Homological containment
A method for characterizing the dimension of a compactum lying in a Euclidean space in terms of metric properties of the complementary space. The essential measure of a cycle
in a compactum
is taken to be the least upper bound of those
for which it is possible to select a compact support
of the cycle
such that the cycle is not homologous to zero in
. The
-dimensional homological diameter
of a cycle
in an open set
is the greatest lower bound of the
-dimensional diameters of the bodies of all cycles that are homologous in
to
. Here, the
-dimensional diameter
of a compactum
is the greatest lower bound of those
for which there exists a continuous
-shift of
in a
-dimensional compactum (and thus in a polyhedron).
Any -dimensional cycle of the open set
which is linked with each point of the compactum
is said to be a pocket around
.
The pocket theorem. Let . Then there exists an
such that any pocket around
has
-dimensional homological diameter larger than
, while the
-dimensional homological diameter of any cycle in
is zero. Pockets around
with arbitrary small essential measure always exist in this situation. On the other hand, if
, then there exists an
such that for any pocket
around
the inequality
is true (here,
and
for any pocket
).
The pocket theorem may be further strengthened using the concept of a zone around a compactum.
The zone theorem. Let be a compactum of dimension
. There exists a
such that for any
and any
there exists in
an
-dimensional cycle
(a zone of dimension
around
), for
bounding in
, for which
,
. Furthermore, for any cycle
homologous to
in the
-neighbourhood of the latter with respect to
the inequality
is valid; for any chain
bounded by the cycle
in
one has
.
On the other hand, if and if
, then, for any
, any
-dimensional cycle
in
for which
is homologous in its
-neighbourhood (with respect to
) to some cycle
with
arbitrary small. Furthermore, if
and if
, then for any
any
-dimensional cycle
, bounding in
, for which
(and
if
) bounds in
a chain
with
. Here
,
, is the greatest lower bound of those
for which there exists an
-shift of the vertices of the chain
by means of which
becomes degenerate up to dimension
;
is the greatest lower bound of those
for which there exists an
-shift of the vertices of
converting
to a zero chain.
References
[1] | P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian) |
Comments
An -shift of a subspace
contained in some Euclidean space
is a mapping
such that the distance of
to
is less than
for all
.
Homological containment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homological_containment&oldid=47255