[FRIAM] Photos of popped balloon

[Barry MacKichan]

When I brought up homology I (for real) thought that in this case there 
are *so many* alternatives I could have used. A short start of a list:

homology and cohomology

homotopy

triangulating the surface and counting vertices, edges, triangles, … 
with signs

looking a functions on the surface and counting its critical points with 
signs based on the types of critical points (max, min, saddle, etc) — 
Morse theory

summing (with signs) the dimensionality of solutions of certain partial 
differential equations on the surface

properties of open coverings of the surface (Čech cohomology)

a bunch of physics stuff once you accept the space-time, or wherever 
your interest lies, might be more complicated.

I tend to think of all these things as the petals of a daisy which all 
intersect at the center — an indication that the ideas here are 
fundamental.


--Barry

On 4 Feb 2019, at 11:32, lrudolph at meganet.net wrote:

>> I think they were cylinders, not spheres, so there were two holes. 
>> This
>> is where we start talking about homology groups.
>
> We don't absolutely *have* to.  The theories of Riemann surfaces and
> algebraic functions got pretty far just having the (proto-homological, 
> but
> very ungroupy) notions of "simple connectivity" vs. "multiple
> connectivity".
>
> [For those readers, possibly consisting of Nick alone, here's what 
> that
> means. Suppose you produce a thin sheet of copper by electroplating 
> onto
> some or all of the surface of a solid piece of wax that you then melt
> away.  For instance, you get a cylindrical surface if you start with a
> solid wax cylinder and only electroplate onto its lateral surface, 
> leaving
> the round disks at its two ends unplated; and it will be possible to 
> melt
> the wax away without cutting a hole in the copper.  On the other hand, 
> you
> get a spherical surface if you start with a solid round ball of wax 
> and
> electroplate onto its entire surface (let's not worry about how you do
> that...); in that case, you'll have to puncture the sphere (maybe 
> cutting
> out a little disk around the south pole) to let the melted wax escape.
> Just make one hole!  (And don't worry about possible difficulties 
> draining
> out all the wax, okay?)  For a third example, start with a piece of 
> wax in
> the shape of a donut (a so-called "solid torus" or, in a charmingly
> antique idiom, an "anchor ring"); the resulting copper surface is a
> "torus" plain and simple.  Again, a single hole will suffice to drain 
> the
> (idealized) wax; again, don't make any others.
>
> Now take your pair of metal shears and start cutting somewhere on an 
> edge
> of the copper sheet.  In the cylinder example, you have two edges, 
> each of
> them a circle at one end of the cylinder.  In the sphere and torus
> examples, you have a single (circular) edge, around the hole you 
> drained
> the wax through.
>
> It is a fact (which I hope you can imagine visually with no trouble,
> because all this electroplating would be expensive and difficult) that 
> no
> matter how you the sphere-with-one-hole with your shears, starting and
> ending at edge points, you will cut the copper into two pieces.  It is
> also a fact that on both the cylinder and the once-punctured (i.e.,
> drained) torus, there are *some* ways to cut from an edge point to 
> another
> edge point that do *not* cut the copper into two pieces.  (On the
> cylinder, you have to start somewhere on one of the two circular edges 
> and
> end somewhere on the other: when you've done that you can unroll the
> cylinder flat onto a table.  On the once-punctured torus, there are 
> many
> very different ways to make such a "non-separating" cut.)
>
> Riemann and Co. described this qualitative distinction between the 
> surface
> of a sphere and the (lateral) surface of a cylinder (and torus, etc.,
> etc.) by calling a sphere "simply connected" and the others "multiply
> connected".  "Simple" here is like 0, and "multiple" like "a strictly
> positive integer", which began the process of refining the qualitative
> distinction into a quantitative distinction.  Very soon the 
> quantitative
> distinction was refined much more by making the various positive 
> integers
> distinct (so the "cut number" of the sphere is 0, the cut number of 
> the
> cylinder is 1, and--as it turns out--the cut number of the torus is 
> 2).
>
> Rather later, this quantitative distinction became more refined.
> Eventually it became *so* refined that "homology groups" appeared as 
> the
> best way to describe the refinements.
>
> It is quite possible that the mathematical physicist John Baez, Joan's
> younger cousin, wrote all this stuff up very clearly 15 or 20 years 
> ago.
> If so, it would be findable with Google.]
>
>
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