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 .