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This is some sort of "follow-up" to the (unanswered) question posted here.

Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and as is shown in the above link, when $n$ is even, $$H^2(B\varphi;Z/2): H^2(BO(4k);Z/2)\rightarrow H^2(BO(4k);Z/2)$$ switches the classes $w_2$ and $w_2+w_1^2$. Since the extension $$Z/2\rightarrow Pin^{-}(4k)\rightarrow O(4k) $$ is classified by the class $w_2$ and $$Z/2\rightarrow Pin^{+}(4k)\rightarrow O(4k) $$ is classified by the class $w_2+w_1^2$ c.f. introduction of Kirby, Taylor, "A Calculation of Pin^{+}bordism groupos" Comment. Math. Helvetici 65 (1990) 434-447, we conclude that $\varphi$ pulls back the one extension to the other. In particular, as Lie groups $Pin^+(4k) $ and $Pin^-(4k) $ are isomorphic.

Now my questions are :

  • Is this isomorphism documented somewhere?
  • Is there a simple way to see this directly from the definition of $Pin$ groups using Clifford algebras?
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    $\begingroup$ The isomorphism Pin^+(4k) = Pin^-(4k) is documented in "Analysis, Manifolds and Physics. Part II" by Y. Choquet-Bruhat and C. De Witt-Morette (Elsevier, 2000). Is this what you're looking for? Note that it is not an isomorphism of groups over O(4k). (The isomorphism, as you point out, covers an interesting outer automorphism of O(4k).) $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 23, 2015 at 16:13
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    $\begingroup$ The unique automorphism of the Clifford algebra ${\rm{C}}(V,q)$ that extends negation on the underlying vector space $V$ of the quadratic space $(V,q)$ also swaps the two associated Clifford norms $\nu_q^{+}, \nu_q^{-}: {\rm{GPin}}(q) \rightarrow {\rm{GL}}_1$. Hence, it swaps their respective kernels ${\rm{Pin}}^{\pm}(q)$. (I am assuming these are the Pin groups you have in mind.) This applies in any dimension. $\endgroup$
    – grghxy
    Commented Jul 23, 2015 at 17:41

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