Ivan Shishkin, Rye (1878)

Problems/TopologyExerciseUnreviewed

A simple example of homotopy

by Ancient Tree·
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Are the maps f,g:R→Rf,g:\R \rightarrow \R given by f(x)=x2f(x)=x^{2} and g(x)=(x+1)2g(x)=(x+1)^{2} homotopic?

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The graph of f (green) and g (red).

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Solution by Ancient Tree

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Yes. Here, the most natural homotopy between ff and gg is simply the map H(t,x)=(x+t)2H(t,x)=(x+t)^{2}.
It is continuous, and it verifies H(0,x)=x2=f(x)H(0,x)=x^{2}=f(x), while H(1,x)=(x+1)2=g(x)H(1,x)=(x+1)^{2}=g(x).

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