Let $M,N$ be (compact) smooth manifolds and consider a triangulation $\mathcal{T}$ of $M.$ Consider two maps $f,g:M\to N$ which agree on the vertices of the triangulation. Is it true that $f$ and $g$ are homotopic relative to the vertices of the triangulation if and only if the the restrictions of $f,g$ to each 1-simplex are homotopic relative to the boundary of the 1-simplex? If not, is it true in low dimension? I can see that it is true in dimension one, but there are no higher simplices there.
2026-03-26 01:11:36.1774487496
Are these maps homotopic relative to a finite set of points?
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