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How can I prove that a contractible manifold $M$ is always deformation retractable to a point $p$? This is an exercise from the book "An introduction to Manifolds" by L. Tu.

Intuitively it seems that, at each stage of the homotopy, it should be possible to construct some kind of a "translation" which brings back the image of $p$ to itself.

Sankhadip Chakraborty
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There isn't much to do here. A space $X$ is contractible if the identity map $id_X:X\rightarrow X$ is null homotopic. That is homotopic to the map that takes $X$ to a point $p\in X$. So $\{p\}$ is a deformation retract of $X$. That's it.

R_D
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  • I think I am talking about strong deformation retraction, i.e. at each stage of the homotopy the image of $p$ should be $p$ itself. Sorry for the confusion. – Sankhadip Chakraborty Nov 25 '15 at 12:09