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Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty_+ O(n)^\vee$ have an $A_\infty$-structure? Perhaps worth noting, since $\Sigma^\infty_+ O(n)$ is self dual, it suffices to show $\Sigma^{-n(n-1)/2}_+ O(n)$ has an $A_\infty$ structure.

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty_+ O(n)^\vee$ have an $A_\infty$-structure? Perhaps worth noting, since $\Sigma^\infty_+ O(n)$ is self dual, it suffices to show $\Sigma^{-n(n-1)/2}_+ O(n)$ has an $A_\infty$ structure.

Edit: I realized this question is very basic; the Spanier-Whitehead dual of any space is canonically $E_\infty$ via the diagonal.

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty_+ O(n)^\vee$ have an $A_\infty$-structure? Perhaps worth noting, since $\Sigma^\infty_+ O(n)$ is self dual, it suffices to show $\Sigma^{-n(n-1)/2}_+ O(n)$ has an $A_\infty$ structure.

Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty_+ O(n)^\vee$ have an $A_\infty$-structure? Perhaps worth noting, since $\Sigma^\infty_+ O(n)$ is self dual, it suffices to show $\Sigma^{-n(n-1)/2}_+ O(n)$ has an $A_\infty$ structure.