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

Evelin M. Barbaresco, Patricia E. Desideri, Pedro L. Q. Pergher

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 $t \ge 1$.

Full Text:

PDF


DOI: http://dx.doi.org/10.7146/math.scand.a-15205

Refbacks

  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.
OK


ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library