On \(\omega\)-homologies with coefficients in co-presheaves.

*(English. Russian original)*Zbl 0576.55004
Sov. Math. 29, No. 1, 30-35 (1985); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No. 1, 25-29 (1985).

In this paper the \(\omega\)-homology groups with coefficients in an arbitrary co-presheaf on a locally compact topological space with a countable basis of open sets are studied. The construction of \(\omega\)- homologies, ”an idea that goes back to P. Aleksandrov [C. R. Acad. Sci., Paris 184, 317-319 (1927)]” was introduced and studied in detail, for constant co-presheaves by E. G. Sklyarenko [Usp. Mat. Nauk 24, No.5 (149), 87-140 (1969; Zbl 0204.560); ibid. 34, No.6 (210), 90-118 (1979; Zbl 0426.55005)].

The author considers the so-called factorable co-presheaves, i.e., inductive systems F of abelian groups on a space X such that for any open set U which is the union of open sets \(U_ 1\) and \(U_ 2\) the mapping \(F(U_ 1)\times F(U_ 2)\to F(U)\) is surjective. For factorable co- presheaves, the author proves that the \(\omega\)-homologies form an exact homological functor, that the \(\omega\)-homologies do not depend on the choice of a regular sequence of covers and the metric determining the topology of the space and that there exists an exact sequence relating \(\omega\)-homologies with homologies of separate covers. Also a connection between \(\omega\)-homologies with coefficients in co-presheaves and homologies of analytic sheaves, introduced by the author [Dokl. Akad. Nauk. SSSR 225, 41-43 (1975; Zbl 0343.32006)] is studied.

The author considers the so-called factorable co-presheaves, i.e., inductive systems F of abelian groups on a space X such that for any open set U which is the union of open sets \(U_ 1\) and \(U_ 2\) the mapping \(F(U_ 1)\times F(U_ 2)\to F(U)\) is surjective. For factorable co- presheaves, the author proves that the \(\omega\)-homologies form an exact homological functor, that the \(\omega\)-homologies do not depend on the choice of a regular sequence of covers and the metric determining the topology of the space and that there exists an exact sequence relating \(\omega\)-homologies with homologies of separate covers. Also a connection between \(\omega\)-homologies with coefficients in co-presheaves and homologies of analytic sheaves, introduced by the author [Dokl. Akad. Nauk. SSSR 225, 41-43 (1975; Zbl 0343.32006)] is studied.

Reviewer: Ioan Pop (Iaşi)

##### MSC:

55N30 | Sheaf cohomology in algebraic topology |

55N40 | Axioms for homology theory and uniqueness theorems in algebraic topology |