TITLE: Does a Homotopy on the Boundary Induce a Homotopy to the Entire Manifold?
QUESTION [8 upvotes]: Let $M$ and $N$ be smooth manifolds where $M$ has a non-empty boundary.
Let $F:M\to N$ be a smooth map, and the restriction of $f$ on $\partial M$ be denoted as $f$.
Let $H:\partial M\times I\to N$ be a homotopy between $f$ and a smooth map $g:\partial M\to N$.

Question. Does there necessarily exist a map $G:M\to N$ such that $G$ is homotopic to $F$ and the restrction of $G$ to $\partial M$ is $g$.

The answer is true if we work in the continuous category rather than the smooth category, that is, $M$ and $N$ are topological manifolds, and $F, H, f$, and $g$ are continuous maps.
This is because if $H:\partial M\times I\to N$ is a homotopy between $f$ and $g$, with $H_0=f$ and $H_1=g$, then we can paste $\partial M\times \{0\}$ along $\partial$ $M$ by identifying $(x, 0)$ with $x$, and get a new map $G:(M\cup (\partial M\times I))/\sim \ \to N$ which has the property that $G|_M = F$ and $G|_{\partial M\times \{1\}} = g$. Now we use the fact that $M$ admits a collar neighborhood homeomorphic to $\partial M\times I$ around its boundary to construct a homotopy between $F$ and $G$.
Of course, this construction, as such, does not work for the smooth case, because there is no guarantee that the $G$ that we get is smooth, provided we give the manifold obtained after pasting a suitable smooth structure.

REPLY [2 votes]: This is true. First note that we can change $H$ to be smooth, since both $f$ and $g$ are smooth. Further, we can assume $H(x,t)$ is constant with respect to $t$ for $t$ near $0$ and $1$ using a suitable smooth map $I \to I$. Then the gluing you describe will keep everything smooth.