Albrecht Dold was born on August 5, 1928 in Triberg (Black Forest), Germany. He studied mathematics and physics at the University of Heidelberg, then worked for some years at the Institute for Advanced Study in Princeton, at Columbia University, New York and at the University of Zurich. In 1963 he returned to Heidelberg, where he has stayed since, declining several offers to attractive positions elsewhere. A. Dold's seminal work in algebraic topology has brought him international recognition beyond the world of mathematics itself. In particular, his work on fixed-point theory has made his a household name in economics, and his book "Lectures on Algebraic Topology" a standard reference among economists as well as mathematicians.

• I Preliminaries on Categories, Abelian Groups, and Homotopy.- §1 Categories and Functors.- §2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- §3 Homotopy.- II Homology of Complexes.- §1 Complexes.- §2 Connecting Homomorphism, Exact Homology Sequence.- §3 Chain-Homotopy.- §4 Free Complexes.- III Singular Homology.- §1 Standard Simplices and Their Linear Maps.- §2 The Singular Complex.- §3 Singular Homology.- §4 Special Cases.- §5 Invariance under Homotopy.- §6 Barycentric Subdivision.- §7 Small Simplices. Excision.- §8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- §1 Standard Maps between Cells and Spheres.- §2 Homology of Cells and Spheres.- §3 Local Homology.- §4 The Degree of a Map.- §5 Local Degrees.- §6 Homology Properties of Neighborhood Retracts in ?n.- §7 Jordan Theorem, Invariance of Domain.- §8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- §1 Cellular Spaces.- §2 CW-Spaces.- §3 Examples.- §4 HomologyProperties of CW-Spaces.- §5 The Euler-Poincaré Characteristic.- §6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- §7 Simplicial Spaces.- §8 Simplicial Homology.- VI Functors of Complexes.- §1 Modules.- §2 Additive Functors.- §3 Derived Functors.- §4 Universal Coefficient Formula.- §5 Tensor and Torsion Products.- §6 Horn and Ext.- §7 Singular Homology and Cohomology with General Coefficient Groups.- §8 Tensorproduct and Bilinearity.- §9 Tensorproduct of Complexes. Künneth Formula.- §10 Horn of Complexes. Homotopy Classification of Chain Maps.- §11 Acyclic Models.- §12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- §1 The Scalar Product.- §2 The Exterior Homology Product.- § 3 The Interior Homology Product (Pontijagin Product).- § 4 Intersection Numbers in ?n.- §5 The Fixed Point Index.- §6 The Lefschetz-Hopf Fixed Point Theorem.- §7 The Exterior Cohomology Product.- § 8 The Interior Cohomology Product (?-Product).- § 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- §10 Hopf Algebras.- §11 The Cohomology Slant Product.- §12 The Cap-Product (?-Product).- § 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- §1 Elementary Properties of Manifolds.- §2 The Orientation Bundle of a Manifold.- §3 Homology of Dimensions ? n in n-Manifolds.- §4 Fundamental Class and Degree.- §5 Limits.- §6 ?ech Cohomology of Locally Compact Subsets of ?n.- §7 Poincaré-Lefschetz Duality.- §8 Examples, Applications.- §9 Duality in ?-Manifolds.- §10 Transfer.- §11 Thom Class, Thom Isomorphism.- §12 The Gysin Sequence. Examples.- §13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- §1 Limits of Functors.- §2 Polyhedrons under a Space, and Partitions of Unity.- §3 Extending Functors from Polyhedrons to More General Spaces.

Albrecht Dold was born on August 5, 1928 in Triberg (Black Forest), Germany. He studied mathematics and physics at the University of Heidelberg, then worked for some years at the Institute for Advanced Study in Princeton, at Columbia University, New York and at the University of Zurich. In 1963 he returned to Heidelberg, where he has stayed since, declining several offers to attractive positions elsewhere. A. Dold's seminal work in algebraic topology has brought him international recognition beyond the world of mathematics itself. In particular, his work on fixed-point theory has made his a household name in economics, and his book "Lectures on Algebraic Topology" a standard reference among economists as well as mathematicians.

• I Preliminaries on Categories, Abelian Groups, and Homotopy.- §1 Categories and Functors.- §2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- §3 Homotopy.- II Homology of Complexes.- §1 Complexes.- §2 Connecting Homomorphism, Exact Homology Sequence.- §3 Chain-Homotopy.- §4 Free Complexes.- III Singular Homology.- §1 Standard Simplices and Their Linear Maps.- §2 The Singular Complex.- §3 Singular Homology.- §4 Special Cases.- §5 Invariance under Homotopy.- §6 Barycentric Subdivision.- §7 Small Simplices. Excision.- §8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- §1 Standard Maps between Cells and Spheres.- §2 Homology of Cells and Spheres.- §3 Local Homology.- §4 The Degree of a Map.- §5 Local Degrees.- §6 Homology Properties of Neighborhood Retracts in ?n.- §7 Jordan Theorem, Invariance of Domain.- §8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- §1 Cellular Spaces.- §2 CW-Spaces.- §3 Examples.- §4 HomologyProperties of CW-Spaces.- §5 The Euler-Poincaré Characteristic.- §6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- §7 Simplicial Spaces.- §8 Simplicial Homology.- VI Functors of Complexes.- §1 Modules.- §2 Additive Functors.- §3 Derived Functors.- §4 Universal Coefficient Formula.- §5 Tensor and Torsion Products.- §6 Horn and Ext.- §7 Singular Homology and Cohomology with General Coefficient Groups.- §8 Tensorproduct and Bilinearity.- §9 Tensorproduct of Complexes. Künneth Formula.- §10 Horn of Complexes. Homotopy Classification of Chain Maps.- §11 Acyclic Models.- §12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- §1 The Scalar Product.- §2 The Exterior Homology Product.- § 3 The Interior Homology Product (Pontijagin Product).- § 4 Intersection Numbers in ?n.- §5 The Fixed Point Index.- §6 The Lefschetz-Hopf Fixed Point Theorem.- §7 The Exterior Cohomology Product.- § 8 The Interior Cohomology Product (?-Product).- § 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- §10 Hopf Algebras.- §11 The Cohomology Slant Product.- §12 The Cap-Product (?-Product).- § 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- §1 Elementary Properties of Manifolds.- §2 The Orientation Bundle of a Manifold.- §3 Homology of Dimensions ? n in n-Manifolds.- §4 Fundamental Class and Degree.- §5 Limits.- §6 ?ech Cohomology of Locally Compact Subsets of ?n.- §7 Poincaré-Lefschetz Duality.- §8 Examples, Applications.- §9 Duality in ?-Manifolds.- §10 Transfer.- §11 Thom Class, Thom Isomorphism.- §12 The Gysin Sequence. Examples.- §13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- §1 Limits of Functors.- §2 Polyhedrons under a Space, and Partitions of Unity.- §3 Extending Functors from Polyhedrons to More General Spaces.

Albrecht Dold was born on August 5, 1928 in Triberg (Black Forest), Germany. He studied mathematics and physics at the University of Heidelberg, then worked for some years at the Institute for Advanced Study in Princeton, at Columbia University, New York and at the University of Zurich. In 1963 he returned to Heidelberg, where he has stayed since, declining several offers to attractive positions elsewhere. A. Dold's seminal work in algebraic topology has brought him international recognition beyond the world of mathematics itself. In particular, his work on fixed-point theory has made his a household name in economics, and his book "Lectures on Algebraic Topology" a standard reference among economists as well as mathematicians.

• I Preliminaries on Categories, Abelian Groups, and Homotopy.- §1 Categories and Functors.- §2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- §3 Homotopy.- II Homology of Complexes.- §1 Complexes.- §2 Connecting Homomorphism, Exact Homology Sequence.- §3 Chain-Homotopy.- §4 Free Complexes.- III Singular Homology.- §1 Standard Simplices and Their Linear Maps.- §2 The Singular Complex.- §3 Singular Homology.- §4 Special Cases.- §5 Invariance under Homotopy.- §6 Barycentric Subdivision.- §7 Small Simplices. Excision.- §8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- §1 Standard Maps between Cells and Spheres.- §2 Homology of Cells and Spheres.- §3 Local Homology.- §4 The Degree of a Map.- §5 Local Degrees.- §6 Homology Properties of Neighborhood Retracts in ?n.- §7 Jordan Theorem, Invariance of Domain.- §8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- §1 Cellular Spaces.- §2 CW-Spaces.- §3 Examples.- §4 HomologyProperties of CW-Spaces.- §5 The Euler-Poincaré Characteristic.- §6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- §7 Simplicial Spaces.- §8 Simplicial Homology.- VI Functors of Complexes.- §1 Modules.- §2 Additive Functors.- §3 Derived Functors.- §4 Universal Coefficient Formula.- §5 Tensor and Torsion Products.- §6 Horn and Ext.- §7 Singular Homology and Cohomology with General Coefficient Groups.- §8 Tensorproduct and Bilinearity.- §9 Tensorproduct of Complexes. Künneth Formula.- §10 Horn of Complexes. Homotopy Classification of Chain Maps.- §11 Acyclic Models.- §12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- §1 The Scalar Product.- §2 The Exterior Homology Product.- § 3 The Interior Homology Product (Pontijagin Product).- § 4 Intersection Numbers in ?n.- §5 The Fixed Point Index.- §6 The Lefschetz-Hopf Fixed Point Theorem.- §7 The Exterior Cohomology Product.- § 8 The Interior Cohomology Product (?-Product).- § 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- §10 Hopf Algebras.- §11 The Cohomology Slant Product.- §12 The Cap-Product (?-Product).- § 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- §1 Elementary Properties of Manifolds.- §2 The Orientation Bundle of a Manifold.- §3 Homology of Dimensions ? n in n-Manifolds.- §4 Fundamental Class and Degree.- §5 Limits.- §6 ?ech Cohomology of Locally Compact Subsets of ?n.- §7 Poincaré-Lefschetz Duality.- §8 Examples, Applications.- §9 Duality in ?-Manifolds.- §10 Transfer.- §11 Thom Class, Thom Isomorphism.- §12 The Gysin Sequence. Examples.- §13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- §1 Limits of Functors.- §2 Polyhedrons under a Space, and Partitions of Unity.- §3 Extending Functors from Polyhedrons to More General Spaces.