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I am reading topology of Lie groups by Mimura and Toda and got to the part where they are beginning to compute $H^*(O(n))$, page 120.

If we let $r_m :S^m \to O(m+1)$ be the map that sends $v$ to the the reflection by the hyperplane perpendicular to $v$ or, r(v)(v')=v'-2 v. Then we can form the map $ s: (D^{m+1}_+,S^m) \xrightarrow{r_{m+1}} (O(m+2),O(m+1)) \xrightarrow{\text{proj}}(O(m+2)/O(m), S^m) $ Now the authors claim that this induces an isomorphism in cohomology. $s^*:H^{m+1}(O(m+2)/O(m), S^m) \xrightarrow{\cong} H^{m+1}(D^{m+1}_+,S^m) $ Could someone illuminate this isomorphism?

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    It is due to the fibration $S^m \to O(m+2)/O(m) \to S^{m+1}$.2012-02-02

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