Bousfield and Kan's Q-completion and fiberwise Q-completion of spaces lead to two different approaches to the rational homotopy theory of nonsimply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of path-connected pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps pi(1)-rational homotopy equivalences. We compare these two notions and show that pi(1)-rational homotopy equivalences correspond to maps that induce Omega-quasi-isomorphisms on the rational singular chains, ie maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphisms and Omega-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative.

The classical Hopf invariant is an invariant of homotopy classes of maps from S4n−1 to S2n, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for En–operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space X and the homotopy groups of X. We will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the En–bar construction on the cochains of X and the homotopy groups of X. This pairing gives us a set of invariants of homotopy classes of maps from Sm to a simplicial set X; this pairing can detect more homotopy classes of maps than the classical Hopf invariant.

The second part of the paper is devoted to combining the En–Hopf invariants with the Koszul duality theory for En–operads to get a relation between the En–Hopf invariants of a space X and the En+1–Hopf invariants of the suspension of X. This is done by studying the suspension morphism for the E∞–operad, which is a morphism from the E∞–operad to the desuspension of the E∞–operad. We show that it induces a functor from E∞–algebras to E∞–algebras, which has the property that it sends an E∞–model for a simplicial set X to an E∞–model for the suspension of X.

We use this result to give a relation between the En–Hopf invariants of maps from Sm into X and the En+1–Hopf invariants of maps from Sm+1 into the suspension of X. One of the main results we show here is that this relation can be used to define invariants of stable homotopy classes of maps. 

In previous works by the authors – [26, 31] – a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of ∞-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such ∞-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently extended to accept ∞-morphisms in both slots simultaneously. We apply this theory to rational models for mapping spaces.

In this thesis we use the theory of algebraic operads to define a complete invariant of real and rational homotopy classes of maps of topological spaces and manifolds. More precisely let f,g : M -> N be two smooth maps between manifolds M and N. To construct the invariant, we define a homotopy Lie structure on the space of linear maps between the homology of M and the homotopy groups of N, and a map mc from the set of based maps from M to N, to the set of Maurer-Cartan elements in the convolution algebra between the homology and homotopy. Then we show that the maps f and g are real (rational) homotopic if and only if mc(f) is gauge equivalent to mc(g), in this homotopy Lie convolution algebra. In the last part we show that in the real case, the map mc can be computed by integrating certain differential forms over certain subspaces of M. We also give a method to determine in certain cases, if the Maurer-Cartan elements mc(f) and mc(g) are gauge equivalent or not.