The homotopy structure of the linear group of a Banach space

The homotopy type of the linear group of an infinite-dimensional Banach space is as important in the theory of Banach manifolds and bundles as is the (stable) homotopy structure of the orthogonal and unitary groups in the theory of finite-dimensional vector bundles and in $K$-theory (for more details see [4]). Kuiper has proved [20] the contractibility of the linear group $GL(H)$ of a Hilbert space $H$, and Neubauer has given a positive answer [34] to the question of the contractibility of $GL(l^p)$, $1\leq p<\infty$, and $gl(c_0)$. At the same time there are examples (the first of which was given by Douady [11]) of Banach spaces with a non-contractible and disconnected linear group. In [30] the author drew attention to the fact that the constructions of Kuiper and Neubauer could be formalized to provide a general procedure for proving (or analysing) the contractibility of the linear group $GL(X)$. This enables us to settle the question of the homotopy structure of the linear groups of many specific Banach spaces. The present paper reviews the results that have been obtained up till now on the contractibility of the linear group of Banach spaces. In § 1 examples are given of Banach spaces with homotopically non-trivial linear groups. The general procedure for analysing the contractibility of $GL(X)$, Theorem 1, is set out in § 2, and the problem of obtaining explicit analytic conditions necessary for the applicability of this procedure is solved in § 3. In §§ 4–6 examples are given of many specific Banach spaces (of smooth and of measurable functions), and the contractibility of their linear groups is proved. § 7 contains remarks on the general procedure and unsolved questions.