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<p dir="auto">When I brought up homology I (for real) thought that in this case there are <em>so many</em> alternatives I could have used. A short start of a list:</p>
<p dir="auto">homology and cohomology</p>
<p dir="auto">homotopy</p>
<p dir="auto">triangulating the surface and counting vertices, edges, triangles, … with signs</p>
<p dir="auto">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</p>
<p dir="auto">summing (with signs) the dimensionality of solutions of certain partial differential equations on the surface</p>
<p dir="auto">properties of open coverings of the surface (Čech cohomology)</p>
<p dir="auto">a bunch of physics stuff once you accept the space-time, or wherever your interest lies, might be more complicated.</p>
<p dir="auto">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.</p>
<p dir="auto">--Barry</p>
<p dir="auto">On 4 Feb 2019, at 11:32, <a href="mailto:lrudolph@meganet.net">lrudolph@meganet.net</a> wrote:</p>
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<div class="plaintext"><blockquote><blockquote><p dir="auto">I think they were cylinders, not spheres, so there were two holes. This<br>
is where we start talking about homology groups.</p>
</blockquote><p dir="auto">We don't absolutely *have* to. The theories of Riemann surfaces and<br>
algebraic functions got pretty far just having the (proto-homological, but<br>
very ungroupy) notions of "simple connectivity" vs. "multiple<br>
connectivity".<br>
<br>
[For those readers, possibly consisting of Nick alone, here's what that<br>
means. Suppose you produce a thin sheet of copper by electroplating onto<br>
some or all of the surface of a solid piece of wax that you then melt<br>
away. For instance, you get a cylindrical surface if you start with a<br>
solid wax cylinder and only electroplate onto its lateral surface, leaving<br>
the round disks at its two ends unplated; and it will be possible to melt<br>
the wax away without cutting a hole in the copper. On the other hand, you<br>
get a spherical surface if you start with a solid round ball of wax and<br>
electroplate onto its entire surface (let's not worry about how you do<br>
that...); in that case, you'll have to puncture the sphere (maybe cutting<br>
out a little disk around the south pole) to let the melted wax escape.<br>
Just make one hole! (And don't worry about possible difficulties draining<br>
out all the wax, okay?) For a third example, start with a piece of wax in<br>
the shape of a donut (a so-called "solid torus" or, in a charmingly<br>
antique idiom, an "anchor ring"); the resulting copper surface is a<br>
"torus" plain and simple. Again, a single hole will suffice to drain the<br>
(idealized) wax; again, don't make any others.<br>
<br>
Now take your pair of metal shears and start cutting somewhere on an edge<br>
of the copper sheet. In the cylinder example, you have two edges, each of<br>
them a circle at one end of the cylinder. In the sphere and torus<br>
examples, you have a single (circular) edge, around the hole you drained<br>
the wax through.<br>
<br>
It is a fact (which I hope you can imagine visually with no trouble,<br>
because all this electroplating would be expensive and difficult) that no<br>
matter how you the sphere-with-one-hole with your shears, starting and<br>
ending at edge points, you will cut the copper into two pieces. It is<br>
also a fact that on both the cylinder and the once-punctured (i.e.,<br>
drained) torus, there are *some* ways to cut from an edge point to another<br>
edge point that do *not* cut the copper into two pieces. (On the<br>
cylinder, you have to start somewhere on one of the two circular edges and<br>
end somewhere on the other: when you've done that you can unroll the<br>
cylinder flat onto a table. On the once-punctured torus, there are many<br>
very different ways to make such a "non-separating" cut.)<br>
<br>
Riemann and Co. described this qualitative distinction between the surface<br>
of a sphere and the (lateral) surface of a cylinder (and torus, etc.,<br>
etc.) by calling a sphere "simply connected" and the others "multiply<br>
connected". "Simple" here is like 0, and "multiple" like "a strictly<br>
positive integer", which began the process of refining the qualitative<br>
distinction into a quantitative distinction. Very soon the quantitative<br>
distinction was refined much more by making the various positive integers<br>
distinct (so the "cut number" of the sphere is 0, the cut number of the<br>
cylinder is 1, and--as it turns out--the cut number of the torus is 2).<br>
<br>
Rather later, this quantitative distinction became more refined.<br>
Eventually it became *so* refined that "homology groups" appeared as the<br>
best way to describe the refinements.<br>
<br>
It is quite possible that the mathematical physicist John Baez, Joan's<br>
younger cousin, wrote all this stuff up very clearly 15 or 20 years ago.<br>
If so, it would be findable with Google.]<br>
<br>
<br>
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