homologically, Alexander duality says this is very boring, but from the perspective of homotopy theory, the story is very interesting indeed.

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NKS, thanks. I'll need to read up a lot obviously, but your answer is phrased in a way that seems very familiar to the way I was analyzing the issue. Betti numbers were a tougher link. You mentioned knot theory, which is also nice. I had wondered whether to bring that up, since as best I can tell knots tend to fall apart as you embed them in higher spaces. –  Terry Bollinger Jun 7 '12 at 15:09

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