Spaces of unbounded Fredholm operators: I. Homotopy equivalences

This paper is devoted to the space of unbounded Fredholm operators equipped with the graph topology, the subspace of operators with compact resolvents, and their subspaces consisting of self-adjoint operators. Our main results are the following: (1) Natural maps between these four spaces and classical spaces of bounded operators representing K-theory are homotopy equivalences. This provides an alternative proof of a particular case of results of Joachim. (2) The subspace of essentially positive Fredholm operators represents odd K-theory. (3) The subspace of invertible operators in each of these spaces of unbounded operators is contractible.