Algebraic Cobordism (eBook)

Introduction 6
Contents 10
1 Cobordism and oriented cohomology 12
1.1 Oriented cohomology theories 12
1.2 Algebraic cobordism 17
1.3 Relations with complex cobordism 24
2 The definition of algebraic cobordism 27
2.1 Oriented Borel-Moore functors 27
2.2 Oriented functors of geometric type 36
2.3 Some elementary properties 39
2.4 The construction of algebraic cobordism 45
2.5 Some computations in algebraic cobordism 50
3 Fundamental properties of algebraic cobordism 60
3.1 Divisor classes 60
3.2 Localization 64
3.3 Transversality 75
3.4 Homotopy invariance 84
3.5 The projective bundle formula 86
3.6 The extended homotopy property 89
4 Algebraic cobordism and the Lazard ring 92
4.1 Weak homology and Chern classes 92
4.2 Algebraic cobordism and K-theory 114
4.3 The cobordism ring of a point 125
4.4 Degree formulas 131
4.5 Comparison with the Chow groups 145
5 Oriented Borel-Moore homology 150
5.1 Oriented Borel-Moore homology theories 150
5.2 Other oriented theories 158
6 Functoriality 168
6.1 Re.ned cobordism 168
6.2 Intersection with a pseudo-divisor 178
6.3 Intersection with a pseudo-divisor II 187
6.4 A moving lemma 193
6.5 Pull-back for l.c.i. morphisms 201
6.6 Refined pull-back and re.ned intersections 210
7 The universality of algebraic cobordism 223
7.1 Statement of results 223
7.2 Pull-back in Borel-Moore homology theories 225
7.3 Universality 229
7.4 Some applications 233
A Resolution of singularities 240
References 243
Index 245
Glossary of Notation 249