Piecewise-smooth paths and thin homotopies

Let $M$ be a smooth manifold and $U\subseteq M$ open. A piecewise-smooth path in $U$ is a continuous map $\gamma :[0,1]\to U$ for which some finite partition $0=t_0<t_1<\cdots <t_n=1$ makes every restriction $\gamma {\mathrm{\mid }}_{[t_{j-1},t_j]}$ smooth as a map of manifolds with boundary. Its source and target are $\gamma (0)$ and $\gamma (1)$. The collection of all such paths is a set: it is a subset, defined by separation, of the function set from $[0,1]$ to $U$.

Two paths ${\gamma }_0,{\gamma }_1$ with the same source $x$ and target $y$ are thin-homotopic, written ${\gamma }_0{\sim }_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}}{\gamma }_1$, if there is a continuous map $H:[0,1{]}^2\to U$ and a finite subdivision of the square into compact curvilinear triangles such that $H$ is smooth up to each triangle's boundary, $H(0,t)={\gamma }_0(t)$, $H(1,t)={\gamma }_1(t)$, $H(s,0)=x$, $H(s,1)=y$, and $\mathrm{rank}(d{H}_{(s,t)})\le 1$ on every smooth triangle. A curvilinear triangle here is a compact region $D$ equipped with a specified homeomorphism $\psi :{\mathrm{\Delta }}^2\to D$ from the standard closed two-simplex which is a smooth embedding on the interior, extends smoothly to a neighborhood of ${\mathrm{\Delta }}^2$ in its affine span, and maps the three edges of ${\mathrm{\Delta }}^2$ to the three boundary arcs of $D$. Saying $H$ is smooth up to $D$'s boundary means that $H\circ \psi$ extends smoothly from ${\mathrm{\Delta }}^2$ to an open neighborhood of ${\mathrm{\Delta }}^2$ in its affine span, with values in $U$; equivalently, it is a smooth singular $2$-simplex under the neighborhood-extension convention of Smooth singular simplex. The rank is the dimension of the image of the differential from the two-dimensional tangent space of the triangle to $T_{H(s,t)}M$, computed through the interior chart $\psi$ and extended to boundary points by these smooth extensions. The quotient by this relation is used only after its equivalence and compatibility properties are proved.

Smooth manifolds and their smooth charts

Smooth maps between manifolds with boundary

Smooth singular simplex

For sets $A$ and $B$ the collection of all functions $A\to B$ is a set, being a subset of $\mathcal{P}(A\times B)$

The Axiom Schema of Separation: for each formula $\phi$, $\mathrm{\forall }\bar{p}\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow (z\in x\wedge \phi (z,\bar{p})))$

Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$

The rank condition is the convention for thinness. The curvilinear triangles allow reparameterization homotopies whose smooth break lines move with the homotopy parameter. This is a local piecewise-smooth convention; it does not assert invariance for arbitrary continuous homotopies.