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4 Bulletin of the Polish Academy of Sciences. Mathematics

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Strong cohomological dimension

Dydak J., Koyama A.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

183-189

We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that Inde X = dim[sub G] X if X is a separable metric ANR and G is a countable Abelian group. Hence dim[sub Z] X = dim X for any separable metric ANR X.

Once more on the Lefschetz fixed point theorem

Górniewicz L., Ślosarski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 2

161-170

An abstract version of the Lefschetz fixed point theorem is presented. Then several generalizations of the classical Lefschetz fixed point theorem are obtained.

Fibrations in the category of absolute neighborhood retracts

Miyata T.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 2

145-154

The category Top of topological spaces and continuous maps has the structures of a fibration category and a cofibration category in the sense of Baues, where fibration = Hurewicz fibration, cofibration = the usual cofibration, and weak equivalence = homotopy equivalence. Concentrating on fibrations, we consider the problem: given a full subcategory C of Top, is the fibration structure of Top restricted to C a fibration category? In this paper we take the special case where C is the full subcategory ANR of Top whose objects are absolute neighborhood retracts. The main result is that ANR has the structure of a fibration category if fibration = map having a property that is slightly stronger than the usual homotopy lifting property, and weak equivalence = homotopy equivalence.

Periodic solutions of periodic retarded functional differential equations

Pawłowski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

353-363

The paper presents a geometric method of finding periodic solutions of retarded functional differential equations (REDE) x'(t) = f(t,x1), where f is T-periodic in t. We construct a pair of subsets of R x R^n called a T-periodic block and compute its Lefschetz number. If it is nonzero, then there exists a T-periodic solution.