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- W2789347224 abstract "We classify $f$ of $S^1$ in a $2$-manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the of double points of a self-transverse $C^1$-approximation of $f$, and the turning $T(ebar f)$ of the immersion $ebar f:S^1to M_fsubsetBbb R^2$, where $bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $left subsetpi_1(M)$. Namely, $f,g:S^1to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $Bbb R P^2$ or the normal bundle $nu(f)$ is non-orientable, then $S(f)=S(g)$, whereas if $Mnot= S^2,Bbb R P^2$ and $nu(f)$, $nu(g)$ have orientations $o$, $o'$, compatible with respect to the homotopy, then $T(e_obar f)=T(e_{o'}bar g)$, where $e_o$ is a standard embedding of the oriented surface $M_f$ (an annulus or a plane) in $Bbb R^2$. In fact, for homotopic $f$, $g$ both $S(f)-S(g)$ and $T(e_obar f)-T(e_{o'}bar g)$ boil down to the turning of a lift of a null-homotopic immersion $f# g^*$ to the universal cover of $M$. Here immersions $S^1to M$ are either smooth or topological; we include a smoothing theorem, which shows that there is no difference. We also classify of a graph in $M$ up to regular homotopy in terms of the invariants $S(f)$ and $T(e_obar f)$ of immersed $S^1$'s. The proofs are based on the h-principle. The point of this unsophisticated note is to simplify [10] and [11], where a classification of of a graph in $M$ was obtained for $MneBbb R P^2$ in terms of a rather laboriously defined winding number of a pair of homotopic $S^1to M$ (rather than of an individual immersion) with respect to a given vector field with zeroes on $M$." @default.
- W2789347224 created "2018-03-29" @default.
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- W2789347224 date "2016-12-01" @default.
- W2789347224 modified "2023-09-27" @default.
- W2789347224 title "Immersions of the circle into a surface" @default.
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