Removing isolated zeroes by homotopy

Adam Coffman and Jiri Lebl

article abstract

Suppose that the inverse image of the zero vector by a continuous map
f from R^n to R^q has an isolated point P.  The existence of a
continuous map g which approximates f but is nonvanishing near P is
equivalent to a topological property we call ``locally inessential,''
generalizing the notion of index zero for vector fields, the q=n case.
For dimensions n, q where the n-1-fundamental group of the q-1-sphere
is trivial, every isolated zero is locally inessential.  We consider
the problem of constructing such an approximation g, and show that
there exists a continuous homotopy from f to g through locally
nonvanishing maps.  If f is a semialgebraic map, then there exists
such a homotopy which is also semialgebraic.  For q=2 and f real
analytic with a locally inessential zero, there exists a Holder
continuous homotopy F(x,t) which, for (x,t) not equal to (P,0), is
real analytic and nonvanishing.  The existence of a smooth homotopy,
given a smooth map f, is stated as an open question.