An n-dimensional μ-component boundary link is a codimension 2 embedding of spheres L=\bigsqcup_μS^n \subset S^{n+2} such that there exist μdisjoint oriented embedded (n+1)-manifolds which span the components of L. An F_μ-link is a boundary link together with a cobordism class of such spanning manifolds. The F_μ-link cobordism group C_n(F_μ) is known to be trivial when n is even but not finitely generated when n is odd.
Our main result is an algorithm to decide whether two odd-dimensional F_μ-links represent the same cobordism class in C_{2q-1}(F_μ) assuming q>1. We proceed to compute the isomorphism class of C_{2q-1}(F_μ), generalizing Levine's computation of the knot cobordism group C_{2q-1}(F_1).
Our starting point is the algebraic formulation of Levine, Ko and Mio who identify C_{2q-1}(F_μ) with a surgery obstruction group, the Witt group G^{(-1)^q,μ}(Z) of μ-component Seifert matrices. We obtain a complete set of torsion-free invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to `algebraically integral' simple self-dual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties.
To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing G^{(-1)^q,μ}(Q) as an infinite direct sum of Witt groups of finite-dimensional division Q-algebras with involution. Numerical invariants of such Witt groups are available in the literature.
We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.
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 $Σ_{g,d}$ 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 Co…
There are two categorifications of the Jones polynomial: "even" discovered by M.Khovanov in 1999 and "odd" dicovered by P.Ozsvath, J.Rasmussen and Z.Szabo in 2007. The first one can be fully constructed in the category of cobordisms (strictly: in the additive closure of that category), where we can build a complex for a given tangle and show its invariance under Reidemeister moves. The even link homology is given by a monoidal functor from cobordisms into modules. However, odd link homology cannot be obtained in this way. In this paper I fill this gap. I enrich cobordisms with chronologies (pr…
We use the Blanchfield-Duval form to define complete invariants for the cobordism group C_{2q-1}(F_μ) of (2q-1)-dimensional μ-component boundary links (for q\geq2). The author solved the same problem in math.AT/0110249 via Seifert forms. Although Seifert forms are convenient in explicit computations, the Blanchfield-Duval form is more intrinsic and appears naturally in homology surgery theory. The free cover of the complement of a link is constructed by pasting together infinitely many copies of the complement of a μ-component Seifert surface. We prove that the algebraic analogue of this const…
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 construct orbifolds with quasitoric boundary and show that they have stable almost complex structure. We show that a quasitoric orbifold is complex cobordant to finite disjoint copies of complex orbifold projective spaces. Finally some computations in the complex cobordism ring for manifolds are given.
We give a necessary and sufficient condition for the orientability of a locally standard 2-torus manifold with a fixed point which generalizes previous results of Nakayama-Nishimura in 2005 and Soprunova-Sottile in 2013. We construct manifolds with boundary where the boundary is a disjoint union of locally standard 2-torus manifolds. We discuss equivariant oriented cobordism class of locally standard 2-torus manifolds.
We give a new construction of oriented manifolds having the boundary $\CC P^{2k+1}$ for each $k \geq 0$. The main tool is the theory of quasitoric manifolds.
We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.
For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism invariants rho_n for closed (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that {rho_n} is a linearly independent set and for each n, the image of rho_n is an infinitely generated and dense subset of R. In their seminal work on knot concordance, T. Cochran, K. Orr, and P. Teichner define a filtration F_(n)^m of the m-component (string) link…
By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic $H$-structure manifold with a higher-group $\mathbb{G}$, and their bordism invariants --- e.g. perturbative anomalies of chiral fermions [originated from …
In this note we classify the diffeomorphism classes rel. boundary of smooth h-cobordisms between two fixed 1-connected 4-manifolds in terms of isometries between the intersection forms.
We construct a cobordism group for embedded graphs in two different ways, first by using sequences of two basic operations, called "fusion" and "fission", which in terms of cobordisms correspond to the basic cobordisms obtained by attaching or removing a 1-handle, and the other one by using the concept of a 2-complex surface with boundary is the union of these knots. A discussion given to the question of extending Khovanov homology from links to embedded graphs, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such graph by using some…
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
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"), the 1-morphisms are cobordisms of manifolds with boundary, and the 2-morphisms are diffeomorphisms of the surface cobordisms. Our methods and techniques are direct generalizations of those used by U. Tillmann in her study of the category of closed strings. We input the striking theorem of Madsen and Weiss regarding the topology of the stable mapping class gr…
We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions, interpreted as the connectivity of a certain map, in which the cobordism class of the "linking manifold" is enough to distinguish the homotopy class of one link map from another.
(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch of the algebraic and differential topological theory of Morse functions and their higher dimensional versions, or the global singularity theory, by Kalmár, Ikegami, Sadykov, Saeki, Wrazidlo, Yamamoto etc. since 2000s. Cobordism relations are in most cases defined as the following for example; two closed manifolds of a fixed dimension or maps on t…
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery …
We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of al…
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an …
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…
