An extension of Brouwer's fixed-point theorem to nonacyclic, set valued functions

If $f$ is a set valued function defined on an $n$-ball such that each $f(x)$ is a subset of the $n$-ball, and the graph of $f$ is closed, then all that is needed to insure that there is a fixed point $(x \in f(x))$ is that the singularity sets not be too high dimensional. I.e., the dimension of $\{ x \in {B^n}\vert{\tilde H^q}(f(x)) \ne 0\}$ is $\leqq n - q - 2$. Examples are given to show that the dimension requirements are the best possible. The proof involves defining an analogue of the retraction in the ``no retraction'' proofs of the Brouwer theorem, and then applying the Leray spectral sequence to the projection of the graph of this retraction onto the $n$-ball.