Module sectional category of products

Extending a result of F\'elix-Halperin-Lemaire on Lusternik-Schnirelmann category of products, we prove additivity of a rational approximation for Schwarz's sectional category with respect to products of fibrations.


Introduction
The sectional category [11] (or Schwarz genus) of a fibration p : E → X is the smallest integer n such that X admits a cover by open sets on each of which a local section for p exits. This homotopy invariant is a generalization of the well known Lusternik-Schnirelmann (LS) category [9] of a path-connected space X, cat(X), as it is the sectional category of the path fibration P X → X, α → α(1), where P X is the space of paths starting at the base point.
One of the most important results of [4] says that, if X and Y are simply connected rational spaces of finite type, then cat(X × Y ) = cat(X) + cat(Y ). This was done through Hess' theorem [8] by proving the analogous result for a lower bound of LS category called module LS category.
Throughout this paper we will consider all spaces to be simply connected CW-complexes of finite type. We will also denote f 0 the rationalisation of a map f . As for LS category, there exists a lower bound of sectional category, called module sectional category [6], for which we have mcat(X) = msecat(P X → X). In this paper we prove Theorem 1. Let f and g be two fibrations. If f 0 admits a homotopy retraction, then msecat(f × g) = msecat(f ) + msecat(g).
As a direct application of Theorem 1, the module invariant associated to (higher) topological complexity, mTC n (X) := msecat(π n ), is additive: Corollary 2. Let X and Y be two spaces. Then mTC n (X × Y ) = mTC n (X) + mTC n (Y ).
The results given are an improvement of [2].

Preliminaries
This section contains a brief summary of the tools that will be used, see [5] for further details. Let (A, d) be a commutative differential graded algebra over Q (cdga). An (A, d)-module is a chain complex (M, d) together with a degree 0 action of A verifying that d(ax) = d(a)x + (−1) deg(a) ad(x). The module M # = hom(M, Q) admits an (A, d)-module structure with action (aϕ)(x) = (−1) deg(a) deg(ϕ) ϕ(ax) and differential d(ϕ) = (−1) deg(ϕ) ϕ • d. If N is an (A, d)-modules, then the module M ⊗ A N admits an (A, d)-module structure with action a(m ⊗ n) = (am) ⊗ n and differential d(m ⊗ n) = d(m) ⊗ n + (−1) deg(m) m ⊗ d(n). An (A, d)-module P is said to be semifree if there exists an increasing filtration P * by (A, d)-submodules such that P k /P k−1 is a free (A, d) module on a basis of cocycles. Every (A, d)-module M admits a semifree resolution, that is a quasi-isomorphism of (A, d) modules, P ≃ −→ M, where P is (A, d)-semifree. If P is (A, d)-semifree and if η is a quasi-isomorphism of (A, d)-modules then η ⊗ A Id P and Id P ⊗ A η are also quasi-isomorphisms. A morphism of (A, d)-modules ϕ : (M, d) → (N, d) is said to have a homotopy retraction if there exists a commutative diagram of (A, d)-modules, We will use the following lemma which is an expression of one of the central ideas of [4].
Proof. Suppose that ϕ admits a homotopy retraction of (A, d)-module. This means that there exists a commutative diagram of (A, d) module of the form semi-free resolution and apply − ⊗ A P to the diagram above. We get K·P and the left hand morphism is the projection ̺ : P → P K·P . The diagram shows that ̺ admits a homotopy retraction of (A, d)-module and therefore that it is injective in homology.
Conversely, suppose that ̺ is homology injective. Since A is of finite type, η # = hom(η, Q) : A → hom(P, Q) is also a semi-free resolution. Since ̺ is homology injective, Then ̺ # •α is homotopic to η # and thus a quasi-isomorphism. To finish the proof, we observe that K · hom P K·P , Q = {0} and thus we have a diagram which yields a homotopy retraction for ϕ as (A, d)-modules.
Let us denote by p n : J n X (E) → X the join of n + 1 copies of a fibration p : E → X. As it is well-known [11], secat(p) ≤ n if and only if p n admits a homotopy section. By definition, msecat(p) is the smallest n such that A P L (p n ) admits a homotopy retraction of A P L (X)-modules, where A P L denotes Sullivan's functor of piecewise linear forms [12].
Recall the following general characterization of msecat(f ) from [6]. Let Then msecat(f ) is the least m such that the following (A, d) semi-free extension admits a retraction of (A, d)-module: Using the following notation (suggested by the standard rules of signs) (iii) the projection P → P K m+1 ·P is injective in homology.

Proof. By [1], there is a diagram
where the left hand triangle is commutative up to a homotopy of (A, d)modules and the right hand triangle is strictly commutative. Applying to previous diagram Id P ⊗ A −, we get the following diagram of (A, d) module: where the left hand triangle is commutative up to a homotopy of (A, d)module and the right hand triangle is strictly commutative. The result then follows from Lemma 3.

The main result
Observe that Proposition 4 together with the strategy of [4] can be used to easily prove Theorem 1 provided that both fibrations admit a homotopy retractions. In line with our statement, we here present a proof of the additivity of module sectional category when only one of the fibrations admits homotopy retraction.
We first notice that one of the inequalities of Theorem 1 follows in general: Proposition 5. Let p : E → X and p ′ : E ′ → X ′ be two fibrations. We have Proof. In [7, Pg. 26], a commutative diagram of the following form is constructed: By applying A P L to this diagram, we can establish that, if msecat(p) ≤ n and msecat(p ′ ) ≤ m then msecat(p × p ′ ) ≤ n + m.
Keeping the notation of Proposition 4, the differential in (A ⊗ (Q ⊕ X), d) can be taken such that d 0 (x) ∈ K. This implies that, in P ⊗(Q⊕s −m X ⊗m+1 ), d 0 (s −m X ⊗m+1 ) ⊂ K m+1 · P . With this in mind we proceed to the Proof of Theorem 1. Take an s-model for f , ϕ and an (A, d) semi-free extension of ϕ, A ⊗ (Q ⊕ X), as in the previous section. Let also (B, d) → (B ⊗ (Q ⊕ Y ), d) a semi-free model of g (where (B, d) is a cdga model of the base of g). Then f × g is modeled by the tensor product of the two semi-free extensions which gives a semi-free extension of (A ⊗ B, d)-modules that we write as follows In order to prove the statement, we suppose msecat(f ) = m and msecat(f × g) ≤ m + p and we establish that msecat(g) ≤ p.
Since msecat(f ) = m we know from Proposition 4 that there exists Ω ∈ H(K m · P ) which is not trivial in H(P ). Then there exist a cocyle ω ∈ K m · P representing Ω in H(P ) and θ ∈ P ⊗ s −(m−1) X ⊗m such that dθ = ω. As a chain complex, we can write P = ω · Q ⊕ S where d(S) ⊂ S, and we define the following linear map of degree −|ω|: I ω : P → Q, I ω (ω) = 1, I ω (S) = 0.
This map commutes with differentials. Now write the element θ ∈ P ⊗ s −(m−1) X ⊗m as with m i ∈ P and x i ∈ X ⊗m . Since dθ = ω we have d + θ = 0 and d 0 θ = ω.
From msecat(f × g) ≤ m + p we know that the morphism admits a retraction r of (A ⊗ B, d)-modules. Finally the composite gives a morphism (of degree 0) of (B, d)-module which is a retraction for the inclusion B → B ⊗ (Q ⊕ s −p Y ⊗p+1 ). This proves that msecat(g) ≤ p.