CYCLIC HOMOLOGY


Jean-Louis LODAY



2nd edition Grundlehren der Mathematischen Wissenschaften, 301. Springer-Verlag, Berlin, 1998. xviii+513 pp.

Contents

Introduction
1. Hochschild homology
2. Cyclic homology of algebras
3. Smooth algebras and other examples
4. Operations on Hochschild and cyclic homology
5. Variations on cyclic homology
6. The cyclic category, Tor and Ext-interpretation
7. Cyclic spaces and S^1-equivariant homology
8. Chern character
9. Classical invariant theory
10. Homology of Lie algebras of matrices
11. Algebraic K-theory
12. Non Commutative Differential Geometry
13. Mac Lane (co)homology (By Jean-Louis Loday and Teimuraz Pirashvili)
Appendices:

Introduction
1. Hochschild homology
0.Chain complexes
1.Hochschild homology
2.The trace map and Morita invariance
3.Derivations, differential forms
4.Non unital algebras and excision
5.Hochschild cohomology, cotrace, duality
6.Simplicial modules
Bibliographical comments

2. Cyclic homology of algebras
1.Definitions of cyclic homology
2.Connes'exact sequence, Morita invariance, excision
3.Differential forms and de Rham cohomology
4.Cyclic cohomology
5.Cyclic modules
6.Non commutative differential forms
Bibliographical comments

3. Smooth algebras and other examples
1.Tensor algebras
2.Symmetric algebras
3.Universal enveloping algebras of Lie algebras
4.Smooth algebras
5.André-Quillen homology
6.Deligne cohomology
Bibliographical comments

4. Operations on Hochschild and cyclic homology
1.Conjugation and derivation
2.Shuffle product in Hochschild homology
3.Cyclic shuffles and Künneth sequence for HC
4.Product, coproduct in cyclic homology
5.l-decomposition for Hochschild homology
6.l-decomposition for cyclic homology
Bibliographical comments

5. Variations on cyclic homology
1.The periodic and negative theories
2.Dihedral and quaternionic homology
3.Differential graded algebras
4.Commutative Differential graded algebras
5.Bivariant cyclic cohomology
6.Topological algebras, entire cyclic cohomology
Bibliographical comments

6. The cyclic category, Tor and Ext-interpretation
1.Connes cyclic category \Delta C and the category \Delta S
2.Tor and Ext interpretation of HH and HC
3.Crossed simplicial groups
4.The category of finite sets and l-decomposition
Bibliographical comments

7. Cyclic spaces and S^1-equivariant homology
1.Cyclic sets and cyclic spaces
2.Cyclic homology and S^1-equivariant homology
3.Examples of cyclic sets and the free loop space
4.Hochschild homology and cyclic homology of group algebras
5.HC– and HCper of cyclic spaces
Bibliographical comments

8. Chern character
1.The classical Chern character ą la Chern-Weil

2.The Grothendiek group K_0
3.The Chern character from K_0 to cyclic homology
4.The Dennis trace map and the generalized Chern character
5.The Bass trace conjecture and the idempotent conjecture
Bibliographical comments

9. Classical invariant theory
1.The fundamental theorems of invariant theory
2.Coinvariant theory and the trace map
3.Cayley-Hamilton and Amitsur-Levitzki formulas
4.Proofs of the fundamental theorems
5.Invariant theory for the orthogonal and symplectic groups
Bibliographical comments

10. Homology of Lie algebras of matrices
1.Homology of Lie algebras
2.Homology of the Lie algebra gl(A)
3.Stability and first obstruction to stability
4.Homology with coefficients in the adjoint representation
5.The symplectic and orthogonal cases
6.Non-commutative homology (or Leibniz homology)
Bibliographical comments

11. Algebraic K-theory
1.The Bass-Whitehead group K_1 and the Milnor group K_2
2.Higher algebraic K-theory
3.Algebraic K-theory and cyclic homology of nilpotent ideals
4.Absolute and relative Chern characters
5.Secondary characteristic classes
Bibliographical comments

12. Non Commutative Differential Geometry
1.Foliations and the Godbillon-Vey invariant
2.Fredholm modules and index theorem
3.Novikov conjecture on higher signatures
4.The K-theoretic analogue of the Novikov conjecture

13. Mac Lane (co)homology (By Jean-Louis Loday and Teimuraz Pirashvili)
1.(Co)homology with coefficinets in additive bimodules
2.Mac Lane (co)homology
3. Stable K-theory and Mac Lane homology
4. Calculations
Bibliographical comments

Appendices:
A.Hopf algebras
B.Simplicial
C.Homology of discrete groups and small categories
D.Spectral sequences
E.Smooth algebras by Mariá O. Ronco

References