Theory of Retracts

Preface
Chapter I. General Properties of r-maps
1. r-maps
2. Retractions
3. Retracts
4. Neighborhood retracts
5. Retraction and extension of maps
6. r-types and r-invariants
7. Fixed point property
8. Some local r-invariants
9. Function spaces
10. Maps of pairs
11. Homotopies
12. Deformation retracts
13. Contractibility
14. h-maps and homotopy domination
15. Local contractibility
16. Homotopically trivial spaces
17. n-connectedness and local n-connectedness
Chapter II. Algebraic Relations Induced by Maps
1. r-homomorphisms
2. Homology and cohomology groups
3. Homology theory in metric spaces
4. Homomorphisms induced by h-maps and r-maps
5. Join of maps
6. Homotopy groups
7. Homomorphisms induced by maps
8. Homomorphism of Hurewicz
9. Maps into manifolds
10. Join of maps into spheres
11. Cohomotopy groups
12. Homomorphisms induced by maps
13. Homotopy dependence of maps
14. λ-morphisms
Chapter III. Extension of Maps
1. Coverings
2. Polytopes
3. Finite and locally finite polytopes
4. Metrizability of polytopes
5. Null-triangulations
6. Nerve of a covering
7. Generalization of Tietze’s Theorem
8. Embedding of a metrizable space in a normed linear space
9. Extension of maps with values belonging to an LCⁿ-space
Chapter IV. Absolute Retracts and Absolute Neighborhood Retracts in Metric Spaces
1. Spaces AR(?) and ANR(?)
2. Elementary properties of spaces AR(?)
3. Elementary properties of spaces ANR(?)
4. Absolute retracts and extension of maps
5. Spaces of maps with values in an ANR(?)-space
6. Addition of AR(?)-spaces and of ANR(?)-spaces
7. Cartesian products of AR(?)-sets and of ANR(?)-sets
8. Extension of a homotopy
9. Contractible ANR(?)-spaces
10. Open subsets of ANR(?)-spaces
Chapter V. Absolute Retracts and Absolute Neighborhood Retracts in Compacta
1. AR-spaces and ANR-spaces
2. Elementary properties of AR-spaces and ANR-spaces
3. A theorem on extension of maps
4. ANR-spaces and polyhedra
5. Approximation of ANR-sets by two prisms
6. Embedding of compacta in AR-spaces
7. A characterization of ANR-spaces
8. Condition of Lefschetz
9. Matching of sets
10. Finite-dimensional ANR-spaces
11. Locally contractible compactum which is not an ANR-space
12. Upper semicontmuous decompositions of ANR’s
13. Plane AR-sets
14. Plane ANR-sets
15. n-dimensional subsets of an n-dimensional ANR-space
16. Umbrellas theorem
Chapter VI. Pathologies among ANR-spaces
1. The singularity of Peano
2. The singularity of Alexandroff
3. The singularity of Brouwer
4. The singularity of Mazurkiewicz
5. Some open problems
6. Families of locally r-incomparable AR-sets
7. Universal retracts
Chapter VII. ANR-spaces satisfying Some Special Conditions
1. Condition (Δ)
2. Maps into a space Y_ε(Δ)
3. Extension of maps with values in a space Y_ε(Δ)
4. Addition of sets satisfying condition (Δ)
5. Cartesian product of sets satisfying condition (Δ)
6. Weak homology. Convergent cycles and their divisors
7. True modular cycles
8. True cycles in spaces satisfying condition (Δ)
9. Modular dimensions of compacta satisfying condition (Δ)
10. Condition (Γ)
11. A lemma on embedding in simplexes
12. Simplicial realizations of finite coverings
13. Spaces satisfying condition (Γ) as deformation retracts of polyhedra
14. Homotopy types of spaces satisfying condition (Γ)
15. Homogeneous ANR-spaces
16. Disconnection of homogeneous ANR-spaces by compacta
Chapter VIII. On classification of Spaces
1. Partial ordering of r-types
2. r-minorants and r-majorants ([61], p. 323)
3. r-increasing and r-decreasing sequences
4. Power of the class of all r-types of compacta
5. r-neighbors
6. AR-space with an infinite number of r-neighbors
7. Dimension of r-neighbors
8. Index of r-proximity
Chapter IX. A Review of Various Results and Problems of the Theory of Retracts
1. Modifications of the basic notions of the theory of retracts
2. Embedding problems
3. Problems on addition and matching
4. Various operations on AR’s and ANR’s
5. Dimension of ANR-spaces
6. AR-sets and ANR-sets in Euclidean spaces
7. Some homotopy invariants
8. Decompositions of ANR-spaces into Cartesian products
9. ANR-spaces and polyhedra
10. Problems of metrization
11. Hyperspaces of ANR-spaces
12. Problems of classification of ANR-spaces
13. Problem of elimination of singularities
14. Fixed points in ANR-spaces
15. Homotopy classification of maps into ANR-spaces
16. Colocalization of topological properties. Spheroidal spaces and r-spaces
Bibliography
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List of Special Symbols
Index
ABCDE
FGHIJKL
MNOPQR
STUVW; (A)ΓΔλε