Module sectional category of products

Adapting a result of Félix–Halperin–Lemaire concerning the Lusternik–Schnirelmann category of products, we prove the additivity of a rational approximation for Schwarz’s sectional category with respect to products of certain fibrations.


Introduction
The sectional category [12] (or Schwarz genus) of a fibration p : E → X , secat( p), is the smallest integer m such that X admits a cover by (m + 1) open sets on each of which a local section for p exists. This homotopy invariant is a generalization of Communicated by Pascal Lambrechts. the well-known Lusternik-Schnirelmann (L.-S.) category [10] of a path-connected space X , cat(X ), as the latter 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 [5] says that, if X and Y are simply connected rational spaces of finite type, then cat(X × Y ) = cat(X ) + cat(Y ). This was achieved by first proving the analogous result for the lower bound module L.-S. category, mcat(X ), of cat(X ) using differential graded (DG) module techniques. It was then lifted to rational category using Hess' theorem [9]. We propose to apply similar DG-module techniques to the lower bound msecat( p) of secat( p) called module sectional category and introduced in [7].
Throughout this paper we consider fibrations whose base and total space have the homotopy type of simply connected CW-complexes of finite type. Our main result is Theorem 1 Let p and p be two fibrations. If either p or p admits a homotopy retraction, then Recall the important particular case of sectional category provided by Farber's (higher) topological complexity [4,11] of a space X , TC n (X ) = secat(π n ), where the considered fibration π n : X [1,n] → X n is given by π n (α) = (α(1), α(2), . . . , α(n)). Consequently, the module invariant associated to (higher) topological complexity, i.e., mTC n (X ) := msecat(π n ), is additive on products. Namely Corollary 2 Let X and Y be two spaces. Then These results are improvements over [2] as only one of the two fibrations of Theorem 1 needs a homotopy retraction and the Poincaré duality assumption is no longer required.

Preliminaries
This section contains a brief summary of the DG-modules techniques that will be used (see [6] 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 satisfying d(ax) where the action is the one of the direct sum, the differential on M is the differential of (M, d), and U admits a direct sum decomposition  (N , d) can be decomposed as (the inclusion of) a semifree extension followed by a quasi-isomorphism as well as the following lifting lemma. Given a solid arrow commutative diagram of (A, d)-modules of the form The following lemma is an adaptation of a central idea of [5].
Proof Suppose that ϕ admits a homotopy retraction of (A, d)-modules. This means that there exists a homotopy commutative diagram of (A, d)-modules of the form Since B and A K are isomorphic cdgas, we have B ⊗ A P = P K ·P . Hence the left hand morphism is simply the projection : P → P K ·P . The diagram shows that admits a homotopy retraction of (A, d)-modules. Hence it is injective in homology.
Conversely, suppose that is injective in homology. Since A is of finite type, η # : A → P # is also an (A, d)-semifree resolution. Moreover, be a decomposition of ϕ as a semifree extension followed by a quasiisomorphism. Applying the lifting lemma to the solid arrow commutative diagram we obtain the desired homotopy (A, d)-module retraction for ϕ.

The invariant msecat( p)
Let us denote by p m : J m X (E) → X the join of m + 1 copies of a fibration p : E → X . As is well-known [12], secat( p) ≤ m if and only if p m admits a homotopy section. By definition, msecat( p) is the smallest m such that A P L ( p m ) admits a homotopy retraction of A P L (X )-modules, where A P L denotes Sullivan's functor of piecewise linear forms [13].
Let ϕ : (A, d) → (B, d) be any cdga model of p and a factorization in the category of (A, d)-modules of ϕ as the inclusion of a semifree extension followed by a quasi-isomorphism ξ . We refer to the inclusion as a semifree We notice that, if ϕ is surjective, then the quasi-isomorphism ξ can be constructed to satisfy ξ(U ) = 0, which implies that d 0 x ∈ ker ϕ for x ∈ U . Recall that the n th -suspension s −n V of a graded vector space V is defined by (s −n V ) i = V i−n . According to [7] (Thm 5.4, p.135), msecat( p) is the least m such that the following admits a retraction of (A, d)-modules, where the differential D is given by for x 0 ,..., x m ∈ U and d + x i = j i a i j i ⊗ x i j i with a i j i ∈ A and x i j i ∈ U .
Using the following notation (suggested by the standard rules of signs) where σ i j i := (−1) |a i j i |(|x 0 |+···+|x i−1 |+m) and τ i := (−1) (|x 0 |+···+|x i−1 |) . When the fibration p : E → X is endowed with a homotopy retraction, there exists a surjective cdga model of p which is a retraction of a cdga cofibration (see, for instance, [3, Section 5.1] for an explicit construction). Such a model is called an s-model. We will use the following result from [1].

Theorem 4 ([1, Theorem 3.3]) Let p be a fibration endowed with a homotopy retraction. For any s-model ϕ : A → A K of p, msecat( p) is the smallest m for which the projection ρ m : A → A
K m+1 admits a homotopy retraction of (A, d)-modules. By using this result together with Lemma 3, we obtain the following new characterization of msecat( p) when p admits a homotopy retraction.
where the map λ m : A → C m is a model of p m : J m X (E) → X , the left hand triangle is commutative up to a homotopy of (A, d)-modules, and the right hand triangle is strictly commutative. Applying Id P ⊗ A − to the previous diagram, we get the following diagram of (A, d)-modules: where the left hand triangle is commutative up to a homotopy of (A, d)-modules and the right hand triangle is strictly commutative, which yields (ii) ⇒ (iii). Finally the implication (iii) ⇒ (ii) follows from Lemma 3 applied to ρ m .

The main result
Finally, we 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 6
Let p : E → X and p : E → X be two fibrations. We have Proof In [8,Section 7.2], maps ψ E,E n,m producing a commutative diagram of the following form are constructed: By applying A P L to this diagram, we can establish that, if msecat( p) ≤ m and msecat( p ) ≤ n then msecat( p × p ) ≤ m + n.
In order to prove our main result (Theorem 1), it remains to establish the inequality msecat( p × p ) ≥ msecat( p) + msecat( p ) under the additional assumption that one of the fibration, say p, admits a homotopy retraction. We notice that, if both fibrations would admit a homotopy retraction, a direct adaptation of the strategy of [5] together with Proposition 5 would give a proof of this inequality. The following less immediate adaptation of [5] provides a proof when only p admits a homotopy retraction.
Proof (Proof of Theorem 1) Take an s-model ϕ for p and an (A, d)-semifree extension a (B, d)-semifree model of p . Then p × p is modeled by the tensor product of the two semifree extensions which gives a semifree extension of (A ⊗ B, d)-modules that we write as follows In order to prove the statement, we suppose msecat( p) = m and msecat( p × p ) = m + n and show that msecat( p ) ≤ n.
From the assumption msecat( p × p ) = m + n 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 −n V ⊗n+1 ). This proves that msecat( p ) ≤ n.