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  "body": "\n## Statement\n\nLet $X$ and $Y$ be topological spaces and let $f, g \\colon X \\to Y$ be continuous maps that are [[homotopic|F2F143KQ2M]].\nThen the induced maps on [[singular homology|M0Q5CFJ7ZH]] are equal: $f_* = g_* \\colon H_n(X) \\to H_n(Y)$ for every $n \\ge 0$.\nIn particular, a [[homotopy equivalence|XZG6MTSWCN]] $f \\colon X \\to Y$ induces isomorphisms $f_* \\colon H_n(X) \\to H_n(Y)$ for every $n$.\n\n## Remarks\n\nThe proof cuts the prism $\\Delta^n \\times [0, 1]$ into $(n+1)$-simplices.\nA homotopy $H \\colon X \\times [0,1] \\to Y$ from $f$ to $g$ then gives a prism operator $P \\colon C_n(X) \\to C_{n+1}(Y)$ with $\\partial P + P \\partial = g_\\# - f_\\#$.\nSo $f_\\#$ and $g_\\#$ are chain homotopic, and chain homotopic maps induce the same map on homology.\nThe second claim follows by applying the first to $g \\circ f \\simeq \\id_X$ and $f \\circ g \\simeq \\id_Y$.\n\nExample: a [[contractible space|R5WP8Z69Q8]] $X$, such as $\\R^n$ or a convex set, is homotopy equivalent to a point.\nHence $H_0(X) \\cong \\Z$ and $H_n(X) = 0$ for $n \\ge 1$.\nThe same holds with any coefficients and for relative homology of pairs, with homotopies of pairs.\n\n## Explanations\n\n### L1 human\n\nHomology does not change when a space or a map is deformed continuously.\nTwo spaces that can be squeezed into each other without tearing, such as a thick annulus and a circle, have the same homology.\nThis is what makes homology a tool for shapes up to deformation rather than for exact geometry.\n\n### L2 human\n\nThis is the homotopy axiom of the Eilenberg–Steenrod axioms for singular homology.\nIt lets one replace a space by a simpler homotopy equivalent one, for example a deformation retract, before computing.\nIt also shows that singular homology factors through the homotopy category of spaces.\nTogether with excision, the long exact sequence of a pair and the homology of a point, it is enough to compute the homology of spheres and of CW complexes.\n",
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