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General homology and cohomology theories. Current state and typical applications. (English. Russian original) Zbl 0793.55004

J. Sov. Math. 57, No. 2, 3009-3066 (1991); translation from Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 27, 125-228 (1989).
The subject of the paper is the current state of the general homology and cohomology theories and their typical applications.
Chapter One is devoted to sheaf cohomology (possibly with supports) in the case of a space, a subspace and a pair of spaces as the cohomology of the complex of sections with respect to an arbitrary acyclic resolution (for example, a resolution of flabby or soft sheaves); the cohomology thus defined is uniquely determined. Some specific resolutions and properties of the cohomology functor are considered. In particular, the existence of injective resolutions gives the possibility of interpreting the sheaf cohomologies as derived functors of the zero-dimensional cohomology functor. The constructions allow one to apply methods of homological algebra to the study of cohomology.
In the second chapter, the author discusses homology theory in the categories of compact, locally compact and paracompact spaces. The role of compact supports, the sheaf of chains, the homology of type, the sheaves of local homologies, etc. is also considered.
The third chapter presents some typical concrete approaches to homology and cohomology: singular theory; Alexander-Spanier cohomology; Massey free cochains and the associated chains; Čech chains and cochains; chains under locally compact spaces; Borel-Moore homology. The author considers some properties of homology and cohomology theories: the universal coefficient formula: the relation between a homology theory and the Čech theory generated by it; the additive property and its connection with uniqueness.
The fourth chapter provides some typical applications: homology and cohomology of a connection; the Mayer-Vietoris sequence; local behaviour; homological dimension; some results by Dranishnikov; the Leray spectral sequence of a map; the Vietoris theorems; Poincaré duality; homology manifolds; homology and cohomology spheres etc.

MSC:

55N30 Sheaf cohomology in algebraic topology
55N05 Čech types
55N07 Steenrod-Sitnikov homologies
55N35 Other homology theories in algebraic topology
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