Topological Geometry

CONTENTS FOREWORD 7 Acknowledgments; references and symbols CHAPTER 0 GUIDE 1 CHAPTER 1 MAPS 4 Membership; maps; subsets and quotients; forwards and back- wards; pairs; equivalences; products on a set; union and intersection; natural numbers; products on ω;Σ and Π; order properties of ω CHAPTER 2 REAL AND COMPLEX NUMBERS 26 Groups; rings; polynomials; ordered rings; absolute value; the ring of integers; fields; the rational field; bounded subsets; the >-> notation; the real field; convergence; the complex field; the exponential maps CHAPTER 3 LINEAR SPACES 53 Linear spaces; linear maps; linear sections; linear subspaces; linear injections and surjections; linear products; linear spaces of linear maps; bilinear maps; algebras; matrices; the algebras 8K; one-sided ideals; modules CHAPTER 4 AFFINE SPACES 74 Affine spaces; translations; affine maps; affine subspaces; affine subspaces of a linear space; lines in an affine space; convexity; affine products; comment CHAPTER 5 QUOTIENT STRUCTURES 85 Linear quotients; quotient groups; ideals; exact sequences; diagram-chasing; the dual of an exact sequence; more diagram-chasing; sections of a linear surjection; analogues for group maps; orbits CHAPTER 6 FINITE-DIMENSIONAL SPACES 100 Linear dependence; the basis theorem; rank; matrices; finite-dimensional algebras; minimal left ideals CHAPTER 7 DETERMINANTS 116 Frames; elementary basic framings; permutations of n; the determinant; transposition; determinants of endomorphisms; the absolute determinant; applications; the sides of a hyperplane; orientation CHAPTER 8 DIRECT SUM 132 Direct sum; 2K-modules and maps; linear complements; complements and quotients; spaces of linear complements; Grassmannians CHAPTER 9 ORTHOGONAL SPACES 146 Real orthogonal spaces; invertible elements; linear correlations; non-degenerate spaces; orthogonal maps; adjoints; examples of adjoints; orthogonal annihilators; the basis theorem; reflections; signature; Witt decompositions; neutral spaces; positive—definite spaces; euclidean spaces; spheres; complex orthogonal spaces CHAPTER 10 QUATERNIONS 174 The algebra H; automorphisms and anti-automorphisms of H; rotations of R4; linear spaces over H; tensor product of algebras; automorphisms and anti-automorphisms of 8K CHAPTER 11 CORRELATIONS 198 Semi-linear maps; correlations; equivalent correlations; algebra anti-involutions; correlated spaces; detailed classification theorems; positive-definite spaces; particular adjoint anti-involutions; groups of correlated automorphisms CHAPTER 12 QUADRIC GRASSMANNIANS 223 Grassmannians; quadric Grassmannians; affine quadrics; real affine quadrics; charts on quadric Grassmannians; Grassmannians as coset spaces; quadric Grassmannians as coset spaces; Cayley charts; Grassmannians as quadric Grassmannians; further coset space representations CHAPTER 13 CLIFFORD ALGEBRAS 240 Orthonormal subsets; the dimension of a Clifford algebra; universal Clifford algebras; construction of the algebras; complex Clifford algebras; involuted fields; involutions and anti-involutions; the Clifford group; the uses of conjugation; the map N; the Pfaffian chart; Spin groups; The Radon—Hurwitz numbers CHAPTER 14 THE CAYLEY ALGEBRA 277 Real division algebras; alternative division algebras; the Cayley algebra; Hamilton triangles; Cayley triangles; further results; the Cayley projective line and plane CHAPTER 15 NORMED LINEAR SPACES 288 Norms; open and closed balls; open and closed sets; continuity; complete normed affine spaces; equivalence of norms; the norm of a continuous linear map; continuous bilinear maps; inversion CHAPTER 16 TOPOLOGICAL SPACES 311 Topologies; continuity; subspaces and quotient spaces; closed sets; limits; covers; compact spaces; Hausdorff spaces; open, closed and compact maps; product topology; connectedness CHAPTER 17 TOPOLOGICAL GROUPS AND MANIFOLDS 336 Topological groups; homogeneous spaces; topological manifolds; Grassmannians; quadric Grassmannians; invariance of domain CHAPTER 18 AFFINE APPROXIMATION 353 Tangency; differentiable maps; complex differentiable maps; properties of differentials; singularities of a map CHAPTER 19 THE INVERSE FUNCTION THEOREM 375 The increment formula; the inverse function theorem; the implicit function theorem; smooth subsets; local maxima and minima; the rank theorem; the fundamental theorem of algebra; higher differentials CHAPTER 20 SMOOTH MANIFOLDS 399 Smooth manifolds and maps; submanifolds and products of manifolds; dimension; tangent bundles and maps; particular tangent spaces; smooth embeddings and projections; embeddings of projective planes; tangent vector fields; Lie groups; Lie algebras BIBLIOGRAPHY 435 LIST OF SYMBOLS 439 INDEX 447