相关论文: Cobordism categories of manifolds with Baas-Sulliv…

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…

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…

Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the…

Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of…

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…

In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in order to formalize the concept of field theories. Our main result identifies the homotopy type of the…

Let $\mathcal{A}_{g,d}$ be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either $S^1 \times I$ with one incoming and one outgoing boundary component or the surface $\Sigma_{g,d}$…

In this paper, we introduce a bordism category $\mathcal{C}_d^{PL}$ whose objects are bundles of closed $(d-1)$-dimensional piecewise linear manifolds and whose morphisms are bundles of $d$-dimensional piecewise linear cobordisms. In the…

We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's…

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…

This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose…

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve…

Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the…

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify…

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle…

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism…

Fix a tangential structure $\theta: B \longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\mathbf{Cob}^{\text{mf}, k}_{\theta}$, where morphisms are given by…

In this paper we study the topology of the cobordism category of open and closed strings. This is a 2-category in which the objects are compact one-manifolds whose boundary components are labeled by an indexing set (the set of "D-branes"),…