论文标题
平滑的单数复合物和差异原理束
Smooth singular complexes and diffeological principal bundles
论文作者
论文摘要
在以前的论文中，我们使用了标准简单$δ^p $ $（p \ ge 0）$，具有具有多种良好属性的差异，以引入差异空间$ x $的单一复杂$ s^\ dcal（x）$。另一方面，Hector和Christensen-Wu使用了标准简单$δ^p _ {\ rm sub} $（p \ ge 0）$，赋予了$ \ rbb^{p+1} $的子 - 散至学，以及标准的offine $ p $ p $ -spaces $ \ abb \ abb^p $ \ abb^p $（p $ ge 0） $ s^\ dcal _ {\ rm sub}（x）$和$ s^\ dcal _ {\ rm aff}（x）$分别为差异空间$ x $。在本文中，我们证明$ s^\ dcal（x）$是$ s^\ dcal _ {\ rm sub}（x）$和$ s^\ dcal _ {\ rm aff}（x）$。该结果很容易意味着$ s^\ dcal _ {\ rm sub}（x）$和$ s^\ dcal _ {\ rm aff}（x）$的同型组对$ x $的平滑同质组是同构的，证明是Chistensen和Wu的标准。此外，我们使用奇异函数$ s^\ dcal _ {\ rm aff} $来表征差异主束（即，在iglesias-zemmour的意义上是主捆）。通过使用这些结果，我们将$ \ dcal $ numbled的主束的特征类扩展到差异学主捆的特征类。
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 approximation both of $S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$. This result easily implies that the homotopy groups of $S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$ are isomorphic to the smooth homotopy groups of $X$, proving a conjecture of Christensen and Wu. Further, we characterize diffeological principal bundles (i.e., principal bundles in the sense of Iglesias-Zemmour) using the singular functor $S^\dcal_{\rm aff}$. By using these results, we extend characteristic classes for $\dcal$-numerable principal bundles to characteristic classes for diffeological principal bundles.
