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Michael Albanese
Michael Albanese
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Theorem: ''Let $EG \to BG$ be a principal bundle such that for each finite CW $X$, the transformation $\eta_X$ from $[X;BG]$ to the set of isomorphism classes of principal bundles is a bijection. Then $EG$ is weakly contractible.''

Proof: Let $P \to X$ be a principal bundle. A bundle map $P \to EG$ is the same as an equivariant map. Put the compactly generated compact-open topology on $\map_G (P;EG)$$map_G (P;EG)$. The assumption says that $map_G(P;EG)$ is nonempty and path-connected (the first property expresses the surjectivity of $\eta_X$, the second one the injectivity). If $P=X \times G$ is a trivial bundle, then $map_G (P;EG)=map(X;EG)$ and you get that $[X;EG]$ is a point for each $X$. If the set of all free homotopy classes $[S^n;X]$ is trivial, then all homotopy groups are trivial).

You see that it is enough to check the criterion for spheres; you get honest contractibility if $EG$ belongs to the class of admissible base spaces.

Theorem: ''Let $EG \to BG$ be a principal bundle such that for each finite CW $X$, the transformation $\eta_X$ from $[X;BG]$ to the set of isomorphism classes of principal bundles is a bijection. Then $EG$ is weakly contractible.''

Proof: Let $P \to X$ be a principal bundle. A bundle map $P \to EG$ is the same as an equivariant map. Put the compactly generated compact-open topology on $\map_G (P;EG)$. The assumption says that $map_G(P;EG)$ is nonempty and path-connected (the first property expresses the surjectivity of $\eta_X$, the second one the injectivity). If $P=X \times G$ is a trivial bundle, then $map_G (P;EG)=map(X;EG)$ and you get that $[X;EG]$ is a point for each $X$. If the set of all free homotopy classes $[S^n;X]$ is trivial, then all homotopy groups are trivial).

You see that it is enough to check the criterion for spheres; you get honest contractibility if $EG$ belongs to the class of admissible base spaces.

Theorem: ''Let $EG \to BG$ be a principal bundle such that for each finite CW $X$, the transformation $\eta_X$ from $[X;BG]$ to the set of isomorphism classes of principal bundles is a bijection. Then $EG$ is weakly contractible.''

Proof: Let $P \to X$ be a principal bundle. A bundle map $P \to EG$ is the same as an equivariant map. Put the compactly generated compact-open topology on $map_G (P;EG)$. The assumption says that $map_G(P;EG)$ is nonempty and path-connected (the first property expresses the surjectivity of $\eta_X$, the second one the injectivity). If $P=X \times G$ is a trivial bundle, then $map_G (P;EG)=map(X;EG)$ and you get that $[X;EG]$ is a point for each $X$. If the set of all free homotopy classes $[S^n;X]$ is trivial, then all homotopy groups are trivial).

You see that it is enough to check the criterion for spheres; you get honest contractibility if $EG$ belongs to the class of admissible base spaces.

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Johannes Ebert
Johannes Ebert
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Theorem: ''Let $EG \to BG$ be a principal bundle such that for each finite CW $X$, the transformation $\eta_X$ from $[X;BG]$ to the set of isomorphism classes of principal bundles is a bijection. Then $EG$ is weakly contractible.''

Proof: Let $P \to X$ be a principal bundle. A bundle map $P \to EG$ is the same as an equivariant map. Put the compactly generated compact-open topology on $\map_G (P;EG)$. The assumption says that $map_G(P;EG)$ is nonempty and path-connected (the first property expresses the surjectivity of $\eta_X$, the second one the injectivity). If $P=X \times G$ is a trivial bundle, then $map_G (P;EG)=map(X;EG)$ and you get that $[X;EG]$ is a point for each $X$. If the set of all free homotopy classes $[S^n;X]$ is trivial, then all homotopy groups are trivial).

You see that it is enough to check the criterion for spheres; you get honest contractibility if $EG$ belongs to the class of admissible base spaces.