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) |
Comments
References
| [a1] | J.G. Hocking, G.S. Young, "Topology" , Addison-Wesley (1961) |
Vietoris homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vietoris_homology&oldid=49145