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Transactions of the American Mathematical Society

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Cyclic Sullivan-de Rham forms


Author: Christopher Allday
Journal: Trans. Amer. Math. Soc. 347 (1995), 3971-3982
MSC: Primary 55N91; Secondary 18G60, 55P62
DOI: https://doi.org/10.1090/S0002-9947-1995-1316843-9
MathSciNet review: 1316843
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Abstract: For a simplicial set $ X$ the Sullivan-de Rham forms are defined to be the simplicial morphisms from $ X$ to a simplicial rational commutative graded differential algebra (cgda)$ \nabla $. However $ \nabla $ is a cyclic cgda in a standard way. And so, when $ X$ is a cyclic set, one has a cgda of cyclic morphisms from $ X$ to $ \nabla $. It is shown here that the homology of this cgda is naturally isomorphic to the rational cohomology of the orbit space of the geometric realization $ \left\vert X \right\vert$ with its standard circle action. In addition, a cyclic cgda $ \nabla C$ is introduced; and it is shown that the homology of the cgda of cyclic morphisms from $ X$ to $ \nabla C$ is naturally isomorphic to the rational equivariant (Borel construction) cohomology of $ \left\vert X \right\vert$.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1316843-9
Article copyright: © Copyright 1995 American Mathematical Society

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