Monoids of moduli spaces of manifolds
Monoids of moduli spaces of manifolds
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流形模空间的幺半群
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发表时间：
2009
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通讯作者：
O. Randal
中科院分区：
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作者：
Søren Galatius;O. Randal
我们从Tillmann和Galatius-Madsen-Tillmann-Weiss的角度研究了d维配边范畴。有一个闭光滑（d-1）-流形和光滑d维协边的范畴C_\theta$，它具有由纤维化$\theta：X \to BO（d）$指定的广义方向。GMTW的主要结果是确定了分类空间BC_\theta$的同伦类型。本文的目的是系统地研究$C_\theta$的子范畴$D$具有与$C_\theta$等价的分类空间同伦，且D$越小越好。
我们证明，在大多数情况下的利益，$D$可以选择是同伦交换幺半群。由此证明了具有$\theta$-结构的曲面的模空间的稳定上同调是某个Thom谱的无限圈空间的上同调.这是已知的某些特殊的$\theta$，使用同调稳定性的结果，我们的工作是独立的，这样的结果，并涵盖了更多的情况。
We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $\theta : X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_\theta$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_\theta$ having classifying space homotopy equivalent to that of $C_\theta$, the smaller such $D$ the better.
We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $\theta$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.