It is known that if f,g : X → Y
are maps of a topological space X into a topological manifold Y , and that f and g
can be deformed by homotopies to maps f′ and gf which are coincidence-free, then
f may be deformed by a homotopy to a map f′′ such that f′′ and g are
coincidence-free. This result is generalized as follows: If f,g : X → Y are maps of a
topological space X into a topological manifold Y and ff and g′ are homotopic to f
and g respectively, then for any homotopy {gt} from g to g′, there is a homotopy
{ft} from ff such that the set of coincidences of ft and g1−t is the same for all
t ∈ [0,1]. Some applications of this result to fixed point theory and root theory are
indicated.
