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) -dimensional -simplex of a subset of a metric space is defined as an ordered subset in subject to the condition . The -chains of are then defined for a given coefficient group as formal finite linear combinations of -simplices with coefficients . The boundary of an -simplex is defined as follows: ; this is an -chain. By linearity, the boundary of any -chain is defined and -cycles are defined as -chains with zero boundary. An -chain of a set is -homologous to zero in (the notation is ) if for a certain -chain in .

A true cycle of a set is a sequence in which is an -cycle in and (). The true cycles form a group, . A true cycle is homologous to zero in if for any there exists an such that all for are -homologous to zero in . One denotes by the quotient group of the group by the subgroup of cycles that are homologous to zero.

A cycle is called convergent if for any there exists an such that any two cycles , are mutually -homologous in if . The group of convergent cycles is denoted by . Let be the corresponding quotient group.

A cycle has compact support if there exists a compact set such that all the vertices of all simplices of all cycles lie in . 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 the transition to cycles and homology with compact support, one obtains the groups and . 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 , , , modulo a subset are also defined. An -cycle of the set modulo is any -chain in for which the chain lies in . In a similar manner, an -cycle modulo is -homologous modulo to zero in if , where and are -chains in , while the chain lies in .

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)