Topological geometry / Ian R. Porteous

This book represents an energetic and determined effort to present to serious undergraduates the essentials of geometric mathematics. As such it provides a sound background for graduate courses in algebraic and differential topology, differential geometry, and algebraic geometry.
About the first third of the book is devoted to material which has become standard in a term course in algebra: Sets and maps; the natural numbers; groups, rings, ordered rings; the integers; the fields of rational, real, and complex numbers. Linear spaces and maps; subspaces, quotients, products and direct sums; spaces of linear maps; exact sequences. Finite-dimensional spaces; bases; matrices; determinants. Elementary geometry emerges in the course of the development: Affine spaces, hyperplanes, orientation, Grassmannians.
The middle third of the book could serve well for a term's course in geometric algebra: Euclidean spaces; adjoint maps; quadratic forms. Spheres. Quaternions and their geometry. Correlations; correlated spaces and their classification. The classical Lie groups; quadrics (as homogeneous spaces). Clifford algebras and their geometry; spinor groups. The Cayley algebra.
The final third (again, material appropriate for a term's course) presents the first steps in the differential calculus in Euclidean (and Banach) spaces, and on manifolds. Normed linear spaces; bounded linear maps and their properties. Topological spaces and continuous maps; Hausdorff spaces; product spaces; compactness; connectedness. Topological groups; homogeneous spaces. Manifolds. Differentiable maps (real and complex) in Banach spaces; composition formulas. The inverse and implicit function theorems; applications. Smooth manifolds and maps; tangent bundles. Smooth embeddings, with examples. Lie groups and Lie algebras. These final sections successfully draw together various strands that have been developed throughout the text.
Surely this book is the product of substantial thought and care, both from the standpoints of consistent mathematical presentation and of students' pedagogical requirements. In particular, everyone should be pleased by the liberal supply of exercises and examples in every chapter. (MathSciNet)

Bibliogr. p. 435-437. Index