The geometric Hopf invariant and surgery theory

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.