Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Connected sums provide a means for combining topological surfaces to create other surfaces through strategic cutting and re-gluing. Now let's be clear about what we intend to do here, because previously we stated that cutting and gluing are not allowed in topology. To be precise, two objects are said to be topologically equivalent, if they can be deformed into one another smoothly without cutting or gluing.
These two objects are considered to be the same in topology because they both have only one hole, even though they look radically different. This is somewhat similar to having three bananas and three oranges; to be sure, bananas are different than oranges, but both groups are examples of the number "3." Similarly, the two tori depicted are different from each other, but both are examples of a genus 1 surface. (Remember, an object's genus is the number of holes that it has.)
Carrying on with the oranges and bananas example, we can take a set of three oranges and combine it with a set of five oranges to get a new set of eight oranges. This act of combining is what we think of as addition, one of the four basic mathematical operations (subtraction, multiplication, and division being the other three). All of these operations take objects, numbers in this case, and do something to them, most of the time giving us a new, different number.
We can think of "cutting and gluing" as an operation in topology. It is a way of taking two topological objects and combining them to make a new and, most of the time, different topological object. We call the result of this operation a "connected sum." Our exercise in the preceding section, in which we turned two projective planes into a Klein bottle, serves as an example.
Before we explore this further, let's establish some notation for convenience. We'll refer to a torus as T2, a sphere as S2, the Klein bottle as K2, the projective plane as P2, a disk as D2, and the Euclidean plane as E2. The number 2s in these designations indicate that they are all two-dimensional surfaces. The symbol we'll use for a connected sum is the pound sign, or number sign, #.
So, for example, if we take two T2s, cut out a disk from each, and then glue them together, we will have T2 # T2, as shown here:
This is a double-holed torus. We won't give it its own symbol; instead, we'll just remember what the operation means. T2 # T2 # T2 would symbolize a three-holed torus.
The example from the previous section, in which two projective planes were joined to create a Klein bottle, would have this notation: P2 # P2 = K2.
Let's return to our fruit example for a simple review of the concept of identity.
We can take a set of three bananas and add a set of zero bananas and end up with just three bananas. Because adding zero doesn't change anything, we refer to "0" as the "additive identity."
In topology there is an identity as well—it is the sphere. If we take the connected sum of any object with a sphere, we end up with the original object. For example, T2 # S2 = T2.
Also, K2 # S2 = K2
To explore connected sums fully, we need one more relationship:
K2 # P2 = T2 # P2. We can get a sense of why this is true by thinking of a projective plane as a reversing region; in other words, anything that passes through it has its orientation reversed.
If we attach a projective plane to a Klein bottle and then maneuver the Klein bottle so that it passes through the projective plane, the Klein bottle turns into a torus and the projective plane remains unchanged. This suggests to us that the connected sum of a Klein bottle and a projective plane is equivalent to the connected sum of a torus and a projective plane.
We also find that the commutative property applies to connected sums. In our fruit example, adding a set of three oranges to a set of five oranges is no different than adding a set of five oranges to a set of three oranges—the order does not matter. Similarly, taking the connected sum of two objects gives the same result no matter what order we do it in. Connected sums also adhere to the associative property, so that T2 # S2 # K2 = K2 # T2 # S2.
So, in topology, we have an identity and both the commutative and associative properties. This suggests that we can use surfaces to do algebra! These basic properties, along with the fact that P2 # P2 = K2 and K2 # P2 = T2 # P2, enable us to deal with really complicated surfaces. Suppose that we have some arbitrary surface, M2. Furthermore, suppose we know that:
M 2 = P2 # T2 # P2 # S2 # P2 # K2
Let's first use the commutative property to rearrange the sequence of this sum:
M 2 = P2 # P2 # P2 # K2 # T2 # S2
Now, because K2 # P2 = T2 # P2, we can make a substitution and write:
M 2 = P2 # P2 # P2 # T2 # T2 # S2
We know that adding a sphere is the identity and changes nothing, so let's drop it:
M 2 = P2 # P2 # P2 # T2 # T2
Now, remembering that P2 # P2 = K2, we can write:
M 2 = P2 # K2 # T2 # T2
Again making use of the fact that P2 # K2 = P2 # T2, we can write:
M 2 = P2 # T2 # T2 # T2
We should recognize this configuration as a three-holed torus with a projective plane attached. Alternatively, because adding a sphere changes nothing, we could view this as a sphere with three "handles," representing the tori, and a projective plane attached.
It is fascinating to realize that any 2-manifold, or two-dimensional surface, that we can envision will always be reducible, using the rules stated above, to a sphere with some number of handles and/or some number of projective planes attached. This very important theorem in topology is known as the "classification of surfaces." It was first proven for orientable surfaces by August Möbius, a German mathematician, physicist, and astronomer, who was a student of Gauss.
We have been examining surfaces and their topological representations, but what about 3-manifolds? The possibilities with these structures are not as straightforward as those involving two-dimensional surfaces, but it is a fascinating story that started in the 19^{th} century and was only resolved in the first decade of the 21^{st} century.
Before we can understand the 3-manifold case, we need one more concept as a tool. Let's return once again to our Flatland explorer. Recall that on her very first journey, she carried with her a length of blue thread. Had she been on the surface of a sphere, she could have, while still holding both ends and without cutting the thread, spooled it all back up, effectively shrinking her loop of thread until it was entirely on her original spool.
However, she is not on the surface of a sphere, but is rather on a torus with some sort of projective plane attached. If she tried to re-spool her blue thread, she would quickly find it to be impossible, because the thread passes through the hole of the torus.
This property, commonly referred to as the "loop-shrinking property," states that, on a sphere, or any surface that is topologically equivalent to a sphere, every loop that is drawn on the surface can be shrunk continuously to a point. If, on the other hand, you are not on a sphere, then it will always be possible to draw a loop that cannot be shrunk to a point, as our intrepid Flatlander discovered with her blue thread.
The great French polymath Henri Poincaré sought a similar property governing 3-manifolds in 1904. He was curious to know whether a 3-manifold could exhibit the loop-shrinking property and not be the three-dimensional equivalent of a sphere (often referred to as a "3-sphere"). His assumption that this was indeed possible became known as Poincaré's Conjecture. This loop-shrinking conjecture has much to do with how 3-manifolds are classified, in much the same way that the two-dimensional loop-shrinking conjecture helps to classify surfaces—that is, it can tell us if we are on a 3-sphere or not.
The proof of Poincaré's Conjecture eluded mathematicians for nearly 100 years and became one of the most-sought-after results in all of mathematics. In the intervening time, a great body of mathematics was developed and explored by many brilliant thinkers, such as Thurston, and Hamilton. Thurston, in particular, established a conjecture that allowed all 3-manifolds to be classified in a similar way to the 2-manifolds. Now referred to as the Geometrization Theorem, it, along with Poincaré's original conjecture, was proven by the reclusive Russian mathematician, Grigory Perelman, at the start of the 21^{st} century.
This great contribution to mathematics, representing the culmination of a century of international efforts, earned Perelman a Fields Medal, which is the mathematical equivalent of the Nobel Prize. In an odd twist, Perelman refused the honor of the Fields Medal in an act that brought a fair amount of controversy to the mathematics community.
Regardless of the dramatic personalities involved, the classification of 3-manifolds has far reaching consequences for mathematics. While it may have some repercussions for the physical sciences, its primary value is in its beauty as a mathematical construction. This is true for most topological exercises, which generally are done not so much for their practical value as for their mathematical and aesthetic value. Mathematics can indeed be as wondrous and beautiful as a great work of art or music or any other achievement of the human mind. Be that as it may, topology is not studied completely for its own sake. In the remainder of this unit, we will examine two practical applications of topological thinking. This first has to do with a rather mundane manufacturing task with a startling topological explanation. The second is an exploration of the shape of our universe.
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