دانلود کتاب Potential Theory and Geometry on Lie Groups (New Mathematical Monographs, Series Number 38)

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Contents
Preface
1 Introduction
1.1 Distance and Volume Growth
1.2 A Classification of Unimodular Locally Compact Groups
1.3 Lie Groups
1.3.1 Convolution powers of measures
1.3.2 The heat diffusion semigroup
1.4 Geometric B–NB Classification of Lie Groups. An Example
1.4.1 Invariant Riemannian structures on G and quasi-isometries
1.4.2 An important example
1.4.3 Isoperimetric inequalities
1.4.4 The polynomial filling property
1.5 A Special Class of Groups and the Metric Classification
1.5.1 The geometric classification for models
1.5.2 The coarse quasi-isometries
1.5.3 Coarse quasi-isometric models
1.5.4 A general connected Lie group
1.5.5 The general metric B–NB classification
1.5.6 The drawback of this metric classification
1.6 Homotopy Retracts
1.6.1 The classical retract
1.6.2 The polynomial retract
1.6.3 The polynomial retract property used in the B–NB classification
1.6.4 The investment/return ratio; or what it takes to prove the (B–NB; Ht) theorem
1.7 Homology on Lie Groups
1.7.1 The de Rham complex
1.7.2 The case of a Lie group
1.7.3 The homological investment/return ratio [sic]
1.8 Cech Cohomology on a Lie Group
1.8.1 A good cover of a Lie group
1.8.2 The Cech complex
1.8.3 The polynomial complex
1.9 The Role of the Algebra in the B–NB Classification
1.10 A Broader Overview and Suggestions for the Reader
PART I ANALYTIC AND ALGEBRAIC CLASSIFICATION
2 The Classification and the First Main Theorem
Part 2.1: Algebraic Definitions and Convolutions of Measures
2.1 Soluble Algebras and Their Roots. The Levi Decomposition
2.1.1 The nilradical
2.1.2 The radical and the Levi decomposition
2.2 The Classification
2.2.1 Soluble algebras
2.2.2 Amenability and the R-condition
2.3 Equivalent Formulations of the Classification and Examples
2.3.1 Affine geometry
2.3.2 Examples
2.3.3 The use of Lie’s theorem
2.3.4 The composite roots
2.3.5 An illustration: the modular function
2.4 Measures on Locally Compact Groups and the C-Theorem
2.4.1 A class of measures
2.5 Preliminary Facts
2.5.1 The Harnack principle for convolution
2.5.2 Applications of Harnack
2.5.3 A technical reduction
2.5.4 Unimodular groups
2.6 Structure Theorems for Lie Groups and the Exact Sequence
2.6.1 The use of structure theory
2.6.2 Special case of the C-theorem
2.6.3 The C-condition on the exact sequence
2.7 Notation, Heuristics and Disintegration of Measures
2.7.1 Probabilistic language
2.7.2 The disintegration of measures, and notation
2.8 Special Properties of the Convolutions in H
2.9 The Reduction to the Random Walk Estimate
2.10 The Random Walk and Proof of the Theorem When G/H =∼ Rd
2.11 The Random Walk and Proof of the C-Theorem in the General Case
2.11.1 The idea of the proof
2.11.2 The proof
2.11.3 An exercise in Lie groups
Part 2.2: The Heat Diffusion Kernel and Gaussian Measures
2.12 The Heat Diffusion Semigroup
2.12.1 Heat diffusion kernel and the Harnack principle
2.12.2 Gaussian measures
2.13 The C-Theorem
2.13.1 The Gaussian C-theorem
2.14 Distances on a Group and the Geometry of Gaussian Measures
2.14.1 Statements of the facts
2.14.2 The distance distortion
2.14.3 The volume growth
2.14.4 The Gaussian estimates for the projected measure μˇ
2.15 The Disintegration of Gaussian Measures
2.16 The Gaussian Random Walk on G/H and Proof of the C-Theorem
2A Appendix: Probabilistic Estimates
2A.1 The sampling for bounded variables
2A.2 A variant in the sampling
2A.3 Gaussian variables and the C-condition
3 NC-Groups
Part 3.1: The Heart of the Matter
3.1 Amenability
3.1.1 Preliminaries
3.1.2 Definition
3.1.3 Remarks
3.1.4 Alternative definition of amenability
3.1.5 Lie groups
3.1.6 Quotients by amenable subgroups
3.2 The NC-Theorem. A Reduction and Examples
3.2.1 The NC-theorem
3.2.2 A reduction
3.2.3 Examples and a special class of groups
3.3 The Principle of the Proof, an Example and the Plan
3.3.1 Convolutions of the kernel
3.3.2 The probabilistic interpretation
3.3.3 General criterion
3.3.4 Illustration of the criterion in a special group
3.3.5 The gambler’s ruin estimate
3.3.6 The gambler’s ruin estimate in a conical domain
3.3.7 Plan of the proof
3.4 The Structure Theorem and Cartan Subgroups
3.4.1 An example: algebraic groups
3.4.2 Cartan subalgebras
3.4.3 A reduction
3.4.4 Proof of the Lie algebras lemma
3.4.5 A lemma in linear algebra
3.5 Proof of the NC-Theorem
3.5.1 The vector space, the roots and the conical domain
3.5.2 Condition §3.3.3(ii) and the conclusion of the proof
Part 3.2: Heat Diffusion Kernel
3.6 Statement of the Results and the Tools
3.6.1 The lifting of the semigroup
3.6.2 The gambler’s ruin estimate
3.7 Proof of the NC-Theorem for the Heat Diffusion Kernel
3.7.1 Condition (ii) of the criterion
Part 3.3: An Alternative Approach
3.8 Algebraic Considerations
3.8.1 A structure theorem
3.8.2 The Lie algebra is soluble
3.8.3 General NC-algebras
3.8.4 The Lie groups and the Ad-mapping
3.9 An Alternative Proof of the NC-Theorem
3.9.1 Products of triangular matrices
3.9.2 Proof for soluble groups
3.9.3 General amenable NC-groups
3A Appendix: The Gambler’s Ruin Estimate
3A.1 One-dimensional case
3A.2 The Markov chain in V = Rd
3A.3 Normal variables
3A.4 Proof of (3.33) for normal variables
3A.5 The construction of the subharmonic function
3A.6 Proof of (3.13)
4 The B–NB Classification
Part 4.1: The Basic Theorem
4.1 The Lie Algebras
4.2 Statement of the Results
4.3 An Overview of the Proofs
4.3.1 Special class of connected Lie groups and the principal bundles
4.4 Left-Invariant Operators on an R-Principal Bundle
4.4.1 The formal definition
4.4.2 The coordinate representation of the operators
4.4.3 Measures, adjoints and Lp-norms
4.4.4 The amenability of R
4.4.5 Convolution operators on a group
4.4.6 The Haar measure on G = RK
4.5 Proof of the B-Theorem 4.6
4.5.1 Theorem 4.6 in the principal bundle
4.5.2 Proof of a weaker form of Theorem 4.6 for the special groups of §4.3.1
4.5.3 Proof of Lemma 4.8
4.6 Structure Theorems; Reduction to the Special Class of Groups
4.6.1 The use of Schur’s lemma
4.6.2 A reduction
4.6.3 An alternative approach to general groups
Part 4.2: The Heat Diffusion Kernel and Gaussian Measures
4.7 Gaussian Left-Invariant Operators
4.8 The Gaussian B-Theorem
4.9 Proof of the Gaussian B-Theorem
4.10 An Explicit Formula and the Proof of Lemma 4.12
5 NB-Groups
Part 5.1
Overview of Part 5.1
5.1 The First Eigenfunction and the Sharp B-Theorem 4.6
5.1.1 Proof of (5.2)
5.2 Proof of the Sharp B-Theorem 4.6
5.3 Symmetric Markovian Operators
5.3.1 Definition and the criterion
5.3.2 A modification of the criterion
5.3.3 The construction of the Markovian operators
5.3.4 Example
5.3.5 General operators
5.3.6 The group case
5.4 Theorem 4.7 for Principal Bundles and the Harnack Estimate
5.5 The Euclidean Bundle
5.5.1 The definition
5.5.2 The conical domain and the gambler’s ruin estimate
5.5.3 Gambler’s ruin estimate
5.6 Proof of Proposition 5.2
5.6.1 Overview of the proof
5.6.2 The Euclidean case (i)
5.6.3 The case R = N ⋌H
5.6.4 The general case (iii)
5.7 Proof of Theorem 4.7
5.7.1 The reduction
5.7.2 The use of positive-definite functions
5.7.3 Proof of the reduction in §5.7.1
5.7.4 Proof of Lemma 5.6
5.8 * The Global Structure of Lie Groups
Part 5.2: The Heat Diffusion Kernel
5.9 Preliminaries and the Reductions
5.10 Gaussian Left-Invariant Operators on Principal Bundles
5.10.1 The Gaussian Euclidean bundle
5.11 Proof of Proposition 5.10
5.11.1 The plan of the proof
5.11.2 Verification of the conditions of Criterion 5.1 on X = R × K,
R = NR ⋌QR
Part 5.3: Proof of the Lower B-Estimate
5.12 Statement of the Results and Plan of the Proof
5.12.1 The Euclidean principal bundle revisited
5.13 Proof of Proposition 5.12. Special Case
5A Appendix: Proof of the Gambler’s Ruin Estimate §5.5.2
5A.1 μ-subharmonic functions
5A.2 Harmonic coordinates on the Euclidean prin- cipal bundle and the gambler’s ruin estimate
5A.3 The change of coordinates on a Euclidean bundle and the correctors
5B Appendix: Proof of (5.74)
5B.1 The Hessian and preliminaries
5B.2 The subharmonic functions
5B.3 Proof of estimate (5.74)
6 Other Classes of Locally Compact Groups
6.1 Connected Locally Compact Groups
6.2 Compact Groups and a Generalisation
6.3 On a Class of Locally Compact Groups
6.4 A Review of Some Results from Algebraic Groups
6.4.1 General definitions
6.4.2 Soluble groups
6.4.3 The commutator subgroup
6.4.4 Nilpotent groups and the exponential mapping
6.5 The C–NC Classification for Solvable Algebraic Groups
6.5.1 The roots
6.5.2 The real roots and their classification
6.5.3 On the definition of the real roots
6.5.4 The real root space decomposition
6.6 Statement of the Theorems
6.6.1 Conditions on the measures and the groups
6.6.2 Statement of the theorems
6.7 Proof of the NC-Theorem
6.8 Proof of the C-Theorem
6.8.1 The construction of the exact sequence
6.9 Final Remarks
Appendix A Semisimple Groups and the Iwasawa Decomposition
A.1 The Levi Decomposition
A.2 Compact Lie Groups
A.3 Non-compact Lie Algebras and the Iwasawa Decom- position
A.4 Uniqueness
A.5 First Step: The Cartan Decomposition and the Choice of k
A.6 Second Step: The Choice of a
A.7 Third Step: The Choice of n
A.8 Uniqueness of the Iwasawa Radical. Borel Subgroups
A.9 The Nilradical of the Iwasawa Radical r
A.10 A Lemma in the Representations of a Semisimple Lie Algebra
Appendix B The Characterisation of NB-Algebras
B.1 Notation
B.2 Further Notation
B.3 Two Lemmas
B.4 Proof of the Lemmas
B.5 The Unimodular Case
B.6 Characterisation of Non-unimodular NB-Algebras
B.7 The Sufficiency of the Condition
B.8 Proof of the Sufficiency
B.9 Additional Remarks on the Sufficiency of the Condition
Appendix C The Structure of NB-Groups
C.1 Simply Connected Groups and Their Centres
C.2 A General NB-Group
C.3 The Quasi-Isometric Modification
Appendix D Invariant Differential Operators and Their Diffusion Kernels
D.1 Definitions and Notation
D.2 The Harnack and the Gaussian Estimates
D.3 Proof of the Gaussian Estimate
D.4 Applications to a Special Class of Operators
D.5 Questions Related to the Lower Gaussian Estimate
Appendix E Additional Results. Alternative Proofs and Prospects
E.1 Small Time Estimates
E.2 General Estimates for the Heat Diffusion Kernel
E.3 Bi-invariant Operators and Symmetric Spaces
E.4 A Fundamentally Different Approach to the B– NB Classification
PART II GEOMETRIC THEORY
7 Geometric Theory. An Introduction
7.1 Basic Definitions and Notation
7.1.1 Manifolds
7.2 Riemannian Structures on Lie Groups
7.2.1 Definitions
7.3 Simply Connected Soluble Groups
7.3.1 Exponential coordinates of the second kind
7.4 Polynomial Homotopy and Geometric Theorems of Soluble Groups
7.4.1 Definitions
7.4.2 Lie groups
7.5 A Polynomial Filling Property
7.5.1 Notation
7.6 Differential Forms
7.6.1 Definitions and notation
7.6.2 The coefficients are smooth: ωI ∈ C∞
7.6.3 The alternative definition
8 The Geometric NC-Theorem
8.1 Differentiation on Lie Groups
8.1.1 A description of the tangent space
8.1.2 The inverse function
8.1.3 The product
8.1.4 Applications
8.2 Strict Exponential Distortion and the Proof of Proposition 8.2
8.2.1 Hyperbolic geometry and ‘heuristics’
8.2.2 Strict exponential distortion
8.2.3 Proof of Proposition 8.2
8.3 Semidirect Products
8.3.1 The definition of the semidirect product
8.3.2 Definition of replicas
8.3.3 The semisimple replica
8.3.4 A class of special soluble groups
8.3.5 Riemannian structures on semidirect products
8.3.6 Quasi-isometric and polynomially equivalent replicas
8.4 Polynomial Sections
8.4.1 Definitions and examples
8.4.2 The strict polynomial section
8.4.3 One-parameter subgroups and notation
8.4.4 Proving that σ is a polynomial section
8.4.5 Proof of the second assertion of (8.62)
8.4.6 Proof that σ is strictly polynomial
8.4.7 A variant of the argument
8.5 Denouement
8.5.1 Proofs of Proposition 8.3 and Theorem 7.10
8.5.2 Comments on the proof of Theorem 7.10
8.5.3 Comments on the definition of a strict section and the polynomial homotopy equivalence
9 Algebra and Geometry on C-Groups
9.1 The Special Soluble Algebras
9.1.1 Algebraic considerations
9.1.2 Bracket-reduced SSA
9.1.3 Combinatorics
9.1.4 A -couples
9.1.5 The A algebras
9.1.6 Proof of the algebraic structure theorem (Theorem 9.7)
9.1.7 The two alternatives
9.2 Geometric Constructions on Special Soluble Groups: Examples
9.2.1 Notation
9.2.2 The special case G2
9.2.3 A first application of the construction
9.2.4 The use of transversality and the filling property
9.2.5 Generalisations and the Heisenberg alternative
9.2.6 The endgame in the Heisenberg case
9.2.7 The five-dimensional example of G3
9.3 The First Basic Construction (I): The Description
9.3.1 An introduction and guide for the reader
9.3.2 Terminology and conventions
9.3.3 The description of the embedded sphere S = Sr−1 ⊂ Gr
9.3.4 The endgame. Whitney regularisation and the F-property
9.3.5 A different strategy
9.4 The First Basic Construction (II): Details and Compu- tations
9.4.1 Notation on the unit cube
9.4.2 The simplicial decomposition of ∂ r
9.4.3 The mappings. General description
9.4.4 The mapping that does the stretching
9.4.5 Affine mappings in conical domains
9.4.6 The choice of the ζI and the transversality condition §9.3.3
9.4.7 The LL(R) property of the mapping f in (9.75)–(9.76)
9.4.8 An alternative description of the construction
9.5 The Second Basic Construction
9.5.1 The general SSG that are C-groups
9.5.2 The new tools. A special case
9.5.3 The construction needed for Proposition 9.34 under Fr for
s = 1
9.5.4 The construction needed for Proposition 9.35 with s 1
9.5.5 The general case s 2
9.5.6 The Heisenberg alternative
9.5.7 A recapitulation
10 The Endgame in the C-Theorem
10.1 An Overview and a Guide for the Reader
10.1.1 Notation and the previous constructions
10.1.2 General soluble simply connected C-groups and their ‘rank’
10.1.3 Coordinates, distances and Riemannian metrics
10.1.4 The transversality condition on the mapping (10.1), (10.4)
10.1.5 The filling function and the key proposition
10.1.6 Deducing the C-theorem (Theorem 7.11)
10.1.7 Guide for the reader
10.1.8 The second approach based on smoothing
10.2 The Use of Stokes’ Theorem
10.2.1 The smooth case
10.2.2 Stokes’ theorem for the general case. The use of currents
10.2.3 The boundary operator on currents
10.2.4 The general proof of Proposition 10.5
10.2.5 Slicing of currents and yet one more proof of Proposition 10.5
10.2.6 Guide to the literature on currents
10.3 The Smoothing of the Mapping f : ∂ d → Q
10.3.1 An overview
10.3.2 Simplexes revisited. Their canonical position in affine space
10.3.3 The tessellation of §9.4.5 revisited
10.3.4 The linearization ‘lemma’ of Ψ near the origin
10.3.5 The linearization of the Key Example 10.12
10.3.6 Smoothing by convolution
10.3.7 Smoothing the second basic construction of §9.5
10.3.8 The smoothing for the Heisenberg alternative
10.3.9 Smoothing of the extension mapping F of §10.1.5
10.4 The Second Proof of Proposition 10.5
10.4.1 The mapping F in §10.1.5 can be made to be an embedding
10.4.2 Use of facts from differential topology
10.4.3 Getting round the constraint (10.87) on the dimensions
11 The Metric Classification
11.1 Definitions and Statement of the Metric Theorems
11.1.1 Definitions of quasi-isometries
11.1.2 The building blocks and the C–NC classifica- tion theorem
11.1.3 The classification theorem for simply con- nected Lie groups
11.1.4 Soluble non-simply-connected groups
11.1.5 One more geometric classification of con- nected Lie groups
11.2 Proof of Theorem 11.12
11.2.1 A reformulation of Theorem 11.12
11.2.2 Proof of Lemma 11.17
11.3 Soluble Connected Groups
11.3.1 The maximal central torus
11.3.2 Nilpotent groups
11.3.3 Tori in connected soluble groups
11.3.4 The role of the fundamental group
11.4 General Groups and Theorems 11.14 and 11.16
11.4.1 Notation and structure theorems
11.4.2 The simply connected covering group
Appendix F Retracts on General NB-Groups (Not Necessarily Simply Connected)
F.1 Introduction
F.2 R-Groups
F.3 Amenable Groups
F.4 Homotopy Retracts for Groups That Are Not Simply Connected
F.5 Proof of the Glueing Lemma
F.5.1 The exponential retract
F.5.2 A special exponential retract
F.5.3 The perturbations and the proof of the glueing lemma
PART III HOMOLOGY THEORY
12 The Homotopy and Homology Classification of Connected Lie Groups
12.1 An Informal Overview of the Chapter and of Part III
12.1.1 A review of what has already been achieved in the geometric classification
12.1.2 The use of the Poincare´ equation
12.1.3 Homology and the Poincare´ equation for general connected Lie groups
12.1.4 The homology on manifolds. The use of currents
12.1.5 Content of Chapter 12
12.2 Definitions and the Main Theorem Related to Homotopy
12.2.1 Homotopies. Homotopic equivalence
12.2.2 Homotopy retracts
12.2.3 Smooth manifolds
12.2.4 Riemannian manifolds and polynomial mappings
12.2.5 Simply connected groups
12.2.6 Retract to a maximal compact subgroup
12.2.7 General connected Lie groups
12.3 A Review of the General Theory of Currents
12.3.1 Scope of §§12.3, 12.4
12.3.2 Even and odd forms
12.3.3 Further notation
12.3.4 An illustration: chains
12.3.5 Currents as forms with distribution coefficients
12.3.6 The differential and the boundary operators
12.3.7 The pullback of forms and the direct image of currents
12.4 Homology. Review of the Definitions of the General Theory
12.4.1 General definitions. Some classical examples
12.4.2 C∞ manifolds and complexes of differential forms
12.4.3 More on singular homology. Connections with algebraic topology
12.4.4 Intermediate spaces and complexes
12.4.5 Examples and further remarks
12.5 The Heart of the Matter. Forms of Polynomial Growth
12.5.1 Riemannian norm on differential forms
12.5.2 Spaces of differential forms on M
12.5.3 The complex of polynomial forms
12.6 Statement of the Homology Theorems
12.6.1 Simply connected groups
12.6.2 General connected Lie groups
12.6.3 The scope and return on investment of Part III
12.7 Banach Spaces of Currents and Their Duals
12.7.1 The total mass norm
12.7.2 Banach spaces of complexes
12.8 Geometric Properties of C(U,pol) and a Review of Currents
12.8.1 Images by polynomial mappings
12.8.2 Polynomial mappings in Lie groups
12.8.3 Double forms and double currents
12.8.4 Application of double currents
12.8.5 The boundary operators on double currents and the ‘produit tensoriel’
12.8.6 Examples of currents in Rn
12.9 The Use of Polynomial Homotopy in the Complexes ΛP(U),ΛP∗(U)
12.9.1 The abstract chain homotopies. The literature
12.9.2 Brief overview of chain homotopy in singular homology
12.9.3 Chain homotopy on E ′. Heuristics
12.9.4 The construction of the chain homotopy on E ′
12.9.5 The chain homotopy on E
12.9.6 Application to the homology theory of Lie groups
12.9.7 The polynomial homotopy and the complexes ΛP, ΛP∗
12.9.8 Applications to the NC-condition
12.10 Regularisation
12.10.1 The setting of the problem
12.10.2 The construction of the regularising operator
12.10.3 Proof of Proposition 12.18
12.11 Duality Theory for Complexes
12.11.1 Notation and definitions
12.11.2 Notation and standard identifications
12.12 The Use of Banach’s Theorem
12.12.1 Banach’s theorem
12.12.2 The diagrams
12.13 The Use of More Sophisticated Topological Vector Spaces
12.13.1 The scope of this section
12.13.2 The natural topologies on ΛP, ΛP∗
12.13.3 An illustration
12.13.4 An unsuccessful but instructive attempt to prove Proposition 12.47 the ‘other way round’
12.13.5 An exercise in topological vector spaces
12.14 The Acyclicity of ΛP∗ and ΛP of §12.7.2
12.14.1 Fr´ echet spaces and their quotients
12.14.2 The acyclicity of ΛP∗
12.15 The Acyclicity of ΛP
12.15.1 Comments on the proof
12.15.2 The diagram and the use of Baire category
12.15.3 Comment on Propositions 12.57 and 12.55
12.16 The Case Where the Homology of ΛP(U) Is Finite- Dimensional
12.17 The Partial Acyclicity of the Complexes
12A Appendix: Acyclicity in Topological Vector Spaces
12A.1 The position of the problem
12A.2 A class of topological vector spaces and ersatz acyclicity
12A.3 The proof
12A.4 The Imb
13 The Polynomial Homology for Simply Connected Soluble Groups
13.1 The Reductions and Notation of Chapters 8–10
13.1.1 The basic reduction
13.1.2 The LL(R) − ∂ r construction in the two alternatives
13.1.3 The second basic construction
13.1.4 The organisation of this chapter
13.1.5 List of special cases
13.2 The Currents Generated by the First Basic Construction
13.2.1 The definition of the currents
13.2.2 The metric properties of the current S
13.3 The Special Case s = 0 and an Acyclic Complex
13.3.1 The contradiction and connections with Chap- ter 10
13.3.2 The properties of the current S
13.3.3 The construction of the differential form and the contradiction
13.3.4 Additional comments and remarks
13.4 Bouquets of Currents
13.4.1 Definition for the Abelian alternative
13.4.2 The Heisenberg alternative
13.4.3 The first illustration of the bouquet of currents
13.5 The Case G = N ⋌(A′ ⊕ A) with A = R and s = 1
13.5.1 Notation
13.5.2 The choice of the index p1
13.5.3 The currents in the second basic construction
13.5.4 The restriction of these currents to the slice
13.5.5 The error term and the estimate of T 1(ID)
13.5.6 The construction of S2
13.5.7 The contradiction and the endgame in the proof
13.5.8 A retrospective examination of the second basic construction
13.6 The Proof of Theorem 12.17 under the Acyclicity Condition in the General Case
13.6.1 The choice of the indices
13.6.2 The Γ-free complex
13.6.3 The acyclicity of ΛP and the mappings S1,S2
13.6.4 The first application of the mapping S2
13.6.5 The mappings Sq; q 1
13.6.6 The end of the proof in the general case
13.7 The Use of Bouquets and the Proof of Theorem 12.17
13.7.1 The construction of S1, S2 for s 1
13.7.2 The construction of S3 and the proof of the theorem when s = 2
13.7.3 The general definition of Sq, 1 q s + 1
13.7.4 The estimate of Ss+1; the principal term; the error term
13.7.5 The endgame and the contradiction
13A Appendix
13A.1 The use of an infinite bouquet
13A.2 The topological homotopy
13A.3 Further comments
14 Cohomology on Lie Groups
14.1 Introduction: Scope and Methods of the Chapter
14.1.1 The de Rham complex revisited
14.1.2 What has been done and what remains to be done
14.1.3 Simply connected groups and the general strategy in the proofs
14.1.4 The pivotal reduction
14.1.5 The methods and the background for the proofs
14.1.6 About the style and the presentation of the chapter
14.2 Notions from Algebraic Topology and Riemannian Geometry
14.2.1 Cohomology attached to a cover
14.2.2 Good covers, notation and fundamental facts
14.2.3 The cohomology presheaf of a bundle
14.2.4 How to construct a good cover
14.2.5 The polynomial ˇ Cech cohomology
14.3 Revisiting Structure Theorems
14.3.1 Soluble groups
14.3.2 The endgame
14.3.3 About 0-distorted discrete subgroups
14.4 Algebraic Tools
14.4.1 A double complex
14.4.2 The two spectral sequences associated with a double complex
14.4.3 Definition and properties of spectral sequences
14.4.4 The limit of the spectral sequence of a double complex
14.4.5 Spectral sequences that degenerate on the second step
14.5 The Cech–de Rham Complex
14.5.1 The double complex
14.5.2 The first spectral sequence
14.5.3 The polynomial ˇ Cech–de Rham complex
14.6 Proof of Proposition 14.2
14.6.1 The de Rham cohomology
14.6.2 The polynomial cohomology
Appendix G Discrete Groups
G.1 Group Action on a Metric Space
G.1.1 The set-up
G.1.2 Covering spaces
G.1.3 Informal description of the problem
G.2 The Group Γ = Z
G.2.1 Elementary complexes
G.2.2 A short digression: homology of the discrete group Γ
G.2.3 Covering spaces with group Γ = Z and the Cartan–Leray spectral sequence
G.3 Discrete Groups
G.3.1 Notation and free resolutions
G.3.2 The B–NB classification for lattices
G.4 Outline of the Proofs
G.4.1 Cech homology
G.4.2 Chains of rapid decay
G.5 A Variation on the Same Theme
G.5.1 The Grothendieck lemma and all that
G.6 Connections, Curvature and Cohomology
Epilogue
References
Index