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A Visual Introduction to Morse Theory


the Reeb graph10
Elementary Applied Topology

Julien Tierny’s
Valerio Pascucci’s
John Milnor’s
Morse theory
Robert Ghrist’s

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Positivity     43.00%   
   Negativity   57.00%
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\mathbb{M}$ a corresponding real-valued measurement $f(x)$.In some sense, these functions are the ‘natural’ thing to study here interested in the level sets of $f$3. the level set of $f$ according to $c$ is defined as$$ $$In other words, the level set contains all those points of the manifold elevation; you can think of the level set as nothing but a contour line $\mathcal{L}_{f}(c)$.A level set of a simple 1D functionUnless there is a horizontal segment in the function, every level set make a stellar comeback towards the end of the article.Level sets might be nice, but at least in our 1D example, they were lacking new types of sets:We call $\mathcal{L}_{f}^{+}(c)$ the superlevel set, while we call indicates our $c$ parameter; everything that is part of the superlevel set is shown in black:A superlevel set of a simple 1D functionBehold, suddenly we obtain segments and not just mere points! selected values:A few superlevel sets of a simple 1D functionWe can see that the segments continue to grow as we decrease $c$. that every point that is part of the superlevel set for $c$ will also be part of the superlevel set of $c - \epsilon$. calculation from the (literal) top (of the function):A few superlevel sets of a simple 1D function, starting with local maximaWe can see that for extremely large values of $c$, there is only one We can thus conclude:Connected components in superlevel sets appear at (local) a connected component, so they are not ‘interesting’.Morse theory uses the term critical points, or critical values (since technically, $c$ determines the superlevel set but and illustrates all the sets we encountered so far:A 2D example of sets encountered in Morse theoryFrom left to right, we see the level set, the superlevel set, and the sublevel set for the same critical value $c$. $\mathbb{M}$ has no interior here, superlevel and sublevel sets can earlier, superlevel sets, for example, contain different connected $\mathbb{M}$ and its critical points, but none of the graphs To calculate a Reeb graph, we first represent $\mathbb{M}$ by its critical points, i.e. its to how the connectivity of the level set changes as we pass a critical set for $c := f(x) = f(y)$.

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