位相幾何学の応用

最近整数ホモロジーに関して読んでいますが遡ってホモロジーの意味と役割を検討したいです。
Homology is a concept relating two k-chains C1 and C2. They are considered homologous when C1+C2 is a boundary. The sum of C1 and C2 is defined to be those k-simplexes in either C1 or C2 but not both. A boundary of a k+1 chain is a k-chain consisting of k-simplexes which are incident on simplexes of the k+1 chain an odd number of times. The definition of homology then leads to that of homology group: the group of k-cycles Zk where homology is used in place of equality is said to be the kth homology group of the complex K that Zk contains cycles of simplexes in. The rank of the kth homology group, hk, is known as the kth Betti number. If G is a group and H a subgroup: if for x and y element of G, x= y+h where h is an element of H, then x and y are homologous mod h. If homology mod H is used in place of equality in G, the resulting group is known as the quotient group of G by H, or G/H. I believe that the notion of quotient groups can be generalized further, i.e. groups where (1=4, 2=5, 3=6, etc) could be a quotient group of integers (Z) by the integers {3n} as equivalence mod 3 replaces equaliy (cf abstract algebra text).
Homology groups and Betti numbers are used in the definition of the Euler characteristic (F-E+V) and in proving that the Euler characteristic is invariant to how the space is divided (e.g. triangulation).
Integral homology involves some new assumptions about our complexes:
1. edges and polygons are assigned directions
2. incidence coefficients (used to determine boundaries) can be negative (between point and edge when point is start of edge and between an edge and face when direction of edge opposes direction of face).
The old homology is known as homology mod 2, as a simplex either occurs in a chain or it does not. In integral homology, though, chains are linear combinations of simplex, such as xA + yB + zC
A K-boundary under the new homology is the boundary of a k+1 chain, as before. However, the boundary of a k+1 simplex S is the k-chain where the coefficients of each k-simplex are the incidence coefficients of S with that simplex (not the k-simplexes which have odd incidence coefficients with S, as in homology mod 2). The boundary of a k+1 chain, then, is the boundary of the linear product of k-simplexes and their coefficients, or the sum of the boundaries of each simplex (times coefficient) in the chain.
i.e. d(aS1+bS2) = ad(S1)+bd(S2) where d is the boundary operator.
The definition of homology is: two K chains C1 and C2 are homologous if C1-C2 is a boundary. Note that this differs from the mod 2 case (where the sum was a boundary). Apparently, this difference is due to the fact that mod 2 chain groups were idemgroups (i.e. A+A = 0), while chain groups in this case are not (A+A = 2A).
Homology groups under integral homology are written Hk=Zk/Bk, where Bk is the set of k-boundaries. Homology groups in the mod 2 definition were also defined as the quotient of group of k-cycles and k-boundaries.
Some questions are:
1. How homology is useful to society besides its applications to conjectures in topology.

2. How integral homology is both different and better than homology.

A more general question might be what the applications of topology itself are. Point set topology and combinatorial topology involve algebra, vectors, differential equations, gradients, and functional analysis. However, can topology be used to solve problems in applied science or engineering, or is it simply a meta-science which describes and explains mathematics? One area of application (at least implicitly) may be graph theory, as we tend to care not about the shapes of edges or the positioning of nodes, but about which edges connect which nodes.
詳細が分からないけど位相幾何学が以下の所に応用されているそうです。
1. topological robotics
2. computational topology - includes computational homology, protein/cell structure, image analysis
3. physics - thermodynamics, entropy, liquid crystals
4. computer graphics - triangulation
5. automobile engineering