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I'm reading the book Geometry of Four Manifolds by Donaldson. In page 179, he mentions that:

There exist a homotopy equivalent $\Omega^n(BSU(2))\cong \Omega^{n-1}(SU(2))$. Here $\Omega^m(X)$ means the mapping space $Map(S^m, X)$ with compact open topology.

The book wrote that it is because bundles over $S^n$ are determined by clutching fuction.

I know the correspondence above give a map from $\Omega^{n-1}(SU(2))\to \Omega^n(BSU(2))$. However seems it is not directly that this map indeed a homotopy equivalent, I try to prove it but I even have no idea to prove the map is continuous, can anyone help me? Thanks!

taiat
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    Does this answer your question? – Michael Albanese Apr 07 '22 at 13:08
  • @MichaelAlbanese Thanks, it really helps, what I want can be obtained by induction procedure, we should only consider two fibration $\Omega^{n-1}(SU(2))\to P\Omega^{n-2}(SU(2))\ to \Omega^{n-2}(SU(2))$ and $\Omega^n(BSU(2))\to P\Omega^{n-1}(BSU(2))\to \Omega^{n-1}(BSU(2))$ instead of $G\to EG\to BG$ and $\Omega(BG)\to PBG\to BG$. – taiat Apr 07 '22 at 15:41

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