论文标题
一维bordism类别的分类空间和空间TC的COBORDISM模型
The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces
论文作者
论文摘要
Bordism类别的同型类别$ HBORD_D $作为对象封闭的$（D-1）$ - 流形和形态差异差异类别为$ d $ d $ dimensional Bordisms。使用新的光纤序列，用于bordism类别，我们计算$ hbord_d $的分类空间，$ d = 1 $，将其显示为圆形捆绑包，超过$ \ mathbb {cp}^\ infty _ infty _ { - 1} $。 作为证据的一部分，我们构建了删除圆的COBORDISM类别的商$ BORD_1^{RED} $。我们表明，此类别对空间进行分类$ω^{\ infty-2} \ mathbb {cp}^\ infty _ { - 1} $，此外，如果一个人将这些界限用映射用映射为单独的$ x $，则是$ bord_1^^{hom osology to tos y tos y tos y tos y tos y tos y to tos in tupistians。 $ tc（\ mathbb {s} [ωx]）$。 在本文的第二部分中，我们构建了一个无限的循环空间图$ b（hbord_1^{red}）\ to q（σ^2 \ mathbb {cp}^\ infty _+）$在此模型中使用它，并将其用于派生的组合公式，用于$ bord_1^$ nill y millair comprational cocycles in $ bord_1^um} $ {red} $ {red mill}， \ in H^{2i+2}（（b（hbord_1）; \ mathbb {q}）$。
The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $Ω^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bord_1^{red}(X)$ can be thought of as a cobordism model for the topological cyclic homology $TC(\mathbb{S}[ΩX])$. In the second part of the paper we construct an infinite loop space map $B(hBord_1^{red}) \to Q(Σ^2 \mathbb{CP}^\infty_+)$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bord_1^{red}$ representing the Miller-Morita-Mumford classes $κ_i \in H^{2i+2}((B(hBord_1); \mathbb{Q})$.
