Topology and Geometry of Intersections of Ellipsoids in R^n

Preface
Contents
Chapter 1 Introduction
1.1 Ellipsoids and Hyperboloids
1.2 Intersections of Two Concentric Ellipsoids
1.3 Intersections of Two Concentric Hyperboloids
1.4 Plan of the Book
Part I General Results
Chapter 2 General Intersections of Quadrics
2.1 Characterization of General Transverse Compact Intersections of Quadrics
2.2 Characterization of General Transverse Intersections of Ellipsoids
2.3 Quadratic Mappings
2.4 Types of Intersections, Their Symmetries and Operations
2.4.1 Intersections of concentric quadrics
2.4.2 Intersections of concentric ellipsoids
2.4.3 Universality of open half intersections of concentric ellipsoids
2.4.4 Intersections of partially coaxial quadrics
2.4.5 Intersections of coaxial quadrics
2.4.6 Intersections of coaxial ellipsoids
2.4.7 Moment-angle manifolds
Chapter 3 General Operations on Intersections of Quadrics
3.1 The Book Construction
3.2 Adding Squares and Operation Ỹ
3.2.1 Adding a real square
3.2.2 Adding a complex square
3.2.3 Operation Ỹ
Chapter 4 Intersections of Coaxial Quadrics
4.1 Symmetry and the Polyhedral Set
4.2 General Properties
4.2.1 Non-emptiness
4.2.2 Compactness
4.2.3 Transversality
4.2.4 Transversality at infinity
4.2.5 Transversality up to infinity
4.2.6 Connectedness
4.2.7 Simple connectedness
4.2.8 Higher connectedness
4.3 Polyhedral Sets and Polytopes. Realization and Operations
4.3.1 Realization
4.4 Truncation
4.4.1 Truncating faces
4.4.2 Truncating vertices
4.4.3 Truncating simplicial faces
4.4.4 Doubles, open books and connected sums
4.4.5 Combining truncations with the book construction
Chapter 5 Intersections of Coaxial Ellipsoids
5.1 General Properties of Intersections of Coaxial Ellipsoids
5.1.1 Non-emptiness
5.1.2 Transversality
5.1.3 Connectedness
5.1.4 Higher connectedness
5.1.5 General properties of moment-angle manifolds
5.2 Primitive Configurations and Multiplicities
5.2.1 Some small configurations
5.3 Truncation of Transverse Intersections of Coaxial Ellipsoids
5.3.1 Some simple vertex truncations
5.3.2 Truncating faces of a product of simplices
5.3.3 A deeper cut
5.4 The Dual Polytope
5.5 Singular Intersections
5.5.1 Cones and their smoothings
5.5.2 Some singular intersections
5.5.3 The link of an isolated singularity
5.5.4 Codimension one singularities
5.5.5 Smoothings and wall-crossing
5.6 The Homology Splitting
5.7 Examples of Homology Computations
5.7.1 Examples of homology computations of singular intersections
5.8 Dualities
5.9 The Sphere and Singularity Theorems
Conclusion
Part II Topological Description of Transverse
Intersections of Concentric Ellipsoids
Chapter 6 Characterization of Connected Sums
Chapter 7 Three Coaxial Ellipsoids
Three Coaxial Ellipsoids
7.1 Main Theorem 7.1
7.2 The Homology for Three Coaxial Ellipsoids
7.3 Proof of the Main Theorem 7.1
7.4 Parallelizability and Euler Characteristic
7.5 Halves
7.5.1 The space
7.5.2 The topology of Z+
7.6 Transverse Intersections of Two Coaxial Hyperboloids
Chapter 8 Three Concentric Ellipsoids
Three Concentric Ellipsoids
8.1 The Normal Form
8.1.1 Normal form of two complex homogeneous quadrics
8.1.2 Linear normal form of two homogeneous real quadrics
8.1.3 Topological normal form of three transverse concentric ellipsoids
8.2 The Main Theorem 8.5
8.3 Preservation of Connected Sums
8.4 Homology
8.4.1 Preservation of the total homology
8.4.2 Computation of the homology
8.5 Proof of the Main Theorem 8.5
Chapter 9 More Than Three Coaxial Ellipsoids
9.1 Dual-Neighborly Polytopes
9.2 The Topology of the Associated Intersection of Coaxial Ellipsoids
9.2.1 The Euler characteristic ?(?(?)) for dual-neighborly polytopes ? of even dimension
9.2.2 The result of other operations
9.2.3 On the sequences of genera
Chapter 10 A Family of Surfaces That Are Intersections of Concentric, Non-Coaxial Ellipsoids
10.1 Actions of 2-Groups on Surfaces with Quotient a Polygon
10.2 The Construction
10.3 Proof of Theorem 10.1 of the Previous Section
Part III Relations With Other Areas of Mathematics
Chapter 11 Dynamical Systems
11.1 Real Linear Dynamical Systems
11.2 Linear Complex Dynamical Systems
11.3 Generalized Hopf bifurcations
11.4 Generalized May–Leonard Systems
Chapter 12 Complex Geometry
12.1 The Main Classical Examples
12.2 Deformations of the Main Classical Examples
12.3 The LVM-manifolds
12.4 Deformations
12.5 Examples
Chapter 13 Contact and Symplectic Geometry
13.1 All Odd-Dimensional Moment-Angle Manifolds Admit Contact Structures
13.2 Large Families of Odd-Dimensional Coaxial Intersections of Ellipsoids Admit Contact Structures
13.3 A Family of Odd-Dimensional Concentric Intersections of Ellipsoids That Admit Contact Structures
13.4 Intersections of Ellipsoids as Lagrangian Submanifolds
Chapter 14 Intersections with Dihedral Symmetry
14.1 Jacobi Formula for the Co-Rank
14.2 Minors of the Vandermonde ?? on the ?-th Roots of Unity
14.2.1 First results
14.2.2 The complementarity theorem
14.2.3 The case ? = ? prime. Chebotaryov’s Theorem
14.2.4 Some cases where ? is a prime power, ? = ?? , ? odd
14.2.5 The Murty–Whang criterion
14.3 Some Complex Varieties With Cyclic and Dihedral Symmetry
14.3.1 Some complex varieties with cyclic symmetry
14.3.2 Some complex varieties with dihedral symmetry
14.4 Intersections of Real Varieties With Dihedral Symmetry
Chapter 15 Polyhedral Products
Part IV Appendices
Appendix A Proof of Theorem 2.1
Appendix B Origins
B.1 From Singularity Theory...
B.2 Dynamical Systems
B.3 ...to the Polyhedral Product Functor
B.3.1 Coxeter groups, small covers and toric manifolds
B.3.2 The polyhedral product functor
Final remarks
Appendix C Complements of Products of Spheres in Spheres
Appendix D Diagonalizability of Matrices
D.1 Generalities
D.2 When the Field is R or C
D.3 Simultaneous Diagonalizability and Commutation
D.4 An Algorithm for Simultaneous Diagonalizability
D.5 Some Algebraic Mappings Between Spaces of Matrices
References
Index