Travail en cours avec M. El Haouari
Let ${\Bbb K}$ be a field of characteristic $p\geq 0$ and $S^1$ the unit circle. We prove that the $shc$-structure on a cochain algebra $(A,d_A)$ induces an associative product on the negative cyclic homology $HC_*^-A$. When the cochain algebra $(A,d_A)$ is the algebra of normalized cochains of the simply connected topological space $X$ with coefficients in ${\Bbb K}$, then $HC_*^-A$ is isomorphic as a graded algebra to $H^{-*}_{S^1}(LX;{\Bbb K})$ the $S^1$-equivariant
cohomology algebra of $LX$, the free loop space of $X$. We use the notion of $shc$-formality to compute the $S^1$-equivariant cohomology algebras of the free loop space of the complex projective space ${\Bbb CP}(n)$
when $n+1 =0$ $[p]$ , the odd sphere $S^{2n=1}$ and of the even spheres $S^{2n}$ when $p = 2$. |
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