We identify the long exact sequence induced on rational homotopy groups by the evaluation map ω: map(X, Y ; f ) → Y , and in particular the rationalization of the evaluation subgroups of f , in terms of derivations of Quillen models and adjoint maps. We consider a generalization of a question of Gottlieb within the context of rational homotopy theory. We also study the rationalization of the G-sequence of a map. In a separate result of independent interest, we give an explicit Quillen minimal model of a product A × X , in the case in which A is a rational co-H -space.
Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A ζ denote the C * -algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA ζ , the group of unitaries of A ζ . The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.
Let Aut(p) denote the space of all self-fibre-homotopy equivalences of a fibration p : E → B. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of this monoid by means of an isomorphism:Here ∧V → ∧V ⊗ ∧W is the Koszul-Sullivan model of the fibration and Der ∧V (∧V ⊗ ∧W ) is the DG Lie algebra of derivations vanishing on ∧V . We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group π 0 (Aut (p)), where Aut (p) is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.
Abstract. Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GL n (A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lc n (A) denote the space of "last columns" of GL n (A). We construct a natural isomorphismȞs+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X, G) for G a Lie group or, more generally, a rational H-space.
We identify the homomorphism induced on rational homotopy groups by the evaluation map ω: map(X, Y ; f ) → Y in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f . This allows us to characterize the rationalized nth evaluation subgroup of f and, more generally, the rationalization of the so-called G-sequence of the map f . We use these results to study the G-sequence in the context of rational homotopy theory.
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