Cobordism Categories Research Articles

Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans

We show that the conditions in Steimle's ‘additivity theorem for cobordism categories’ can be weakened to only require locally (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.

The homotopy type of the topological cobordism category

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}MT\mathrm{Top}(d) , where MT\mathrm{Top}(d) is the Thom spectrum of the inverse of the canonical bundle over B\mathrm{Top}(d) . We also give versions for manifolds with tangential structures and/or boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius–Madsen–Tillman–Weiss.

A Lagrangian pictionary

We describe a dictionary of geometry ⟷ algebra in Lagrangian topology. As a by-product, we obtain a tautological (in a particular sense which we explain) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called surgery models.

An additivity theorem for cobordism categories

Using methods inspired from algebraic $K$-theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the B\"okstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach generalizes to other cobordism-like categories of interest. Indeed we argue that the Genauer fibration sequence is an analogue, in the setting of cobordism categories, of Waldhausen's Additivity theorem in algebraic $K$-theory.

Manifolds and Groups

The workshop concentrated on the interplay of advances in the understanding of manifolds and geometric group theory. In particular, we discussed mapping class groups and moduli spaces of manifolds (also of high dimension) and aspects of homological stability related to them; cobordism categories and their applications; L^2 -invariants, simplicial volume, and their applications; and in general, the phenomena of rigidity versus flexibility in geometry and topology.

Cobordism categories and moduli spaces of odd dimensional manifolds

We prove that the stable moduli space of (n−1)-connected, n-parallelizable, (2n+1)-dimensional manifolds is homology equivalent to an infinite loopspace for n≥4,n≠7. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.

The Circle Transfer and Cobordism Categories

AbstractThe circle transfer $Q\Sigma (LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note, we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let 𝒞1(X) denote the one-dimensional cobordism category and let Circ(X) ⊂ 𝒞1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) → Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into 𝒞1(LX) is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the two-dimensional cobordism category 𝒞2(X) and find that it is null-homotopic when X is a point.

2-Segal sets and the Waldhausen construction

It is known by results of Dyckerhoff–Kapranov and of Gálvez-Carrillo–Kock–Tonks that the output of the Waldhausen S•-construction has a unital 2-Segal structure. Here, we prove that a certain S•-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.

Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$-orientation from the infinite loop space of the Madsen--Tillmann--Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$-dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov--Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah--Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral-flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen--Weiss theorem, proven by Galatius and the third named author.

On Open and Closed Strings

Cobordism categories are highly complicated structures that can be analyzed by way of their classifying spaces. In the case of surfaces, meaning $2$-dimensional cobordisms, this has led to many important results in recent years. This paper studies the subcategory of open strings of the category of open and closed strings as introduced by Baas, Cohen and Ramírez and identifies the homotopy type of its classifying spaces.

On the Homotopy Type of Certain Cobordism Categories of Surfaces

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 $\Sigma_{g,d}$ of genus $g$ and $d$ boundary components that are all incoming. In this paper, we study the homotopy type of the classifying space of the cobordism categories $\mathcal{A}_{g,d}$ and the associated (ordinary) cobordism categories of their connected components $\mathbb{A}_d$. $\mathcal{A}_{0,2}$ is the cobordism category of complex annuli that was considered by Costello and $\mathbb{A}_2$ is homotopy equivalent with the positive boundary 1-dimensional embedded cobordism category of Galatius-Madsen-Tillmann-Weiss. We identify their homotopy type with the infinite loop spaces associated with certain Thom spectra.

Localisations of cobordism categories and invertible TFTs in dimension two

Cobordism categories have played an important role in clas- sical geometry and more recently in mathematical treatments of quantum eld theory. Here we will compute localisations of two-dimensional discrete cobordism categories. This allows us, up to equivalence, to determine the category of invertible two- dimensional topological eld theories in the sense of Atiyah. We are able to treat the orientable, non-orientable, closed and open cases.

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 ⟩ \text {Cob}_{d,\langle k\rangle } is the category whose objects are a fixed dimension d d , with corners of codimension ≤ k \leq \;k , then we identify the homotopy type of the classifying space B Cob d , ⟨ k ⟩ B\text {Cob}_{d,\langle k\rangle } as the zero space of a homotopy colimit of a certain diagram of the Thom spectra. We also identify the homotopy type of the corresponding cobordism category when an extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss (2009), and their proofs are an adaptation of the methods of their paper. As an application we describe the homotopy type of the category of open and closed strings with a background space X X , as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.

Geometry, Quantum Fields, and Strings: Categorial Aspects

Currently, in the interaction between string theory, quantum field theory and topology, there is an increased use of category-theoretic methods. Independent developments (e.g. the categorificiation of knot invariants, bundle gerbes and topological field theories on extended cobordism categories) have put higher categories in the focus.The workshop has brought together researchers working on diverse problems in which categorical ideas play a significant role.

Weak identity arrows in higher categories

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where, instead, the notion of identity arrow is weakened—these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category Δ is replaced by a certain “fat” delta of “coloured ordinals,” where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categories in general. It is explained how the definition of fair n-category is almost forced upon us by three standard ideas. The second part states some basic results about fair categories, and give examples, including Moore path spaces and cobordism categories. The category of fair 2-categories is shown to be equivalent to the category of bicategories with strict composition laws. Fair 3-categories correspond to tricategories with strict composition laws. The main motivation for the theory is Simpson's weak-unit conjecture according to which n-groupoids with strict composition laws and weak units should model all homotopy n-types. A proof of a version of this conjecture in dimension 3 is announced, obtained in joint work with A. Joyal. Technical details and a fuller treatment of the applications will appear elsewhere.

Embedded Cobordism Categories and Spaces of Submanifolds

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 manifolds, as developed by the author and Galatius, to identify the homotopy type of the cobordism category with objects (d-1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincare duality, relating a space of submanifolds of M to a space of functions on M.

Spin Cobordism Categories in Low Dimensions

The Madsen-Tillmann spectra defined by categories of three- and four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here.

A NOTE ON DUALITY, FROBENIUS ALGEBRAS AND TQFTS

Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle). A self-dual definition of a Frobenius object without the use of a prefered dual is considered. The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors. Hermitian structures are modeled as a conjugation compatible with duality. It is the structure cobordism categories posses. A definition of generalized cobordism categories is proposed.

Orbifolding Frobenius Algebras

We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super-graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi-homogeneous singularities and their symmetries.