Homological containment

A method for characterizing the dimension of a compactum lying in a Euclidean space $ \mathbf R ^ {n} $ in terms of metric properties of the complementary space. The essential measure of a cycle $ z $ in a compactum $ \Phi \subset \mathbf R ^ {n} $ is taken to be the least upper bound of those $ \epsilon > 0 $ for which it is possible to select a compact support $ \Phi _ {1} \subseteq \Phi $ of the cycle $ z $ such that the cycle is not homologous to zero in $ O ( \Phi _ {1} , \epsilon ) $. The $ p $- dimensional homological diameter $ \alpha _ \Gamma ^ {p} z $ of a cycle $ z $ in an open set $ \Gamma = \mathbf R ^ {n} \setminus \Phi $ is the greatest lower bound of the $ p $- dimensional diameters of the bodies of all cycles that are homologous in $ \Gamma $ to $ z $. Here, the $ p $- dimensional diameter $ \alpha ^ {p} X $ of a compactum $ X \subset \mathbf R ^ {n} $ is the greatest lower bound of those $ \epsilon > 0 $ for which there exists a continuous $ \epsilon $- shift of $ X $ in a $ p $- dimensional compactum (and thus in a polyhedron).

Any $ ( n - 1 ) $- dimensional cycle of the open set $ \Gamma = \mathbf R ^ {n} \setminus \Phi $ which is linked with each point of the compactum $ \Phi $ is said to be a pocket around $ \Phi $.

The pocket theorem. Let $ r = \mathop{\rm dim} \Phi \leq n - 1 $. Then there exists an $ \alpha > 0 $ such that any pocket around $ \Phi $ has $ ( r - 1) $- dimensional homological diameter larger than $ \alpha $, while the $ r $- dimensional homological diameter of any cycle in $ \Gamma $ is zero. Pockets around $ \Phi $ with arbitrary small essential measure always exist in this situation. On the other hand, if $ \mathop{\rm dim} \Phi = n $, then there exists an $ \alpha > 0 $ such that for any pocket $ z ^ {n-1} $ around $ \Phi $ the inequality $ \mu z ^ {n-1} > \alpha $ is true (here, $ \alpha _ \Gamma ^ {n-2} z ^ {n-1} > 0 $ and $ \alpha _ \Gamma ^ {n-1} z ^ {n-1} = 0 $ for any pocket $ z ^ {n-1} $).

The pocket theorem may be further strengthened using the concept of a zone around a compactum.

The zone theorem. Let $ \Phi \subset \mathbf R ^ {n} $ be a compactum of dimension $ r $. There exists a $ \gamma > 0 $ such that for any $ k = 1 \dots r + 1 $ and any $ \epsilon > 0 $ there exists in $ \Gamma = \mathbf R ^ {n} \setminus \Phi $ an $ ( n - k) $- dimensional cycle $ v $( a zone of dimension $ n - k $ around $ \Phi $), for $ k> 1 $ bounding in $ \Gamma $, for which $ \beta ^ {r-k+ 1} v < \epsilon $, $ \tau v < \epsilon $. Furthermore, for any cycle $ w $ homologous to $ v $ in the $ \gamma $- neighbourhood of the latter with respect to $ \Gamma $ the inequality $ \beta ^ {r-n+ 1} w > \gamma $ is valid; for any chain $ x $ bounded by the cycle $ v $ in $ \Gamma $ one has $ \beta ^ {r-n+ 1} x > \gamma $.

On the other hand, if $ s > r $ and if $ k = 1 \dots s + 1 $, then, for any $ \gamma > 0 $, any $ ( n - k) $- dimensional cycle $ z $ in $ \Gamma $ for which $ \tau z < \gamma $ is homologous in its $ \gamma $- neighbourhood (with respect to $ \Gamma $) to some cycle $ z ^ \prime $ with $ \beta ^ {s-k} z ^ \prime $ arbitrary small. Furthermore, if $ s > r $ and if $ k = 2 \dots s + 1 $, then for any $ \gamma > 0 $ any $ ( n - k ) $- dimensional cycle $ z $, bounding in $ \Gamma $, for which $ \beta ^ {s-k+ 1} z < \gamma $( and $ \tau z < \gamma $ if $ s = n - 1 $) bounds in $ \Gamma $ a chain $ x $ with $ \beta ^ {s-n+ 1} x < \gamma $. Here $ \beta ^ {p} x $, $ p \geq 0 $, is the greatest lower bound of those $ \epsilon > 0 $ for which there exists an $ \epsilon $- shift of the vertices of the chain $ x $ by means of which $ x $ becomes degenerate up to dimension $ p $; $ \tau x $ is the greatest lower bound of those $ \epsilon > 0 $ for which there exists an $ \epsilon $- shift of the vertices of $ x $ converting $ x $ 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 $\epsilon$-shift of a subspace $ X $ contained in some Euclidean space $ \mathbf R ^ {m} $ is a mapping $ f: X \rightarrow \mathbf R ^ {m} $ such that the distance of $ x $ to $ f( x) $ is less than $ \epsilon $ for all $ x \in X $.