A comparison between two de Rham complexes in diffeology

Simplicial cochain algebras for diffeological spaces

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are 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 for every diffeological 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 consequ…

Smooth singular complexes and diffeological principal bundles

In previous papers, we used the standard simplices $Δ^p$ $(p\ge 0)$ endowed with diffeologies having several good properties to introduce the singular complex $S^\dcal(X)$ of a diffeological space $X$. On the other hand, Hector and Christensen-Wu used the standard simplices $Δ^p_{\rm sub}$ $(p\ge 0)$ endowed with the sub-diffeology of $\rbb^{p+1}$ and the standard affine $p$-spaces $\abb^p$ $(p\ge 0)$ to introduce the singular complexes $S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$, respectively, of a diffeological space $X$. In this paper, we prove that $S^\dcal(X)$ is a fibrant approxima…

Foliated Stratified Spaces and a De Rham Complex Describing Intersection Space Cohomology

The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with …

扭曲Morse复形

Twisted Morse complexes

本文研究闭有限维光滑流形上的扭曲Morse同调与上同调，证明相关定理与多类同调/上同调的同构关系，并通过多个计算实例验证了扭曲Morse复形的有效性。

广义$n$-级数与de Rham复形

Generalized $n$-series and de Rham complexes

本文研究了交换环上广义$n$-级数的代数与组合性质，证明了经典组合恒等式的$s$-类似物，并定义了导数的$s$-类似物以证明仿射直线上的Poincaré引理与Cartier同构。

混合维与等维建模的抽象逼近工具

An abstract approximation tool for mixed-dimensional and equidimensional modeling

本文构造重叠开覆盖和有界上链映射，建立单纯形de Rham复形与Čech-de Rham复形子复形间的同构，作为比较混合维与等维问题的抽象逼近工具。