Mapping spaces between manifolds and the evaluation map

Abstract

Abstract. Let f : M → N be a map between simply connected n-dimensional manifolds. We suppose that deg f = 0. Then the injection of aut 1 (N ) into the component Map(M, N; f ) of the mapping space containing f induces an injection on the rational homotopy groups, and the evaluation at the base point map(M, N; f ) → N is zero on the rational homotopy groups of even dimension.

Hochschild cohomology of Sullivan algebras and mapping spaces

“…Remark 8. If ω B is a cocycle representing the fundamental class of (B, d) and f is surjective, then there exists is a semi-free resolution of ∧V as a ∧V ⊗∧V -module, whereV = sV [5]. Moreover, the pushout If γ ∈ Hom ∧V (∧V ⊗ ∧V , A), then…”

“…Consider the inclusion i : CP n → CP n+k . As complex projective spaces are formal, a cdga model of the inclusion is is a semi-free resolution of ∧V as a ∧V ⊗∧V -module, where V = sV [5]. Moreover, the pushout As the differential of D on ∧V ⊗ ∧ V satisfies…”

“…We consider a morphism f : (A, d) → (B, d) of commutative differential graded algebras which models the embedding e, where (A, d) and (B, d) are Poincaré duality algebras [5]. We show that there is an A-linear shriek map f !…”