Stable Homotopy and Generalised Homology
9780226005249
Stable Homotopy and Generalised Homology
J. Frank Adams, the founder of stable homotopy theory, gave a lecture series at the University of Chicago in 1967, 1970, and 1971, the well-written notes of which are published in this classic in algebraic topology. The three series focused on Novikov’s work on operations in complex cobordism, Quillen’s work on formal groups and complex cobordism, and stable homotopy and generalized homology. Adams’s exposition of the first two topics played a vital role in setting the stage for modern work on periodicity phenomena in stable homotopy theory. His exposition on the third topic occupies the bulk of the book and gives his definitive treatment of the Adams spectral sequence along with many detailed examples and calculations in KU-theory that help give a feel for the subject.
Table of Contents
Preface
Pt. I: S.P. Novikov’s Work on Operations on Complex Cobordism
2: Cobordism groups
3: Homology
4: The Conner-Floyd Chern classes
5: The Novikov operations
6: The algebra of all operations
7: Scholium on Novikov’s exposition
8: Complex manifolds
Pt. II: Quillen’s Work on Formal Groups and Complex Cobordism
1: Formal groups
2: Examples from algebraic topology
3: Reformulation
4: Calculations in E-homology and cohomology
5: Lazard’s universal ring
6: More calculations in E-homology
7: The structure of Lazard’s universal ring L
8: Quillen’s theorem
9: Corollaries
10: Various formulae in [pi][subscript *](MU)
11: MU[subscript *](MU)
12: Behaviour of the Bott map
13: K[subscript *](K)
14: The Hattori-Stong theorem
15: Quillen’s idempotent cohomology operations
16: The Brown-Peterson spectrum
17: KO[subscript *](KO)
Pt. III: Stable Homotopy and Generalised Homology
2: Spectra
3: Elementary properties of the category of CW-spectra
4: Smash products
5: Spanier-Whitehead duality
6: Homology and cohomology
7: The Atiyah-Hirzebruch spectral sequence
8: The inverse limit and its derived functors
9: Products
10: Duality in manifolds
11: Applications in K-theory
12: The Steenrod algebra and its dual
13: A universal coefficient theorem
14: A category of fractions
15: The Adams spectral sequence
16: Applications to [pi][subscript *](bu[actual symbol not
reproducible]X): modules over K[x, y]
17: Structure of [pi][subscript *](bu[actual symbol not
reproducible]bu)~
Pt. I: S.P. Novikov’s Work on Operations on Complex Cobordism
2: Cobordism groups
3: Homology
4: The Conner-Floyd Chern classes
5: The Novikov operations
6: The algebra of all operations
7: Scholium on Novikov’s exposition
8: Complex manifolds
Pt. II: Quillen’s Work on Formal Groups and Complex Cobordism
1: Formal groups
2: Examples from algebraic topology
3: Reformulation
4: Calculations in E-homology and cohomology
5: Lazard’s universal ring
6: More calculations in E-homology
7: The structure of Lazard’s universal ring L
8: Quillen’s theorem
9: Corollaries
10: Various formulae in [pi][subscript *](MU)
11: MU[subscript *](MU)
12: Behaviour of the Bott map
13: K[subscript *](K)
14: The Hattori-Stong theorem
15: Quillen’s idempotent cohomology operations
16: The Brown-Peterson spectrum
17: KO[subscript *](KO)
Pt. III: Stable Homotopy and Generalised Homology
2: Spectra
3: Elementary properties of the category of CW-spectra
4: Smash products
5: Spanier-Whitehead duality
6: Homology and cohomology
7: The Atiyah-Hirzebruch spectral sequence
8: The inverse limit and its derived functors
9: Products
10: Duality in manifolds
11: Applications in K-theory
12: The Steenrod algebra and its dual
13: A universal coefficient theorem
14: A category of fractions
15: The Adams spectral sequence
16: Applications to [pi][subscript *](bu[actual symbol not
reproducible]X): modules over K[x, y]
17: Structure of [pi][subscript *](bu[actual symbol not
reproducible]bu)~