Algebraic Hopf invariants and rational models for mapping spaces

The main goal of this paper is to define an invariant \(mc_{\infty }(f)\) of homotopy classes of maps \(f:X \rightarrow Y_{\mathbb {Q}}\), from a finite CW-complex X to a rational space \(Y_{\mathbb {Q}}\). We prove that this invariant is complete, i.e. \(mc_{\infty }(f)=mc_{\infty }(g)\) if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad \(\mathcal {C}\) to an operad \(\mathcal {P}\), a \(\mathcal {C}\)-coalgebra C and a \(\mathcal {P}\)-algebra A, then there exists a natural homotopy Lie algebra structure on \(Hom_\mathbb {K}(C,A)\), the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to model mapping spaces. More precisely, suppose that C is a \(C_\infty \)-coalgebra model for a simply-connected finite CW-complex X and A an \(L_\infty \)-algebra model for a simply-connected rational space \(Y_{\mathbb {Q}}\) of finite \(\mathbb {Q}\)-type, then \(Hom_\mathbb {K}(C,A)\), the space of linear maps from C to A, can be equipped with an \(L_\infty \)-structure such that it becomes a rational model for the based mapping space \(Map_*(X,Y_\mathbb {Q})\).