The Geometric Hopf Invariant and Surgery Theory

Description

Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds. Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists. Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. Front Matter ....Pages i-xvi The Difference Construction (Michael Crabb, Andrew Ranicki)....Pages 1-15 Umkehr Maps and Inner Product Spaces (Michael Crabb, Andrew Ranicki)....Pages 17-38 Stable Homotopy Theory (Michael Crabb, Andrew Ranicki)....Pages 39-75 \({\pmb {\mathbb {Z}}}_2\)-Equivariant Homotopy and Bordism Theory (Michael Crabb, Andrew Ranicki)....Pages 77-125 The Geometric Hopf Invariant (Michael Crabb, Andrew Ranicki)....Pages 127-208 The Double Point Theorem (Michael Crabb, Andrew Ranicki)....Pages 209-295 The \(\pi \)-Equivariant Geometric Hopf Invariant (Michael Crabb, Andrew Ranicki)....Pages 297-304 Surgery Obstruction Theory (Michael Crabb, Andrew Ranicki)....Pages 305-327 Back Matter ....Pages 329-397