Resumen de Involutions whose fixed set has three or four components: a small codimension phenomenon
E. M. Barbaresco, Patricia E. Desideri
Let T:M→M be a smooth involution on a closed smooth manifold and F=⋃nj=0Fj the fixed point set of T, where Fj 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=Fn∪F3∪F2∪{point}; 2) F=Fn∪F3∪F2; 3) F=Fn∪F3∪{point}; or 4) F=Fn∪F3. Also, suppose that the normal bundles of Fn, F3 and F2 in M do not bound. If k denote the codimension of Fn, 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.
