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- W2951215610 abstract "The Reeb graph $mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $fcolon Mto mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{!sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $Gamma(f)$ embedded into $M$ which is homotopy equivalent to $mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$." @default.
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- W2951215610 date "2014-05-19" @default.
- W2951215610 modified "2023-09-27" @default.
- W2951215610 title "On Representation of the Reeb Graph as a Sub-Complex of Manifold" @default.
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