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We know that if a space $X$ has the property that $id_X$, the identity map on $X$, is null-homotopic, then $X$ is contractible.

My question is what happen to a space $X$ where the identity map on $X$ is NOT null-homotopic, why can't it be contractible? What goes wrong? Any examples?

Here contractible refers to a property of a space having the homotopy type of a point.

Could anyone help me clarify this confusion? Thanks.

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  • $\begingroup$ ... what is your definition of the word "contractible"? $\endgroup$ – Zev Chonoles Aug 15 '16 at 5:05
  • $\begingroup$ @ZevChonoles A space having the homotopy type of a point. $\endgroup$ – user338393 Aug 15 '16 at 5:06
  • $\begingroup$ Maybe think about $X= S^1$. $\endgroup$ – Justin Young Aug 23 '16 at 17:31
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(Instead of talking about "identity map not null-homotopic" $\implies$ "not contractible", I'll use the contrapositive.)

If $X$ is contractible, which in your definition means it has the homotopy type of a point, then letting $Y$ be a topological space with one point, there are continuous maps $f:X\to Y$ and $g: Y\to X$ such that $g\circ f\simeq \mathrm{id}_X$ and $f\circ g\simeq \mathrm{id}_Y$.

The map $g\circ f:X\to X$ maps all of $X$ to a single point of $X$. Thus the fact that $g\circ f\simeq \mathrm{id}_X$ says that the identity map of $X$ is null-homotopic.

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