A function space model approach to the rational evaluation subgroups

Let $ f : U \to 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 [B-S]. 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 homotogy group of $ X$. The Gottlieb triviality of a fibration in the sense of Lupton and Smith [L-S] is also discussed from the function space model point of view. Moreover, we proceed to consideration of an 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.


[B-S] Brown Jr, E. H. and Szczarba, R. H.: Rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349(1997), 4931-4951.

[L-S]Lupton, G and Smith, S.B.: The evaluation subgroup of a fibre inclusion, preprint (2006).