Abstract

Topology is often useful in showing that equations have solutions without necessarily finding out what the solutions are. The first example of this is the intermediate value theorem: a continuous function f: R → R which takes both positive and negative values must take the value 0; the topological input is that R is connected. The second of these is the fundamental theorem of algebra: a polynomial function p: C → C of positive degree must take the value 0; the topological input is the calculation of the fundamental group of the circle.

I will explain these as well as the less-well known example of solving equations in groups: given a “polynomial” such as w(x)=g_1 x^{2 g_2 x}(-4) g_3 x^3 with g_1, g_2, g_3 in a group G and x a formal symbol, is there a bigger group H ≥ G and a h ϵ H such that w(h)=e ϵ H?