Differential Algebraic Topology: From Stratifolds to Exotic Spheres (Graduate Studies in Mathematics, 110)

This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincare duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry. Contents......Page 6 INTRODUCTION......Page 10 0. A quick introduction to stratifolds......Page 14 1. A word about structures......Page 18 2. Differential spaces......Page 19 3. Smooth manifolds revisited......Page 21 4. Exercises......Page 24 1. Stratifolds......Page 28 2. Local retractions......Page 31 3. Examples......Page 32 4. Properties of smooth maps......Page 38 5. Consequences of Sard’s Theorem......Page 40 6. Exercises......Page 42 3. Stratifolds with boundary: c-stratifolds......Page 46 1. Exercises......Page 51 1. Motivation of homology......Page 52 2. Z/2-oriented stratifolds......Page 54 3. Regular stratifolds......Page 56 4. Z/2-homology......Page 58 5. Exercises......Page 64 1. The Mayer-Vietoris sequence......Page 68 2. Reduced homology groups and homology groups of spheres......Page 74 3. Exercises......Page 77 1. Brouwer’s fixed point theorem......Page 80 2. A separation theorem......Page 81 3. Invariance of dimension......Page 82 4. Exercises......Page 83 1. The fundamental class......Page 84 2. Z/2-homology of projective spaces......Page 85 3. Betti numbers and the Euler characteristic......Page 87 4. Exercises......Page 90 1. Integral homology groups......Page 92 2. The degree......Page 96 3. Integral homology groups of projective spaces......Page 99 4. A comparison between integral and Z/2-homology......Page 101 5. Exercises......Page 102 1. The axioms of a homology theory......Page 106 2. Comparison of homology theories......Page 107 3. CW-complexes......Page 111 4. Exercises......Page 112 1. The cross product......Page 116 2. The K ̈unneth theorem......Page 120 3. Exercises......Page 122 1. Lens spaces......Page 124 2. Milnor’s 7-dimensional manifolds......Page 128 3. Exercises......Page 130 1. Cohomology groups......Page 132 2. Poincarי duality......Page 134 3. The Mayer-Vietoris sequence......Page 136 4. Exercises......Page 138 1. Transversality for stratifolds......Page 140 2. The induced maps......Page 142 3. The cohomology axioms......Page 145 4. Exercises......Page 146 1. The cross product and the K ̈unneth theorem......Page 148 2. The cup product......Page 150 3. The Kronecker pairing......Page 154 4. Exercises......Page 158 15. The signature......Page 160 1. Exercises......Page 165 1. The Euler class......Page 166 2. Euler classes of some bundles......Page 168 4. Exercises......Page 172 17. Chern classes and Stiefel-Whitney classes......Page 174 1. Exercises......Page 178 1. Pontrjagin classes......Page 180 2. Pontrjagin numbers......Page 183 3. Applications of Pontrjagin numbers to bordism......Page 185 4. Classification of some Milnor manifolds......Page 187 5. Exercises......Page 188 1. The signature theorem and exotic 7-spheres......Page 190 2. The Milnor spheres are homeomorphic to the 7-sphere......Page 194 3. Exercises......Page 197 1. SHk(X) is isomorphic to Hk(X; Z) for CW-complexes......Page 198 2. An example where SHk(X) and Hk(X) are different......Page 200 3. SH.(M) is isomorphic to ordinary singular cohomology......Page 201 4. Exercises......Page 203 1. The product of two stratifolds......Page 204 2. Gluing along part of the boundary......Page 205 3. Proof of Proposition 4.1......Page 207 Appendix B. The detailed proof of the Mayer-Vietoris sequence......Page 210 Appendix C. The tensor product......Page 222 Bibliography......Page 228 Index......Page 230 This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincaré duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry. "This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinarysingular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincar��e duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres."--Publisher's description. "This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincaré duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres."--Publisher's description Offers a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. This title introduces a class of stratified spaces, so-called stratifolds. It derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality.