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in Algebraic Topology |
The regular polyhedra were known at least since the time of the ancient Greeks. The names of the more complex ones are purely Greek. But despite their being known for close to two millenia no one apparently noticed the fact that the sum of the number of faces F and the number of vertices V less the number of edges E is equal to two for all of them; i.e.,
It was the Swiss mathematician Leonhard Euler who recognized and published this fact. The value of two is said to be the Euler characteristic of each of the polyhedra. This value is not changed by stretching or shrinking any side or face or even shrinking a side or face to zero. This means that the Euler characteric is a topological invariant because it is not altered by any continuous mapping.
The quantity (V-E+F) is called the Euler characteristic of the polyhedron and is denoted as by the Greek letter Chi, Χ. Thus Χ(cube)=2.
The French-Swiss mathematician, Simon Lhuilier (1750-1840), found a slight generalization of Euler's formula to take into account polyhedra having holes. Lhuilier's formula is
where G is the number of holes in the polyhedron. Thus the Euler characteristic is 2 for a regular polyhedron but 0 for a torus-like polyhedron.
For a simple treatment of the effect of holes and handles on the Euler characteristic see Euler Characteristic.
It was the French mathematician, Henri Poincaré, who fully generalized Euler's formula.
Consider an n-dimensional polytope K. It is made up of simplexes of equal or lesser dimensions. A two dimensional simplex (2-simplex) is a triangle; a one dimensional simplex (1-simplex) is a line and a zero dimensional simplex (0-simplex) is a point. Let α_{m} be the number of m-dimensional simplexes in K, where m runs from 0 to n. For example, the surface of a tetrahedron is two dimensional and it is composed of 4 triangles (2-simplexes), 6 line segment edges (1-simplexes) and 4 points (0-simplexes).
Consider a sequential labeling of the m-simplexes of K taking into account their orientation; i.e., choose an m-simplex with a particular orientation and call it number 1, another as number 2 and so on. Now construct vectors with integer components of the form
where the c_{i}'s are integers (c_{i}∈Z),
One of these vectors is called a chain and the set of them form a mathematical group under the operation of component-wise addition with the identity being the vector of zeroes and the inverse of a vector being the vector of the negatives of the components. The group is abelian and its dimension is α_{m}.
A chain may be represented as a formal sum of the form
where σ_{i}^{m} is the unit vector for the i-th simplex; i.e.,
This is essentially the same as representing a vector in 3-space as as the formal sum
where i, j and k are the unit vectors in the three directions and equivalent to (1,0,0), (0,1,0) and (0,0,1).
The boundary for σ_{i}^{m} is defined as
where [σ_{i}^{m}, σ_{j}^{m-1}] is the incidence number of the σ_{i}^{m} simplex with the σ_{j}^{m-1} simplex; i.e.,
with the sign of the incidence number depending upon the orientation of σ_{j}^{m-1} with respect to the orientation of σ_{i}^{m}.
The boundary of C = Σc_{i}σ_{j}^{m} is then
Because of a property of the incidence numbers that for all i and k
it follows that
In other words, boundaries have no boundary.
The boundary operation defines a homomorphism of the group of chains on m-simplexes, C_{m}(K) onto the group of chains on the (m-1)-simplexes, C_{m-1}(K):
Within the group of chains on m-simplexes there are special chains that have an empty boundary; i.e. ∂(C)=0. These boundary-less chains are called m-cycles and they form a subgroup of C_{m}(K) denoted as Z_{m}(K).
Within the group of m-cycles there are the special cycles that are the boundary of some chain in C_{m+1}(K). They form a subgroup of Z_{m}(K) and hence a subgroup of C_{m}(K). They are called m-boundaries and their group is denoted as B_{m}(K).
The subgroup relationship of C_{m}(K), Z_{m}(K) and B_{m}(K) determines a factor group of Z_{m}(K) with respect to B_{m}(K). This is called the m-th homology group of the polytope K and is denoted as H_{m}(K)
The boundary operator ∂ maps C_{m}(K) onto B_{m-1}(K)(K) and it maps Z_{m}(K) onto the identity element of B_{m-1}(K), which is also the identity element of C_{m-1}(K) and Z_{m-1}(K). Z_{m}(K) is said to be the kernal of the homomorphism. This means that Z_{m}(K) is a factor group of C_{m}(K) with respect to B_{m-1}(K); i.e.,
From the theorems for abelian free groups it follows then for m>0
Letting rank(Z_{m}(K))=ζ_{m}, rank(C_{m}(K))=α_{m}, and rank(B_{m}(K))=β_{m} the above condition is
For the special case of m=0, rank(B_{m-1}(K))=0 so
Let rank(H_{m}(K))=p_{m}. The number p_{m} is known as the m-th Betti number of K, so named after the Italian mathematician Enrico Betti (1923-1892) who believed that the structure of a topological space could be completely characterized by a set of numerical invariants. Now the characterization of a topological space is in terms of the topologically invariant groups H_{m}(K). Since H_{m}(K)=Z_{m}(K)-B_{m}(K) then
The relevant equations for m>0 placed together are:
Subtracting the second equation from the first gives
Consider now the alternating sum
The right hand side of the above reduces to
Since C_{n+1}(K)=0 it follows that B_{n}(K)=0 and thus β_{n}=0 so
This latter equation is the Euler-Poincaré Formula. The left hand side is the Euler characteristic of the polytope K.
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