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