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

Abstract:

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 $\{ x \in {B^n}|{\tilde H^q}(f(x)) \ne 0\}$ 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.