The Geometric Hopf Invariant

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.