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

Authors

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

$t \ge 1$ .