Simplicial cochain algebras for diffeological spaces
Simplicial cochain algebras for diffeological spaces
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DOI：
10.1016/j.indag.2020.08.002
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发表时间：
2019-02
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影响因子：
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通讯作者：
Katsuhiko Kuribayashi
中科院分区：
文献类型：
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作者：
Katsuhiko Kuribayashi
Souriau的原始de Rham上同调与代数学中的奇异上同调一般不同构。本文引入了一个奇异的de Rham复形，它被赋予到奇异上链复形中的一个积分映射，从而给出了任意拓扑空间的de Rham定理。我们还证明了从原de Rham复形到新的de Rham复形的因子映射是流形的拟同构，更一般地说，是具有奇点的空间的拟同构。此外，陈的迭代积分是在微分框架中考虑的。由此，我们推出了单连通拓扑空间的原de Rham复形的杆复形与拓扑自由圈空间的奇异de Rham复形是拟同构的，只要其下拓扑空间的因子映射是拟同构的.证明该断言的过程产生了代数学中的Leray-Serre谱序列和Eilenberg-Moore谱序列。
The original de Rham cohomology due to Souriau and the singular cohomology indiffeologyare not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem foreverydiffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen’s iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray–Serre spectral sequence and the Eilenberg–Moore spectral sequence in diffeology.