Vietoris homology

One of the first homology theories (cf. Homology theory) defined for the non-polyhedral case. It was first considered by L.E.J. Brouwer in 1911 (for the case of the plane), after which the definition was extended in 1927 by L. Vietoris to arbitrary subsets of Euclidean (and even metric) spaces.

An (ordered) $ n $- dimensional $ \epsilon $- simplex $ t ^ {n} $ of a subset $ A $ of a metric space $ X $ is defined as an ordered subset $ ( e _ {0} \dots e _ {n} ) $ in $ A $ subject to the condition $ \mathop{\rm diam} \{ e _ {0} \dots e _ {n} \} < \epsilon $. The $ \epsilon $- chains of $ A $ are then defined for a given coefficient group $ G $ as formal finite linear combinations $ \sum g _ {i} t _ {i} $ of $ \epsilon $- simplices $ t _ {i} ^ {n} $ with coefficients $ g _ {i} \in G $. The boundary of an $ \epsilon $- simplex $ t ^ {n} = ( e _ {0} \dots e _ {n} ) $ is defined as follows: $ \Delta t ^ {n} = \sum _ {i} (- 1) ^ {i} ( e _ {0} \dots {\widehat{e} } _ {i} \dots e _ {n} ) $; this is an $ \epsilon $- chain. By linearity, the boundary of any $ \epsilon $- chain is defined and $ \epsilon $- cycles are defined as $ \epsilon $- chains with zero boundary. An $ \epsilon $- chain $ x ^ {n} $ of a set is $ \eta $- homologous to zero in $ A $( the notation is $ x ^ {n} \sim 0 $) if $ x ^ {n} = \Delta y ^ {n+ 1 } $ for a certain $ \eta $- chain $ y ^ {n+ 1 } $ in $ A $.

A true cycle of a set $ A $ is a sequence $ z ^ {n} = \{ z _ {1} ^ {n} \dots z _ {k} ^ {n} ,\dots \} $ in which $ z _ {k} ^ {n} $ is an $ \epsilon _ {k} $- cycle in $ A $ and $ \epsilon _ {k} \rightarrow 0 $( $ k \rightarrow \infty $). The true cycles form a group, $ Z ^ {n} ( A, G) $. A true cycle $ z $ is homologous to zero in $ A $ if for any $ \epsilon > 0 $ there exists an $ N $ such that all $ z _ {k} ^ {n} $ for $ k \geq N $ are $ \epsilon $- homologous to zero in $ A $. One denotes by $ \Delta ^ {n} ( A, G) $ the quotient group of the group $ Z ^ {n} ( A, G) $ by the subgroup $ H ^ {n} ( A, G) $ of cycles that are homologous to zero.

A cycle $ z $ is called convergent if for any $ \epsilon > 0 $ there exists an $ N $ such that any two cycles $ z _ {k} ^ {n} $, $ z _ {m} ^ {n} $ are mutually $ \epsilon $- homologous in $ A $ if $ k, m \geq N $. The group of convergent cycles is denoted by $ Z _ {c} ^ {n} ( A, G) $. Let $ \Delta _ {c} ^ {n} ( A, G) = Z _ {c} ^ {n} ( A, G) / H _ {c} ^ {n} ( A, G) $ be the corresponding quotient group.

A cycle $ z $ has compact support if there exists a compact set $ F \subseteq A $ such that all the vertices of all simplices of all cycles $ z _ {k} ^ {n} $ lie in $ F $. One similarly modifies the concept of a cycle being homologous to zero by requiring the presence of a compact set on which all the homology-realizing chains lie; convergent cycles with compact support can thus be defined. By denoting with a subscript $ k $ the transition to cycles and homology with compact support, one obtains the groups $ \Delta _ {k} ^ {n} ( A, G) $ and $ \Delta _ {ck} ^ {n} ( A, G) $. The latter group is known as the Vietoris homology group. If the polyhedron is finite, the Vietoris homology groups coincide with the standard homology groups.

Relative homology groups $ \Delta ^ {n} ( A, B, G) $, $ \Delta _ {c} ^ {n} ( A, B, G) $, $ \Delta _ {k} ^ {n} ( A, B, G) $, $ \Delta _ {ck} ^ {n} ( A, B, G) $ modulo a subset $ B \subseteq A $ are also defined. An $ \epsilon $- cycle of the set $ A $ modulo $ B $ is any $ \epsilon $- chain $ x ^ {n} $ in $ A $ for which the chain $ \Delta x ^ {n} $ lies in $ B $. In a similar manner, an $ \epsilon $- cycle $ x ^ {n} $ modulo $ B $ is $ \eta $- homologous modulo $ B $ to zero in $ A $ if $ x ^ {n} = \Delta y ^ {n+} 1 + w ^ {n} $, where $ y ^ {n+} 1 $ and $ w ^ {n} $ are $ \eta $- chains in $ A $, while the chain $ w ^ {n} $ lies in $ B $.

References

[1]	P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)

References

[a1]	J.G. Hocking, G.S. Young, "Topology" , Addison-Wesley (1961)