论文标题
捆绑家庭的特征类
Characteristic classes for families of bundles
论文作者
论文摘要
带有纤维$ m $的歧管捆绑包的广义米勒 - 莫里塔 - 穆尔福德类仅取决于基本的$τ_m$纤维纤维，这意味着由纤维的切线束形成的矢量捆绑包。这激发了对$τ_m$纤维的分类空间的仔细研究，$ baut（τ_m）$及其共同体戒指，即$τ_m$ - 纤维的特征类别的环。对于简单连接的Poincaré二元空间上的捆绑$ξ$，我们为通用定向$ξ$纤维纤维构建了一个相对的Sullivan模型，以及在其总空间上的特征性类别的显式共生代表。这产生了用于计算$ BAUT（ξ）$的理性共同体学环以及广义Miller-Morita-Mumford类产生的子行为的工具。为了说明，我们对球体和复杂的投影空间进行了样本计算。我们讨论了简单连接的流形的重言式环的应用，以及决定给定的$τ_m$ - 核对是否来自歧管捆绑包的问题。
The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $τ_M$-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for $τ_M$-fibrations, $Baut(τ_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $τ_M$-fibrations. For a bundle $ξ$ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable $ξ$-fibration together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of $Baut(ξ)$ as well as the subring generated by the generalized Miller-Morita-Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given $τ_M$-fibration comes from a manifold bundle.
