Topological Geometry

FOREWORD
CH 0 GUIDE
CH 1 MAPS
Membership; maps; subsets and quotients; forwards and backwards;
pairs; equivalences; products on a set; union and intersection;
natural numbers; products on ω; ∑; and ∏; order
properties of ω
CH 2 REAL AND COMPLEX NUMBERS
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
CH 3 LINEAR SPACES
Linear spaces; linear maps; linear sections; linear sub spaces;
linear injections and surjections; linear products; linear spaces
of linear maps; bilinear maps; algebras; matrices; the algebras
_sK; one-sided ideals; modules
CH 4 AFFINE SPACES
Affine spaces; translations; affine maps; affine subspaces; affine
subs paces of a linear space; lines in an affine space; convexity;
affine products; comment
CH 5 QUOTIENT STRUCTURES
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
CH 6 FINITE-DIMENSIONAL SPACES
Linear dependence; the basis theorem; rank; matrices; finite dimensional
algebras; minimal left ideals
CH 7 DETERMINANTS
Frames; elementary basic framings; permutations of n; the
determinant; transposition; determinants of endomorphisms;
the absolute determinant; applications; the sides of a hyperplane;
orientation
CH 8 DIRECT SUM
Direct sum; _2K-modules and maps; linear complements; compleme.
nts and quotients; spaces of linear complements; Grassmanmans
CH 9 ORTHOGONAL SPACES
Real orthogonal spaces; invertible elements; linear correlations;
non-degenerate spaces; orthogonal maps; adjoints; examples of
ad joints; orthogonal annihilators; the basis theorem; reflections;
signature; Witt decompositions; neutral spaces; positive-definite
spaces; euclidean spaces; spheres; complex orthogonal spaces
CH 10 QUATERNIONS
The algebra H; automorphisms and anti-automorphisms of H;
rotations of R_4 ; linear spaces over H; tensor product of algebras;
automorphisms and anti-automorphisms of _sK
CH 11 CORRELATIONS
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
CH 12 QUADRIC GRASSMANNIANS
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
CH 13 CLIFFORD ALGEBRAS
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
CH 14 THE CAYLEY ALGEBRA
Real division algebras; alternative division algebras; the Cayley
algebra; Hamilton triangles; Cayley triangles; further results;
the Cayley projective line and plane
CH 15 NORMED LINEAR SPACES
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
CH 16 TOPOLOGICAL SPACES
Topologies; continuity; subspaces and quotient spaces; closed
sets; limits; covers; compact spaces; Hausdorff spaces; open,
closed and compact maps; product topology; connectedness
CH 17 TOPOLOGICAL GROUPS AND MANIFOLDS
Topological groups; homogeneous spaces; topological manifolds;
Grassmannians; quadric Grassmannians; in variance of
domain
CH 18 AFFINE APPROXIMATION
Tangency; differentiable maps; complex differentiable maps;
properties of differentials; singularities of a map
CH 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
CH 20 SMOOTH MANIFOLDS
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
LIST OF SYMBOLS
INDEX