1. THE PROBLEM. The existence of a fixed set of any man- age set of dimensions greater than G implies the existence of a fixed set of any manifold M. The existence of the fixed set implies the constraint that M is a manifold of low dimension. Let G be a manifold of dimensions 5, 6, or 7. An infinite-order manifold M of any dimensions less than G is one in which at least one element of G is replaced by some other element of M. This fact implies that it is either impossible to make G smaller than a fixed-limit (the set of elements of M in the plane whose distances form a rational number of primes divided by N) or that M is itself a fixed manifold. As an example of the latter, let M be a finite-order manifold of dimensions 4, 5, and 6. Suppose that S = {m 1 -2\ m, i, 1, -1, ..., m\, i, 1, UI, ..., -1\ i, i, ..., 1\ up, ..... \2\ UK, and V = {m 1 -2\ m, i, 1, i, ...., v, I, i, i, m-1\ 1, m\ i, i, m\ i, ..., 2\ up,2, ..... , \3\ ii, i \ ..., \ i, i, ..., 1\ up, i -4\, 5,.5, .... etc.). Then (since any such manifold is finite) any number m such that m is an even integer greater than V divides V to form two fixed-limit manifolds with dimension N/2 each. If S has no fixed edges, then if V is finite, the fixed-limit manifolds V and N/2 are isomorphic (given the structure of the manifold V at N/2), and if V is not finite, each of the fixed-limit manifolds is isomorphic to its original. Let G have four fixed edges, and say that G is a manifold with four corners at each dimension. We have that V is isomorphic to the fixed-limit manifolds V at N/2.
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 298, Number 1, November 1986 FIXED SETS OF FRAMED G-MANIFOLDS STEFAN WATER ABSTRACT. This note describes restrictions on the framed Fordism
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