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I am not able to construct explicitly a deformation retract of the 3-simplex [0, 1, 2, 3] onto a union of its faces [0 ,1, 2] and [1, 2, 3]. it seems easy, but everything I try fails because I can't figure out a way to deform points onto these faces in such a way that the points already on the face remain fixed. Is it possible? For example I could construct a map $\phi : \Delta^3 \times I \to I$ given by $(a, b, c, d, t) \mapsto (a-ta, b+ta, c+td, d-td)$ (deforming everything onto edge [1, 2]) but obviously this doesn't work.

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Try $r(a,b,c,d)=(a-\lambda,\ b+\lambda,\ c+\lambda,\ d-\lambda)$ where $\lambda=\textrm{min}(a,d)$. This is a retraction $r$ of $\Delta^3$ to the faces. Since $\Delta^3$ is convex, you can just slide each point $x$ along the straight path to $r(x)$ during the time interval $I$.

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  • $\begingroup$ ah..nice, that works! $\endgroup$
    – shurik
    Mar 22 '13 at 21:11

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