ZZ/2 - Homotopy Theory London Mathematical Society Lecture Note Series

This account is a study of twofold symmetry in algebraic topology. The author discusses specifically the antipodal involution of a real vector bundle - multiplication by - I in each fibre; doubling and squaring operations; the symmetry of bilinear forms and Hermitian K-theory. In spite of its title, this is not a treatise on equivariant topology; rather it is the language in which to describe the symmetry. Familiarity with the basic concepts of algebraic topology (homotopy, stable homotopy, homology, K-theory, the Pontrjagin?Thom transfer construction) is assumed. Detailed proofs are not given (the expert reader will be able to supply them when necessary) yet nowhere is credibility lost. Thus the approach is elementary enough to provide an introduction to the subject suitable for graduate students although research workers will find here much of interest.

Acknowledgements; 1. Introduction; 2. The Euler class and obstruction theory; 3. Spherical fibrations; 4. Stable cohomotopy; 5. Framed manifolds; A. Appendix: on the Hopf variant; 6. K-theory; 7. The image of J; 8. The Euler characteristic; 9. Topological Hermitian K-theory; 10. Algebraic Hermitian K-theory; B. Appendix: on the Hermitian J-homomorphism; Bibliography; Index.