Cobordism categories of manifolds with Baas-Sullivan singularities, Part 2

Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part I

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectrum $\text{MT}(d)$ introduced in [arXiv:math/0605249]. We obtain an analogue of the Bockstein-Sullivan exact couple that arises between the classical bordism theories $MO$ and $MO_{P}$ on the level of cobordism categories.

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In \cite{baker-ozel}, by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In \cite{cenap-isr}, by using Quinn's Transversality Theorem \cite{Quinn}, it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisf…

Cobordism categories of manifolds with corners

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the pr…

On the homotopy type of the space of Sullivan diagrams

We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Moreover, we extend the genus stabilization maps of moduli spaces to the harmonic compactification and show that the connectivity of these maps increases with the genus a…

Manifolds with singularities accepting a metric, of positive scalar curvature

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov-Lawson Conjecture for such manifolds in the case of particular type of singularities. We give an affirmative answer when such manifolds with singularities accept a metric of positive scalar curvature in terms of the index of the Dirac operator valued in the corresponding "K-theories with singularities". The key ideas are based on the construction due to Stolz, …

图 cobordisms 的字符串拓扑

String topology and graph cobordisms

本文通过图 cobordisms 构造了字符串操作，并利用图场论和开放-封闭场论扩展了 R-定向 M-纤维的特征类研究。