[FRIAM] Photos of popped balloon

lrudolph at meganet.net lrudolph at meganet.net
Mon Feb 4 11:32:40 EST 2019

> 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

[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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