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.
According to Ando's theorem, the oriented bordism group of fold maps of n-manifolds into n-space is isomorphic to the stable n-stem. Among such fold maps we define two geometric operations corresponding to the composition and to the Toda bracket in the stable stem through Ando's isomorphism. By using these operations we explicitly construct several fold maps with convenient properties, including a fold map which represents the generator of the stable 6-stem.
We denote by π k (R n ) the k-th homotopy group of the n-th rotation group R n and π k (R n : 2) the 2-primary components of it. We determine the group structures of π k (R n : 2) for k = 23 and 24 by use of the fibration R n+1 Rn −→S n . The method is based on Toda's composition methods.
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