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Algebraic Topology studies shapes and forms by converting them into groups, and then using group theory.
As a simple example, consider a circle. Just the circumference, not the interior (or exterior).
Now think about a string going around the circle in an anti-clockwise direction and returning to where it started. There's no way you can shrink that without the string leaving the circle.
Call that "1".
Consider another copy of that, and join the ends together, "nose to tail", as it were. Now you have a string going around the loop twice. We've added together two copies of "1", and we call the result "2".
Interesting thing, if you take a string going clockwise, and connect that to a "2", we get back to a "1". In a sense, the clockwise string acts as if we were subtracting a "1" from a "2". Well, instead, think of it as adding a "-1", and that works.
In this way, strings that wrap around the circle act like the integers, with anti-clockwise being positive, clockwise being negative, and "adding" being the operation of joining nose-to-tail.
Thus we have converted a topological object, the circle, into a group object, $Z.$
Performing the same operation on a cylinder (which is $S^1\times[0,1]$ ) gives the same result.
Next to talk about is doing the same with the torus.
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