Involutions whose fixed set has three or four components: a small codimension phenomenon

Abstract

Let $T:M \to M$ be a smooth involution on a closed smooth manifold and $F = \bigcup_{j=0}^n F^j$ 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 $n \ge 4$ is even and $F$ has one of the following forms: 1) $F=F^n \cup F^3 \cup F^2 \cup \{{\operatorname {point}}\}$; 2) $F=F^n \cup F^3 \cup F^2 $; 3) $F=F^n \cup F^3 \cup \{{\operatorname{point}}\}$; or 4) $F=F^n \cup F^3$. 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 $k \le 4$. 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 $t \ge 1$.