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I did not find a direct definition so I will give the context :

$X$ is compact manifold with boundary. There is no smooth map $f:X\rightarrow \partial X$ that leaves $\partial X$ pointwise fixed.

Here what is meant by pointwise fixed?
I found that fixed means $f(x)=x$ so pointwise means this must be true for all $x$? I am not sure.

mathemather
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1 Answers1

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It means that every individual $x \in \partial X$ is fixed by $f$, i.e. $f(x) = x$ for all $x \in \partial X$.

In contrast, fixing $\partial X$ "setwise" would mean that $f(x) \in \partial X$ for all $x \in \partial X$.