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  "body": "\n## Statement\n\nLet $G$ be a finite group, let $BG$ be its [[classifying space|7XRBR7AV3C]], and let $X$ be a finite-dimensional [[CW complex|5W95QFHXPR]] with a basepoint.\nThen the space $\\operatorname{Map}_*(BG, X)$ of [[pointed|T9DG8DDE75]] continuous maps from $BG$ to $X$ is weakly contractible.\nThat is, the map from it to a point is a [[weak homotopy equivalence|4Q3WY1RX91]].\n\n## Remarks\n\nThis form was proved by Haynes Miller in 1984, and is often called Miller's theorem.\nThe proof uses the category of unstable modules over the Steenrod algebra: one shows that the relevant Ext groups vanish and uses the unstable Adams spectral sequence.\nThe hypothesis that $X$ is finite-dimensional cannot be dropped: if $G \\ne 1$, the identity map of $BG$ is not null-homotopic, since $\\pi_1(BG) \\cong G$ is nontrivial.\n\nAt work: every map from $B\\Z/p$ to a finite-dimensional complex, such as a sphere $S^n$, is null-homotopic.\nSullivan's conjecture also has a fixed-point form, proved independently by Carlsson, by Lannes, and by Dwyer, Miller and Neisendorfer.\nIt says that, for a finite $p$-group acting on a finite complex, the fixed points agree with the homotopy fixed points after $p$-completion, under suitable hypotheses.\n\n## Explanations\n\n### L1 human\n\nThe classifying space of a finite group is an infinite-dimensional space with complicated cohomology.\nThe theorem says that it cannot be mapped in an essential way into any finite-dimensional complex.\nEvery pointed map can be deformed to the constant map, and in a coherent way.\nSo finite groups are invisible to finite complexes in this homotopical sense.\n\n### L2 human\n\nMiller's theorem was the starting point of the theory of Lannes' $T$ functor and unstable modules over the Steenrod algebra.\nThese methods give a homotopy-theoretic description of maps out of $BG$ in many cases.\nIts fixed-point form compares the fixed points $X^G$ of a group action with the homotopy fixed points $X^{hG}$, and is parallel to the [[Segal conjecture|DE29ZC399Z]] for spectra.\n",
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