Semi-free $R^1$ action and Bott map

Abstract

Let $M^{n}$ be a compact closed manifold of dimension at least 3. We study the $R^1$-Bott functions on $M^{n}$. Separately investigated $R^1$-invariant Bott functions on $M^{2n}$ with a semi-free circle action which has finitely many fixed points. The aim of this paper is to find exact values of minimal numbers of singular circles of some indices of $R^1$-invariant Bott functions on $M^{2n}$.

Closely related to $ R^1 $-Bott function on a manifold $ M^n $ is a more flexible object, the decomposition of round handle of $ M^n $.
In its turn, to study the round handles decomposition of $M^n $ we use a diagram, i.e. a graph which carries the information about the handles.