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# Contents

## Idea

In physics and data analysis, where an observable is typically a function $X \overset{f}{\longrightarrow} \mathbb{R}^n$ from a given data set $X$ to the set of real numbers $\mathbb{R}^1$, or to an n-tuple of such (varying in a Cartesian space $\mathbb{R}^n$), an iso-hypersurface or level-hypersuface is a level set of this observable, hence the subset of $X$ consisting all those points $x \in X$ on which this observable $f$ takes the same given value $f(x) = c$.

For example, if $X$ is a model for the Earth’s atmosphere, and if $f$ assigns to each point $x \in X$ the atmospheric pressure $p(x)$ at this point – in some suitable physical units and to some suitable approximation –, then an iso-surface for $p$ is an isobar: a surface of constant pressure.

For such a level set to actually be a hypersurface, hence a differentiable/smooth submanifold, some regularity conditions on $f$ need to be satisfied, such as that $f$ is a differentiable function to suitable degree, and, crucially, that its value $c$ (whose pre-image is formed) is a regular value.

Phrased this way, the construction of iso-hypersurfaces turns out to be a central topic also in areas of pure mathematics, such as in differential topology and cobordism theory, where the formation of pre-images of regular values inside $\mathbb{R}^n$ is known as part of the Pontryagin construction.

Curiously, this means thatPontryagin's theorem applies to iso-hypersurfaces, saying here that, as the value $c$ of the observable $f$ varies, the shape (topology) of the corresponding iso-hypersurfaces changes (at most) by a cobordism, and that the resulting (normally framed) cobordism class of all these hypersurfaces corresponds to the class of $f$ in the $n$-Cohomotopy theory of the data set $X$.

While Pontryagin’s theorem is ancient, the idea that it thus implies the possibility of topological data analysis via Cohomotopy theory of observables and cobordism theory of their iso-hypersurfaces appears only recently (Franek-Krčál 16, Franek-Krčál 17).