Homology of a polyhedron

A homology theory of a topological space which is a polyhedron (cf. Polyhedron, abstract). Homology of a polyhedron first appeared in the works of H. Poincaré (1895) in a study of manifolds in Euclidean spaces. He considered $ r $-dimensional closed submanifolds of a given manifold, known as $ r $-dimensional cycles. If the manifold includes a bounded $ ( r + 1 ) $-dimensional submanifold with as boundary a given $ r $-dimensional cycle, this cycle is said to be homologous to zero in the given manifold. Thus, a circle which is concentric with the circles bounding an annulus is not homologous to zero, whereas the circle forming the boundary of a disc contained in the annulus is homologous to zero in this annulus. The initial analytic definition of a manifold was replaced by Poincaré by its representation by simplices (or simplexes) with adjacent boundaries, forming a complex. Such a method for studying homology may be applied to any space that can be triangulated as a simplicial complex, i.e. that can be seen as rectilinear polyhedra, or their homeomorphic images — curvilinear polyhedra. The geometrical meaning of cycles and their homology is preserved. Thus, a one-dimensional cycle is a closed polygonal line with one-dimensional simplices as its segments. It is homologous to zero if it is the boundary of a two-dimensional subcomplex of the given complex. Two cycles of equal dimension are homologous to each other if, taken together, they bound a subcomplex of the given complex. This is an equivalence relation the result of which is a subdivision of the set of cycles with the same dimension into classes. An algebraic structure may be introduced into the set of classes if the sum of two classes is the class containing the sum of two cycles arbitrarily chosen out of the classes being added. The introduction of a direction of traversal, i.e. of oriented simplices, leads to the concept of the inverse class. A strict interpretation of these illustrative concepts makes it possible to define the concept of the homology groups of a polyhedron.

Let there be given a triangulation $ K $ of a polyhedron $ P $ and an Abelian group $ G $. An $ r $-dimensional chain of the complex $ K $ over the coefficient group $ G $ is an arbitrary function $ c _ {r} $ that assigns to each oriented $ r $-dimensional simplex $ t ^ {r} $ from $ K $ a certain element of $ G $, and that is non-zero only for a finite number of simplices; moreover, $ c _ {r} ( - t ^ {r} ) = - c _ {r} ( t ^ {r} ) $. By adding $ r $-dimensional chains as linear forms one obtains the Abelian group $ C _ {r} ( K, G) $ of all $ r $-dimensional chains of $ K $ over $ G $. Starting from the concept of the boundary of a simplex, and defining the boundary of a chain by additivity, one arrives at a homomorphism

$$ \partial _ {r} : C _ {r} ( K, G) \rightarrow C _ {r - 1 } ( K, G) $$

with the property $ \partial _ {r- 1} \partial _ {r} = 0 $, and the chain complex

$$ \{ C _ {r} ( K, G), \partial _ {r} \} . $$

A chain $ c _ {r} $ is called a cycle if its boundary is the zero chain: $ \partial _ {r} c _ {r} = 0 $. A cycle $ z _ {r} $ is said to be bounding (a boundary) if $ K $ contains an $ ( r + 1) $-dimensional chain $ c _ {r+ 1} $ such that $ z _ {r} = \partial _ {r+ 1} c _ {r+ 1} $. The kernel of the homomorphism $ \partial _ {r} $, i.e. the group $ Z _ {r} ( K, G) $ of all $ r $-dimensional cycles, contains the image under the homomorphism $ \partial _ {r+ 1} $, i.e. the subgroup $ B _ {r} ( K, G) $ of all bounding $ r $-dimensional cycles. The quotient group $ H _ {r} ( K, G) $ of $ Z _ {r} ( K, G) $ by $ B _ {r} ( K, G) $ is the $ r $-dimensional homology group of $ K $ over $ G $. It is taken as the $ r $-dimensional homology group $ H _ {r} ( P, G) $ of the polyhedron $ P $ over $ G $, since it can be proved that all triangulations of $ P $ have isomorphic $ r $-dimensional homology groups over $ G $. In view of the universal coefficient theorem, the group $ H _ {r} ( P, G) $ is defined, for an arbitrary $ G $, by integer groups $ H _ {s} ( P, \mathbf Z ) $, where $ \mathbf Z $ is the group of integers. Furthermore, if the polyhedron is finite, the integer group, which is an Abelian group with a finite number of generators, has a complete system of numerical invariants — the Betti number and the torsion coefficients, i.e. the rank and the torsion coefficients of the group $ H _ {r} ( P, \mathbf Z ) $.

References

[1]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[2]	L.S. Pontryagin, "Grundzüge der kombinatorischen Topologie" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3]	H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German)
[4]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)