New or old idea?: $n-2$ distinct genus 1 hole types for $n$-space

2012-06-02T19:03:49Z

Using only simple translation, rotation, and slicing methods, I concluded several years ago that to characterize fully the properties of a compact $n$-space solid containing a genus 1 hole, the hole itself must be further classified into one of $(n-2)$ unique $M$-types. The corresponding $m$ signature of a hole is a 2-tuple $m=(n,M){\vert}M{\in\lbrace}1,\ldots,n-2{\rbrace}$, a value that is fully orthogonal to the genus of the object.

My question is this: What is the standard name, if any, for the $M$-type of a genus 1 hole? If there is none, should I take the time to write a paper on $M$-type classification?

Update: Since, alas, it looks like I'm on my way out (-2, I know how it goes in SE after that), I would ask anyone still reading this to consider briefly this theorem:

Chains (interlinked forms spanning one direction) in odd-number space require only 1 type of link, while chains in even-numbered spaces require at least 2 distinct link types to stay together.

If the above theorem is well understood somewhere in topology, then my questions very likely are trivial overall, and I definitely qualify as a newbie who needs some classes.

However, if the above theorem is not known and not proven, I would humbly suggest that some of what I'm saying really may merit a somewhat more careful look. I can assure you there is some real thought behind it, however poor my ability to present it effectively in this particular forum. And in any case, my thanks to everyone who took the time to read any of it, including especially the down-voters who likely actually took the most time out of a weekend to look at such a mess of non-standard perspectives!

Cheers, Terry Bollinger

If any of this has piqued your interest, my recent query to math.SE (unanswered so far) is available here. Some generalizations and possible uses of $M$-types are also given below.

The full generalization of $M$ includes three pseudo holes: voids, $M=(*,0)$, which enclose empty regions of $n$ space in the same fashion as $(n-1)$-spheres; splits, $M=(*,n-1)$, which are solids that have been broken into two disconnected pieces; and erasures, $M=(*,n)$, which are solids that have been entirely erased. This gives a nicely regular sequence of $(n+1)$ generalized holes with signatures $(n,m)\vert{m}\in\lbrace{0,\ldots,n}\rbrace$. Removing the three pseudo holes leaves the $(n-2)$ true holes $(n,m)\vert{m}\in\lbrace{1,\ldots,n-2}\rbrace$.

Some hole properties are invariant by column, $M$, while others are invariant by diagonal, $(n-m-1)$.

For example, all holes are associated diagonally with $(n-m-1)$-spheres, which can be interpreted as their reduced forms. Alternatively, a $(n,m)$-hole can be viewed as not much more than an $(n-m-1)$-sphere that has been embedded into a space of dimensionality $n$. The complexity of such simple embeddings of ordinary $n$-spheres in higher or lower spaces arises from the constraints that different embedding spaces place on how spheres can interact with each other and other objects in those spaces. The effects include various forms of attachment, exclusion of some arrangements, and symmetry breaking.

The $1$-sphere is an easily visualized example. When embedded in $n=2$ the $1$-sphere fully encloses a region of $n=2$ space, and so qualifies as an $M=0$ void with signature $m=(2,0)$. If the same $2$-sphere is instead embedded in $n=3$, it takes on the behavior of a ring, that is, of a compact $n=3$ solid of with a genus 1 true hole of type $M=1$, with signature $(3,1)$, the only true hole possible for $n=3$. Attachment and symmetry breaking are demonstrated by considering possible relations for two rings colored red and blue. For $n=3$ only one form of attachment is possible: fully symmetric chaining of the two rings. For $n=2$ chaining is forbidden, but two unique versions of containment that were not available in 3D become possible: blue-in-red, or red-in-blue. Symmetry breaking is demonstrated by considering the three lower symmetry options for moving two rings that are coaxial and fully symmetric in $n=3$ into $n=2$, without breaking either ring: separate, blue-in-red, or red-in-blue. These transformations can also be expressed in terms of different hole types, e.g., only voids can contain, and only specific combination of true holes can chain.

Classifying holes by $M$ columns and $(m-n-1)$ diagonals is useful for recognizing shared hole properties, since some hole properties are invariant across $n$-spaces via the columns, and others are invariant via the diagonals. For example, independently of $n$ the $M$ columns characterize the dominant dimensionality of objects that can pass through the hole, as discussed under pointing theorem in the earlier link to Math.SE). The most important invariant of the diagonals is of course sphere equivalence.

As shown by the earlier rings example, $M$-types appear to be useful for describing and predicting how diverse $n$-objects will can attach to each other within a specific $n$-dimensional embedding space. Another example is my even-odd-$n$ chain theorem, which goes like this: For an $n$-space, the minimum number of unique $M$-links (compact solids with genus 1 true holes of type $M\vert{M}\in\lbrace{1},\ldots,n-2\rbrace$ and signature $(n,M)$ ) needed to form a chain attachment (think towing) is 1 for odd $n$ and 2 for even $n$.

Because they address how objects "connect" and transform in higher spaces, it is possible $M$-types could have applications in theoretical physics. See for example my earlier discussion of n-abaci.

New or old idea?: $n-2$ distinct genus 1 hole types for $n$-space - MathOverflow [closed]

New or old idea?: $n-2$ distinct genus 1 hole types for $n$-space