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Keywords:
string topology; algebraic topology; operads; free loop space
string topology; algebraic topology; operads; free loop space
Summary:
We show that the $E_1$-equivalence $C^{\bullet}(S^{2}) \simeq H^{\bullet}(S^{2})$ does not intertwine the inclusion of constant loops into the free loop space $S^2 \rightarrow LS^2$. That is, the isomorphism $HH_{\bullet}(H^{\bullet}(S^2)) \simeq H^{\bullet}(LS^2)$ does not preserve the obvious maps to $H^{\bullet}(S^{2})$ that exist on both sides. We give an explicit computation of the defect in terms of the $E_{\infty}$-structure on $C^{\bullet}(S^{2})$. Finally, we relate our calculation to recent work of Poirier-Tradler on the string topology of $S^2$.
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