  In low dimensional topology, we have some invariants defined by using
solutions of some nonlinear elliptic operators. The invariants could be
understood as Euler class or degree in the ordinary cohomology, in infinite
dimensional setting. Instead of looking at the solutions, if we can regard some
kind of homotopy class of the operator itself as an invariant, then the refined
version of the invariant is understood as Euler class or degree in cohomotopy
theory. This idea can be carried out for the Seiberg-Witten equation on
4-dimensional manifolds and we have some applications to 4-dimensional
topology.
