arXiv subjects
Publications and source records attributed to Samuel B. Smith.
We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. (2000), 747-768], which is used there as a crucial step in establishing their main finiteness result.
Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.
Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).
Let $A$ be a unital $C^*$-algebra. Its unitary group, $UA$, contains a wealth of topological information about $A$. However, the homotopy type of $UA$ is out of reach even for $A = M_2(\CC)$. There are two simplifications which have been considered. The first, well-traveled road, is to pass to $π_*(U(A\otimes \KK ))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has led to spectacular success in many arenas, as is well-known. A different approach is to consider $π_*(UA)\otimes\QQ $, the rational homotopy of $UA$. In joint work with G. Lupton and N. C. Phillips we have calculated this functor for the cases $A = C(X)\otimes M_n(\CC)$ and $A$ a unital continuous trace $C^*$-algebra. In this note we look at some concrete examples of this calculation and, in particular, at the $\ZZ$-graded map \[ π_*(UA)\otimes\QQ \longrightarrow K_{*+1}(A)\otimes\QQ . \]
We extend the standard localization theory for function and section spaces due to Hilton-Mislin-Roitberg and Moller outside the CW category to the case of compact metric domain in the presence of a grouplike structure. We study applications in two cases directly generalizing the gauge group of a principal bundle. We prove an identity for the monoid of fibre-homotopy self-equivalences of a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini in the CW category -- in the compact case. This leads to an extended localization result for this monoid. We also obtain an extended localization theory for groups of sections of a fibrewise group. We give two applications in rational homotopy theory.
Let ζbe an n-dimensional complex matrix bundle over a compact metric space X and let A_ζdenote 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 ζand 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 $A$ be a unital commutative Banach algebra with maximal ideal space $X.$ We determine the rational H-type of the group $GL_n (A)$ of invertible n by n matrices with coefficients in A, in terms of the rational cohomology of $X.$ We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of "last columns" of $GL_n (A).$ For $n > 1 + s/2,$ we construct a natural isomorphism from the rational Cech cohomology group $H^s (X; Q)$ to the rational homotopy group $π_{2 n - 1 - s} (Lc_n (A)) \otimes Q,$ which shows that the rational cohomology groups of $X$ 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.