James R. Munkres served on the MIT Mathematics Faculty from 1960-2000, and continues as Senior Lecturer. He received the PhD in Mathematics from the University of Michigan under the supervision of Edwin Moise in 1956. Professor Munkres is a differential topologist and is also responsible for the Munkres assignment algorithm. He authored numerous texts. His distinctions include the MIT School of Science Teaching Prize for Undergraduate Education in 1984, and an Honorary Doctorate from Nebraska Wesleyan University in 1989.Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He received his Ph.D. from Princeton University and since then has taught at UCLA, Princeton University, and Pennsylvania State University. Dr. Krantz has written over 175 scholarly papers and more than 65 books. He is the founding editor of the Journal of Geometric Analysis. He was named a fellow of the American Mathematical Society and has received the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize.Harold R. Parks received his Ph.D. in mathematics from Princeton University. He was a J. D. Tamarkin Instructor at Brown University. He then assumed a tenure-track position at Oregon State University. He was promoted to Professor in 1989. He spent the academic year 1982-83 at Indiana University as a Visiting Associate Professor. During his time in the Mathematics Department of Oregon State, he served at various times as Assistant Department Chair, Associate Department Chair, and Department Chair. He has written 8 books and 42 scholarly papers. He edits 2 journals. He is an AMS Fellow.

• 1 Homology Groups of a Simplicial Complex1.1 Introduction1.2 Simplices1.3 Simplicial Complexes and Simplicial Maps1.4 Abstract Simplicial Complexes1.5 Review of Abelian Groups1.6 Homology Groups1.7 Homology Groups of Surfaces1.8 Zero-Dimensional Homology1.9 The Homology of a Cone1.10 Relative Homology1.11 ∗Homology with Arbitrary Coefficients1.12 ∗The Computability of Homology Groups1.13 Homomorphisms Induced by Simplicial Maps1.14 Chain Complexes and Acyclic Carriers2 Topological Invariance of the Homology Groups2.1 Introduction2.2 Simplicial Approximations2.3 Barycentric Subdivision2.4 The Simplicial Approximation Theorem2.5 The Algebra of Subdivision2.6 Topological Invariance of the Homology Groups2.7 Homomorphisms Induced by Homotopic Maps2.8 Review of Quotient Spaces2.9 ∗Application: Maps of Spheres2.10 ∗The Lefschetz Fixed Point Theorem3 Relative Homology and the Eilenberg–Steenrod Axioms3.1 Introduction3.2 The Exact Homology Sequence3.3 The Zig-Zag Lemma3.4 The Mayer–Vietoris Sequence3.5 The Eilenberg–Steenrod Axioms3.6 The Axioms for Simplicial Theory3.7 ∗Categories and Functors4 Singular Homology Theory4.1 Introduction4.2 The Singular Homology Groups4.3 The Axioms for Singular Theory4.4 Excisionin Singular Homology4.5 ∗Acyclic Models4.6 Mayer–Vietoris Sequences4.7 The Isomorphism Between Simplicial and Singular Homology4.8 ∗Application: Local Homology Groups and Manifolds4.9 ∗Application: The Jordan Curve Theorem4.10 The Fundamental Group4.11 More on Quotient Spaces4.12 CW Complexes4.13 The Homology of CW Complexes4.14 ∗Application: Projective Spaces and Lens Spaces5 Cohomology5.1 Introduction5.2 The Hom Functor5.3 Simplicial Cohomology Groups5.4 Relative Cohomology5.5 Cohomology Theory5.6 The Cohomology of Free Chain Complexes5.7 ∗Chain Equivalences in Free Chain Complexes5.8 The Cohomology of CW Complexes5.9 Cup Products5.10 Cohomology Rings of Surfaces6 Homology with Coefficients6.1 Introduction6.2 Tensor Products6.3 Homology with Arbitrary Coefficients7 Homological Algebra7.1 Introduction7.2 The Ext Functor7.3 The Universal Coefficient Theorem7.4 Torsion Products7.5 The Universal Coefficient Theorem for Homology7.6 ∗Other Universal Coefficient Theorems7.7 Tensor Products of Chain Complexes7.8 The Künneth Theorem7.9 TheEilenberg–Zilber Theorem7.10 ∗The Künneth Theorem for Cohomolgy7.11 ∗Application: The Cohomology Ring of a Product Space8 Duality in Manifolds8.1 Introduction8.2 The Join of Two Complexes8.3 Homology Manifolds8.4 The Dual Block Complex8.5 Poincaré Duality8.6 Cap Products8.7 A Second Proof of Poincaré Duality8.8 ∗Application: Cohomology Rings of Manifolds8.9 ∗Application: Homotopy Classification of Lens Spaces8.10 Lefschetz Duality8.11 Alexander Duality8.12 Natural Versions of Duality8.13 Čech Cohomology8.14 Alexander–Pontryagin Duality

James R. Munkres served on the MIT Mathematics Faculty from 1960-2000, and continues as Senior Lecturer. He received the PhD in Mathematics from the University of Michigan under the supervision of Edwin Moise in 1956. Professor Munkres is a differential topologist and is also responsible for the Munkres assignment algorithm. He authored numerous texts. His distinctions include the MIT School of Science Teaching Prize for Undergraduate Education in 1984, and an Honorary Doctorate from Nebraska Wesleyan University in 1989.Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He received his Ph.D. from Princeton University and since then has taught at UCLA, Princeton University, and Pennsylvania State University. Dr. Krantz has written over 175 scholarly papers and more than 65 books. He is the founding editor of the Journal of Geometric Analysis. He was named a fellow of the American Mathematical Society and has received the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize.Harold R. Parks received his Ph.D. in mathematics from Princeton University. He was a J. D. Tamarkin Instructor at Brown University. He then assumed a tenure-track position at Oregon State University. He was promoted to Professor in 1989. He spent the academic year 1982-83 at Indiana University as a Visiting Associate Professor. During his time in the Mathematics Department of Oregon State, he served at various times as Assistant Department Chair, Associate Department Chair, and Department Chair. He has written 8 books and 42 scholarly papers. He edits 2 journals. He is an AMS Fellow.

• 1 Homology Groups of a Simplicial Complex1.1 Introduction1.2 Simplices1.3 Simplicial Complexes and Simplicial Maps1.4 Abstract Simplicial Complexes1.5 Review of Abelian Groups1.6 Homology Groups1.7 Homology Groups of Surfaces1.8 Zero-Dimensional Homology1.9 The Homology of a Cone1.10 Relative Homology1.11 ∗Homology with Arbitrary Coefficients1.12 ∗The Computability of Homology Groups1.13 Homomorphisms Induced by Simplicial Maps1.14 Chain Complexes and Acyclic Carriers2 Topological Invariance of the Homology Groups2.1 Introduction2.2 Simplicial Approximations2.3 Barycentric Subdivision2.4 The Simplicial Approximation Theorem2.5 The Algebra of Subdivision2.6 Topological Invariance of the Homology Groups2.7 Homomorphisms Induced by Homotopic Maps2.8 Review of Quotient Spaces2.9 ∗Application: Maps of Spheres2.10 ∗The Lefschetz Fixed Point Theorem3 Relative Homology and the Eilenberg–Steenrod Axioms3.1 Introduction3.2 The Exact Homology Sequence3.3 The Zig-Zag Lemma3.4 The Mayer–Vietoris Sequence3.5 The Eilenberg–Steenrod Axioms3.6 The Axioms for Simplicial Theory3.7 ∗Categories and Functors4 Singular Homology Theory4.1 Introduction4.2 The Singular Homology Groups4.3 The Axioms for Singular Theory4.4 Excisionin Singular Homology4.5 ∗Acyclic Models4.6 Mayer–Vietoris Sequences4.7 The Isomorphism Between Simplicial and Singular Homology4.8 ∗Application: Local Homology Groups and Manifolds4.9 ∗Application: The Jordan Curve Theorem4.10 The Fundamental Group4.11 More on Quotient Spaces4.12 CW Complexes4.13 The Homology of CW Complexes4.14 ∗Application: Projective Spaces and Lens Spaces5 Cohomology5.1 Introduction5.2 The Hom Functor5.3 Simplicial Cohomology Groups5.4 Relative Cohomology5.5 Cohomology Theory5.6 The Cohomology of Free Chain Complexes5.7 ∗Chain Equivalences in Free Chain Complexes5.8 The Cohomology of CW Complexes5.9 Cup Products5.10 Cohomology Rings of Surfaces6 Homology with Coefficients6.1 Introduction6.2 Tensor Products6.3 Homology with Arbitrary Coefficients7 Homological Algebra7.1 Introduction7.2 The Ext Functor7.3 The Universal Coefficient Theorem7.4 Torsion Products7.5 The Universal Coefficient Theorem for Homology7.6 ∗Other Universal Coefficient Theorems7.7 Tensor Products of Chain Complexes7.8 The Künneth Theorem7.9 TheEilenberg–Zilber Theorem7.10 ∗The Künneth Theorem for Cohomolgy7.11 ∗Application: The Cohomology Ring of a Product Space8 Duality in Manifolds8.1 Introduction8.2 The Join of Two Complexes8.3 Homology Manifolds8.4 The Dual Block Complex8.5 Poincaré Duality8.6 Cap Products8.7 A Second Proof of Poincaré Duality8.8 ∗Application: Cohomology Rings of Manifolds8.9 ∗Application: Homotopy Classification of Lens Spaces8.10 Lefschetz Duality8.11 Alexander Duality8.12 Natural Versions of Duality8.13 Čech Cohomology8.14 Alexander–Pontryagin Duality

James R. Munkres served on the MIT Mathematics Faculty from 1960-2000, and continues as Senior Lecturer. He received the PhD in Mathematics from the University of Michigan under the supervision of Edwin Moise in 1956. Professor Munkres is a differential topologist and is also responsible for the Munkres assignment algorithm. He authored numerous texts. His distinctions include the MIT School of Science Teaching Prize for Undergraduate Education in 1984, and an Honorary Doctorate from Nebraska Wesleyan University in 1989.Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He received his Ph.D. from Princeton University and since then has taught at UCLA, Princeton University, and Pennsylvania State University. Dr. Krantz has written over 175 scholarly papers and more than 65 books. He is the founding editor of the Journal of Geometric Analysis. He was named a fellow of the American Mathematical Society and has received the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize.Harold R. Parks received his Ph.D. in mathematics from Princeton University. He was a J. D. Tamarkin Instructor at Brown University. He then assumed a tenure-track position at Oregon State University. He was promoted to Professor in 1989. He spent the academic year 1982-83 at Indiana University as a Visiting Associate Professor. During his time in the Mathematics Department of Oregon State, he served at various times as Assistant Department Chair, Associate Department Chair, and Department Chair. He has written 8 books and 42 scholarly papers. He edits 2 journals. He is an AMS Fellow.

• 1 Homology Groups of a Simplicial Complex1.1 Introduction1.2 Simplices1.3 Simplicial Complexes and Simplicial Maps1.4 Abstract Simplicial Complexes1.5 Review of Abelian Groups1.6 Homology Groups1.7 Homology Groups of Surfaces1.8 Zero-Dimensional Homology1.9 The Homology of a Cone1.10 Relative Homology1.11 ∗Homology with Arbitrary Coefficients1.12 ∗The Computability of Homology Groups1.13 Homomorphisms Induced by Simplicial Maps1.14 Chain Complexes and Acyclic Carriers2 Topological Invariance of the Homology Groups2.1 Introduction2.2 Simplicial Approximations2.3 Barycentric Subdivision2.4 The Simplicial Approximation Theorem2.5 The Algebra of Subdivision2.6 Topological Invariance of the Homology Groups2.7 Homomorphisms Induced by Homotopic Maps2.8 Review of Quotient Spaces2.9 ∗Application: Maps of Spheres2.10 ∗The Lefschetz Fixed Point Theorem3 Relative Homology and the Eilenberg–Steenrod Axioms3.1 Introduction3.2 The Exact Homology Sequence3.3 The Zig-Zag Lemma3.4 The Mayer–Vietoris Sequence3.5 The Eilenberg–Steenrod Axioms3.6 The Axioms for Simplicial Theory3.7 ∗Categories and Functors4 Singular Homology Theory4.1 Introduction4.2 The Singular Homology Groups4.3 The Axioms for Singular Theory4.4 Excisionin Singular Homology4.5 ∗Acyclic Models4.6 Mayer–Vietoris Sequences4.7 The Isomorphism Between Simplicial and Singular Homology4.8 ∗Application: Local Homology Groups and Manifolds4.9 ∗Application: The Jordan Curve Theorem4.10 The Fundamental Group4.11 More on Quotient Spaces4.12 CW Complexes4.13 The Homology of CW Complexes4.14 ∗Application: Projective Spaces and Lens Spaces5 Cohomology5.1 Introduction5.2 The Hom Functor5.3 Simplicial Cohomology Groups5.4 Relative Cohomology5.5 Cohomology Theory5.6 The Cohomology of Free Chain Complexes5.7 ∗Chain Equivalences in Free Chain Complexes5.8 The Cohomology of CW Complexes5.9 Cup Products5.10 Cohomology Rings of Surfaces6 Homology with Coefficients6.1 Introduction6.2 Tensor Products6.3 Homology with Arbitrary Coefficients7 Homological Algebra7.1 Introduction7.2 The Ext Functor7.3 The Universal Coefficient Theorem7.4 Torsion Products7.5 The Universal Coefficient Theorem for Homology7.6 ∗Other Universal Coefficient Theorems7.7 Tensor Products of Chain Complexes7.8 The Künneth Theorem7.9 TheEilenberg–Zilber Theorem7.10 ∗The Künneth Theorem for Cohomolgy7.11 ∗Application: The Cohomology Ring of a Product Space8 Duality in Manifolds8.1 Introduction8.2 The Join of Two Complexes8.3 Homology Manifolds8.4 The Dual Block Complex8.5 Poincaré Duality8.6 Cap Products8.7 A Second Proof of Poincaré Duality8.8 ∗Application: Cohomology Rings of Manifolds8.9 ∗Application: Homotopy Classification of Lens Spaces8.10 Lefschetz Duality8.11 Alexander Duality8.12 Natural Versions of Duality8.13 Čech Cohomology8.14 Alexander–Pontryagin Duality