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Both Point Set Topology and Algebraic Topology attempt to describe and analyze the properties of geometric objects which are invariant under continuous mappings. Algebraic Topology does this using an indirect approach. Each geometric object is associated with a set of algebraic structures, usually groups. Questions about the geometric objects are converted into questions about the associated groups. This strategy is analogous to the way transform methods are used to solve differential equations.
Laplace and Fourier transforms are used to convert a differential equation into a strictly algebraic equation. Algebraic methods are used to solve for the transform of the solution to the differential equation. The inverse transform is applied to the transform of the solution to get the solution to the differential equation.
In algebraic topology each topological space X has a group (or a sequence of groups) associated with it, say G(X) (or {G_{n}(X); n=0,1,2,..N}). Some information is lost in moving from the geometric object to the associated group. An example will be given below to illustrate what all of this means.
Let X be a sphere in 3space and let Y be a torus also in 3space. The groups to be constructed are based upon loops that begin and end at some point P outside of X and Y. A loop is just a directed path in which the the beginning point and the end point are the same point. Two loops, x_{1} and x_{2}, can be added by attaching the beginning part of the second loop to the ending part of the first loop. There is also a zero loop, one that does not leave the point P. The situation is illustrated in the diagram below.
One loop is equivalent to another loop if it can continuously transformed into the second loop without intersecting the geometric object. The name for a continuous transformation is homotopy. In the case of the sphere any loop may be shrunk into the point P, the zero loop. Thus all loops are homotopically equivalent to the zero loop.
The group elements are the equivalence classes of the loops. As indicated above, for the sphere there is only one equivalence class of loops, which includes the zero loop. Therefore group for the sphere is the trivial group, the group comprised of only one element [0] and such that [0]+[0]=[0].
For the torus the loops that do not go through the hole of the torus are all homotopically equivalent to the zero loop. But the loops that go through the hole of the torus once are not homotopically equivalent to zero. However all loops that pass through the hole of the torus once in the same direction are homotopically equivalent to each other. For the torus the equivalence classes correspond to the number of times the loop passes through the hole of the torus in a designated direction. Thus the equivalence class of the loops which pass through the torus hole n times going from above to below could be represented as [n]. The equivalence class for those passing through the hole n times in the opposite direction would be represented as [n].
If a loop in [n] is concatenated with one in [m] the result is a loop in the class [n+m]. Likewise if a loop in the class [m] is concatenated with one in the class [n] the result is a loop in the class [nm].
From the above it is clear that the group associated with the torus is the group of integers with the operation of addition. The zero loop corresponds to 0 and the inverse of an equivalence class [n] is just [n].
To sum up:
Clearly the torus is not topologically equivalent to the sphere because their first homotopy groups are different. In this case the result is obvious but there are other cases in which it is not easy to tell whether or not the geometric objects are equivalent. However, if the associated groups of two geometric objects are equivalent it does not follow that the objects are topologically equivalent. For example, a hollow cylinder segment and a Möbius strip have the same first homotopy group; i.e, that of the integers under addition, but they are not topologically equivalent; the hollow cylinder is twosided whereas the Möbius strip is onesided.
The association of topological spaces with sequences of groups may be done in more than one way. The above example concerned homotopy but there is another important assignment called simplicial homology. Any particular assignment of group sequences to topological spaces is called a functor. The assignments are not arbitrary. The assignment has to be such that if X and Y are topological spaces and f is a continuous function that maps X into Y (X ^{f}→ Y) then there exists a homomorphic mapping F of G_{n}(X) into G_{n}(Y). A mapping F of groups is homeomorphic if for every x_{1} and x_{2} in G(X), F(x_{1}*x_{2}) = F(x_{1})+F(x_{2}), where * is the binary operation in G(X) and + is the binary operation in G(Y).
Consider a set of vertices (points) labled 0 to p. An msimplex is a sequence of (m+1) of these vertices. An msimplex is said to be positively oriented if the sequence of labels for its vertices is an even permutation of <0,1,2,...,p> and a negatively oriented simplex if it is an odd permutation. Thus an msimplex is positively on negatively oriented on the basis of the evenness or oddness of the number of transpositions required to put the sequence of its vertices into the natural ordering of the vertices of which it is made up. The number of vertices in a simplex is equal to its dimensionality plus one. A simplex is labeled by its dimensionality. A simplex denoted σ^{m} is a simplex of dimension m.
A complex is a set of msimplexes for all possible values of m.
The incidence number [σ^{m}_{i}, σ^{m1}_{j}] for an msimplex with an (m1)simplex is 0 or ±1 depending upon whether or not the (m1)simplex is a face of the msimplex according to the rule shown below.
The incidence numbers for (m1)simplexes in msimplexes thus form a matrix. Likewise there is a matrix of the incidence numbers of (m2)simplexes in (m1)simplexes and so on down to 0simplexes (vertices) in 1simplexes (edges). The remarkable fact is that there is a simple, strong relationship between the incidence matrices at one level and the the incidence matrices for the next level down.
For all simplexes σ^{m}_{i} and σ^{m2}_{j}
The chain groups of a polyhedron are not very interesting, being isomorphic to ntuples of the coefficient group, where n is the number of simplexes of a particular dimension. The topological content is in the sets of cycles and boundaries. A cycle is a chain which has an empty boundary. Boundaries in particular have no boundary so the set of boundaries are a subset of the set of cycles. The group of boundaries is a subgroup of the group of cycles. What is more, the group of boundaries is a normal subgroup of the group of cycles and hence there is a factor group of cycles with respect of its normal subgroup of boundaries.
The Betti groups of a polyhedron are these factor groups. Enrico Betti (18231892) was an Italian mathematician who combined research in pure mathematics with his career as a university professor and administrator and later as a politician. In 1871 he published a treatise in which he attempted to identify the topology of figures with a set of numbers which were determined by the homotopy groups of the figure. Poincaré called these numbers the Betti numbers of the figure. It turned out that the Betti numbers were not adequate to properly characterize the topology of a figure. However the approach of characterizing the topology of a figure by its groups remained a viable mathematical pursuit. Groups became analogous to the integers.
(To be continued.)
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