A function space model approach to the rational evaluation subgroups

Abstract

Abstract. Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G * (U, X; f ), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba [5]. In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith [27] is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S 1 -bundle over the n-dimensional torus is determined explicitly in the non-rational case.

Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

“…) is an explicit model for the connected component F (X, Y ; f ); see [5,Theorem 6.1] and [17,Section 3]. The proof of [19,Proposition 4.3] and [17,Remark 3.4] allow us to deduce the following proposition; see also [7].…”

“…We now provide an overview of the rest of the paper. In Section 2, we recall a model for the evaluation map of a function space from [19], [7] and [17]. In Section 3, a rational model for the map λ G,M mentioned above is constructed.…”

“…Example 2.4. Let M be a space whose rational cohomology is isomorphic to the truncated algebra Q[x]/(x m ), where deg x = l. Recall the model (E/M u , δ) for aut 1 (M ) mentioned in [17,Example 3.6]. Since the minimal model for M has the form (∧(x, y), d) with dy = x m , it follows that…”

On extensions of a symplectic class

“…An outline of this paper is as follows. In Section 2, we recall briefly a model for the evaluation map of a function space from [3], [14] and [18] of which we make extensive use . Section 3 is devoted to proving Theorem 1.8, Propositions 1.9, 1.10 and Corollary 1.11.…”

Mapping spaces between manifolds and the evaluation map

“…On the other hand, by a result of G. Lupton and the author, H * (ev; Q) = 0 if χ(N ) = 0 [4]. See also [10], [11], [6] for more recent results on Gottlieb groups.…”