ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 55 Mdzinarishvili, Leonard, Georgian Technical University, Georgia Partially continuous Alexander-Spanier cohomology theory Our aim is to introduce a new cohomology of Alexander-Spanier type with coefficients in an arbitrary topological abelian group. This cohomology will be denoted by h̄^{∗}. On the category A_{p} of paracompact Hausdorff pairs the cohomology h̄^{∗} with coefficients in an arbitrary topological abelian group G satisfies all axioms of Eilenberg-Steenrod, except the excision axiom, and the additivity axiom. If G is an absolute retract, then h̄^{∗} is isomorphic to ordinary Alexander-Spanier cohomology on A_{p}. If G = S^{1}, then

h̄^{m}(S^{n}, G) ≈ \{ \begin{matrix}S^{1}, & m = 0, n ≥ 1 \\ S^{1} + S^{1}, & m = n, n = 0 \\ 0, & m ≠ n, 0 \\ S^{1}, & m = n, n = 1 \\ ℝ, & m = n, n ≥ 2 \end{matrix}

Reference: L. Mdzinarishvili, Partially continuous Alexander-Spanier cohomology theory; Grüne Preprintreiche der Universität Heidelberg, Heft 130(1996).