Involutions whose fixed set has three or four components: a small codimension phenomenon - Barbaresco, Evelin M.; Desideri, Patricia E.; Pergher, Pedro L. Q. (MATHEMATICA SCANDINAVICA, 2012)
Let T:M → M be a smooth involution on a closed smooth manifold and F = _j=0^n F^j the fixed point set of T, where F^j denotes the union of those components of F having dimension j and thus n is the dimension of the component of F of largest dimension. In this paper we prove the following result, which characterizes a small codimension phenomenon: suppose that n 4 is even and F has one of the following forms: 1) F=F^n F^3 F^2 {point }; 2) F=F^n F^3 F^2 ; 3) F=F^n F^3 {point }; or 4) F=F^n F^3. Also, suppose that the normal bundles of F^n, F^3 and F^2 in M do not bound. If k denote the codimension of F^n, then k 4. Further, we construct involutions showing that this bound is best possible in the cases 2) and 4), and in the cases 1) and 3) when n is of the form n=4t, with t 1.