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  "body": "\n## Summary\n\nA homotopy between two [[continuous maps|0VGFHTM75Y]] $f, g \\colon X \\to Y$ is a continuous deformation of $f$ into $g$, parametrised by a time $t \\in [0, 1]$.\nFormally it is a continuous map $H \\colon X \\times [0, 1] \\to Y$ with $H(\\cdot, 0) = f$ and $H(\\cdot, 1) = g$.\nHomotopy is the basic notion of algebraic topology.\nHomotopic maps induce the same maps on homology and cohomology, and homotopies that fix a base point give the same maps on homotopy groups.\n\n## Explanations\n\n### L1 human\n\nThink of $f$ as a picture drawn in $Y$, and of the homotopy as a film that moves this picture continuously until it becomes $g$.\nAt each moment $t$ the frame $H(\\cdot, t)$ is again a continuous map.\nFor example, in the plane $\\R^2$ any two maps $f, g \\colon X \\to \\R^2$ are joined by the straight-line homotopy $H(x, t) = (1 - t) f(x) + t g(x)$.\nIn a space with a hole, such as the plane without the origin, a loop around the hole cannot be moved to a constant loop.\n",
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