##
**The geometric Hopf invariant and surgery theory.**
*(English)*
Zbl 1404.55001

Springer Monographs in Mathematics. Cham: Springer (ISBN 978-3-319-71305-2/hbk; 978-3-319-71306-9/ebook). xvi, 397 p. (2017).

The main goal of this book is to tie together the work of the two authors, Crabb and Ranicki. Crabb was concerned with \({\mathbb Z}_{2}\)-equivariant homotopy theory and simply connected manifolds, whereas Ranicki was concerned with chain complexes and the surgery theory of non-simply-connected manifolds. Thus the book shows how the geometric Hopf invariant of a stable map \(F: \Sigma^{\infty}X \rightarrow \Sigma^{\infty}Y\) for pointed spaces is the stable \({\mathbb Z}_{2}\)-equivariant map
\[
h(F) = (F \wedge F) \Delta_{X} - \Delta_{Y}F:\Sigma^{\infty}X \rightarrow \Sigma^{\infty}(Y \wedge Y)
\]
that measures the failure of \(F\) to preserve the diagonal maps of \(X\) and \(Y\). The algebraic theory of surgery of Ranicki was based on the quadratic construction on a chain complex \(C\), the chain complex defined by
\[
Q_{\bullet}(C) = W \otimes_{{\mathbb Z}[{\mathbb Z}_{2}]}(C \otimes_{\mathbb Z} C)
\]
with \(W = C(S(\infty))\) a free \({\mathbb Z}[{\mathbb Z}_{2}]\)-module reduction of \(\mathbb Z\). A stable map \(F: \Sigma^{\infty}X \rightarrow \Sigma^{\infty}Y\) has been shown to induce a natural chain homotopy class of chain maps
\[
\phi_{F} : \dot{C}(X) \rightarrow Q_{\bullet}(\dot{C}(Y)) = \dot{C}(Q_{\bullet}(Y))
\]
called the quadratic construction on \(F\). Once the authors noticed that \(\phi_{F}\) is induced by the geometric Hopf invariant \(h(F)\) it became possible to unify the two theories. The book is devoted to doing this.

The eight chapters of the book along with three appendices give the details of this unification. The first four describe stable and \({\mathbb Z}_{2}\)-equivariant homotopy and bordism theory. Chapter five introduces the geometric Hopf invariant and the quadratic construction. Chapters six, seven and eight are byways of applications of the preceding ones. Thus Chapter six applies the theory to double points of immersions. Chapter seven considers the the \(\pi\)-equivariant version of the invariant, and Chapter eight makes the connection to surgery obstruction theory.

The eight chapters of the book along with three appendices give the details of this unification. The first four describe stable and \({\mathbb Z}_{2}\)-equivariant homotopy and bordism theory. Chapter five introduces the geometric Hopf invariant and the quadratic construction. Chapters six, seven and eight are byways of applications of the preceding ones. Thus Chapter six applies the theory to double points of immersions. Chapter seven considers the the \(\pi\)-equivariant version of the invariant, and Chapter eight makes the connection to surgery obstruction theory.

Reviewer: Jonathan Hodgson (Swarthmore)

### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55Q10 | Stable homotopy groups |

55Q25 | Hopf invariants |

57R65 | Surgery and handlebodies |

57R67 | Surgery obstructions, Wall groups |