The geometric Hopf invariant and surgery theory

TL;DR

This paper introduces a geometric Hopf invariant for stable maps, unifies previous treatments of double points in homotopy theory, and connects it with algebraic surgery to provide foundations for non-simply-connected geometric surgery.

Contribution

It develops a $ ext{Z}_2$-equivariant geometric Hopf invariant that captures double points and links homotopy theory with algebraic surgery for non-simply-connected spaces.

Findings

The geometric Hopf invariant acts as an obstruction to desuspending maps.

It unifies various homotopy theoretic approaches to double points.

It relates the Hopf invariant to Wall's surgery obstruction in algebraic topology.

Abstract

The first author's geometric Hopf invariant of a stable map $F:\Sigma^{\infty}X \to \Sigma^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):\Sigma^{\infty}X \to \Sigma^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)\Delta_X - \Delta_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$ up to homotopy. The explicit nature of the construction allows for a $\pi$-equivariant version of $h(F)$ in the case of a $\pi$-equivariant $F$, with $\pi$ a discrete group. In earlier joint work we applied the $\pi_1(N)$-equivariant geometric Hopf invariant of the Umkehr map $F:\Sigma^{\infty}N^+ \to \Sigma^{\infty}T(\nu_f)$ of an immersion $f:M \to N$ to capture the double points of $f$ in ${\mathbb Z}_2$-equivariant homotopy theory. In this manuscript we use the $\pi$-equivariant geometric Hopf invariant $h(F)$ to unify all the previous homotopy theoretic treatments of double points. Furthermore, $h(F)$ is combined with the second author's algebraic surgery theory of chain complexes with Poincar\'e duality to provide the homotopy theoretic foundations for non-simply-connected geometric surgery. For an $n$-dimensional normal map $(f,b):M \to X$ the $\pi_1(X)$-equivariant geometric Hopf invariant $h(F)$ of the Umkehr map $F:\Sigma^{\infty}X^+ \to\Sigma^{\infty}M^+$ is shown to induce the $\pi_1(X)$-equivariant quadratic structure $\psi_F$ on the chain complex kernel $C$ of $(f,b)$. Previously $\psi_F$ had only been constructed using the chain complex analogue of the functional Steenrod squares. The Wall surgery obstruction $\sigma_*(f,b)=(C,\psi_F) \in L_n({\mathbb Z}[\pi_1(X)])$ is the cobordism class of the corresponding $n$-dimensional quadratic Poincar\'e complex $(C,\psi_F)$, as in the original theory.