The Geometric Hopf Invariant

Abstract

The V-coefficient symmetric Q-groups $Q^*_V(C)$ and quadratic Q-groups $Q_*^V(C)$ of a chain complex C, with V an inner product space. The space level V-coefficient symmetric construction $\phi _V(X)$ on a space X, inducing $\phi _V(X):H_*(X) \rightarrow Q^*_V(C(X))$ on the chain level. The geometric Hopf invariant $h_V(F)$ of a V-stable map $F:V^{\infty } \wedge X \rightarrow V^{\infty } \wedge Y$. The V-coefficient quadratic construction $\psi _F(X)$ of a V-stable map $F:V^{\infty } \wedge X \rightarrow V^{\infty } \wedge Y$, inducing $\psi _V(F):H_*(X) \rightarrow Q_*^V(C(Y))$ on the chain level. The ultraquadratic construction $\widehat{\psi }(F)$ of a 1-stable map $F:\varSigma X \rightarrow \varSigma Y$, inducing $\widehat{\psi }(F):H_*(X) \rightarrow H_*(Y \wedge Y)$ on the chain level. The spectral quadratic construction $\psi _V(F)$ for a semistable map $F:X \rightarrow V^{\infty } \wedge Y$. The application of the geometric Hopf invariant to the classification of stably trivialized vector bundles.