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The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism in which the cycles carry a vector bundle on the source as additional data. We show that, over a field of characteristic 0, this extends the analogous homological theory of Lee and Pandharipande. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and employ the obtained structural results to prove a weak version of the projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce the notion of precobordism theories for quasi-projective derived schemes over an arbitrary Noetherian ring of finite Krull dimension as a reasonable class of theories where the proof can be carried out, and prove some of their basic properties. These results can be considered as the first steps towards a Levine-Morel style algebraic cobordism over a base ring that is not a field of characteristic 0. [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

We prove that the Stiefel–Whitney classes of a moment-angle manifold, not necessarily smooth, are trivial. We also consider Stiefel–Whitney classes of the partial quotient of a moment-angle manifold. [ABSTRACT FROM AUTHOR]

Quantum computing is captured in the formalism of the monoidal subcategory of Vect C generated by C 2 — in particular, quantum circuits are diagrams in Vect C — while topological quantum field theories, in the sense of Atiyah, are diagrams in Vect C indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call F Vect C. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to F Vect C , which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category. [ABSTRACT FROM AUTHOR]

In this paper, we study the concordance and cobordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere. [ABSTRACT FROM AUTHOR]

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functorZWB ! yA, where B is the category of bottom tangles in handlebodies and yA is the degree-completion of the category A of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, Ais the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor Z induces a canonical isomorphism grB Š A, where grB is the associated graded of the Vassiliev-Goussarov filtration on B. To each Drinfeld associator ' we associate a ribbon quasi-Hopf algebra H' in yA, and we prove that the braided Hopf algebra resulting from H' by "transmutation" is precisely the image by Z of a canonical Hopf algebra in the braided category B. Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the nonnegative scalar curvature (NNSC)-cobordism of Bartnik data (∑ 1 n − 1 , γ 1 , H 1) and (∑ 2 n − 1 , γ 2 , H 2) . We prove that given two metrics γ1 and γ2 on Sn−1 (3 ⩽ n ⩽ 7) with H1 fixed, then (Sn−1, γ1, H1) and (Sn−1, γ2, H2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature ∫ S 2 H 2 d μ γ 2 is large enough rules out NNSC-cobordisms (see Theorem 1.2); if we require the Gaussian curvature of γ2 to be positive, we get a criterion for nonexistence of the trivial NNSC-cobordism by using the Hawking mass and the Brown-York mass (see Theorem 1.1). For the general topology case, we prove that (Σ 1 n − 1 , γ 1 , 0) and (Σ 2 n − 1 , γ 2 , H 2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.5). [ABSTRACT FROM AUTHOR]

In this paper we study smooth orientation-preserving free actions of the cyclic group ℤ/m on a class of (n −1)-connected 2n-manifolds, ♯♯g(Sn × Sn)Σ, where Σ is a homotopy 2n-sphere. When n = 2, we obtain a classification up to topological conjugation. When n = 3, we obtain a classification up to smooth conjugation. When n ≥ 4, we obtain a classification up to smooth conjugation when the prime factors of m are larger than a constant C(n). [ABSTRACT FROM AUTHOR]

Our goal is to systematically compute the ℂ P 2 -genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the ℂ P 2 -genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to ℂ P 2 . There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the ℂ P 2 -genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the ℂ P 2 -genus of all but 6 of these knots, where the ℂ P 2 -genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the ℂ P 2 -genus of each knot differs from that of its mirror. [ABSTRACT FROM AUTHOR]

In this note we show that there exist cusped hyperbolic 3-manifolds that embed geodesically but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds and by Kolpakov, Reid, and Slavich on embedding arithmetic hyperbolic manifolds. [ABSTRACT FROM AUTHOR]

We describe an effective method for simultaneously computing d -invariants of infinite families of Brieskorn spheres Σ (p , q , r) with p q + p r − q r = 1. [ABSTRACT FROM AUTHOR]

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces. [ABSTRACT FROM AUTHOR]

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical SU (2) -frame. To start we notice that the set of homotopy classes of SU (2) -frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a Z -torsor. Then we define the E ^ -invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The E ^ -invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities. [ABSTRACT FROM AUTHOR]

We compute the homology of the spaces in the Omega spectrum for BoP. There is no torsion in H∗(BoP i) for i⩾2, and things are only slightly more complicated for i<2. We find the complete homotopy type of BoP----i for i⩽6 and conjecture the homotopy type for i>6. This completes the computation of all H∗(MSU-----∗). [ABSTRACT FROM AUTHOR]

We show that the number of isometry classes of cusped hyperbolic 3-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings. [ABSTRACT FROM AUTHOR]

We show that two Pin-structures on a surface differ by a diffeomorphism of the surface if and only if they are cobordant (for comparison, the analogous fact has already been shown for Spin-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of Pin-structures on a surface in a given cobordism class. [ABSTRACT FROM AUTHOR]

We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration. [ABSTRACT FROM AUTHOR]

Let M be a 2n-dimensional closed unitary manifold with a Tn−1-action with only isolated fixed points. In this paper, we first prove that the equivariant cobordism class of a unitary Tn−1-manifold M is just determined by the equivariant Chern numbers c ω T n − 1 [M], where ω = (i1, i2,..., i6) are the multi-indexes for all i1, i2,..., i6 ∈ ℕ. Then we show that if M does not bound equivariantly, then the number of fixed points is greater than or equal to ⌈n/6⌉ + 1, where ⌈n/6⌉ denotes the minimum integer no less than n/6. [ABSTRACT FROM AUTHOR]

The 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 S 1 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. [ABSTRACT FROM AUTHOR]

The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure. [ABSTRACT FROM AUTHOR]

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner. [ABSTRACT FROM AUTHOR]

Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension d = 1, all the spheres are commutative Frobenius objects in categories whose arrows are (d + 1)-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres-they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer [ABSTRACT FROM AUTHOR]

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold becoming diffeomorphic to after a single stabilization by connected summing with or , and almost complete decomposability of for many almost completely decomposable , such as the elliptic surfaces. [ABSTRACT FROM AUTHOR]

In this paper we show that for m > n the set of cobordism classes of maps from m -sphere to n -sphere is trivial. The determination of the cobordism homotopy groups of spheres admits applications to the covers for spheres. [ABSTRACT FROM AUTHOR]

We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of k-fold simple branched coverings between n-manifolds form an abelian group $${{\rm Cob}^1(n, k)}$$ . Moreover, $${{\rm Cob}^1(*, k) = \bigoplus_{n = 0}^{\infty}{\rm Cob}^1(n, k)}$$ is a module over $${\Omega^{SO}_{*}}$$ . We construct a universal k-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups $${{\rm Cob}^1(n, k)}$$ . In the case n = 2 we compute the group $${{\rm Cob}^1(2, k)}$$ , give a complete set of invariants and construct generators. [ABSTRACT FROM AUTHOR]