Application of evaluation map, dual Gottlieb groups and terminal cells (Q1313781)

Let \(X\) be a simply-connected, pointed CW-complex with finite Betti numbers, the authors compare some natural subvector spaces of the cohomology of \(X\) with coefficients in a field \(k\). For the first vector space, denoted by \(E(X;k)\), recall the existence of an evaluation map, \(\text{ev}: \text{Ext}_ A(k,A) \to H^*(A)\), for any augmented cochain algebra \(A\) [the first and the third named author with \textit{S. Halperin}, Adv. Math. 71, No. 1, 92-112 (1988; Zbl 0659.57011) and \textit{A. Murillo}, Trans. Am. Math. Soc. 339, No. 2, 611- 622 (1993; Zbl 0796.55005)]. By definition, \(E(X,k)\) is the image of ev when \(A\) is the algebra of singular cochains of \(X\). The second vector space, denoted by \(\check{G}(X;k)\) and called dual of Gottlieb group, is the set of cohomology classes representable by a map \(f: X \to K(k,n)\) such that \(f \times \text{id}: X \to K(k,n) \times X\) can be factorised through the wedge \(K(k,n) \vee X\). The last one, \(T(X;k)\), is the subspace generated by ``terminal cells''. Among the results, quote: (1) \(\check {G}(X;k)\) is an invariant of the homotopy type; (2) the subspaces previously defined satisfy \(T(X;k) \subset \check{G}(X; k) \subset E(X;k)\).