Related papers: Independence of Iterated Whitehead Doubles
We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an…
We present three large families of new examples of plumbed 3-manifolds that bound rational homology 4-balls. These are constructed using two operations, also defined here, that preserve the lack of a lattice embedding obstruction to…
By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp(S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in…
We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite…
A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question…
A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in…
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives…
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated…
The notion of a (stably) decomposable fiber bundle is introduced. In low dimensions, for torus fiber bundles over a circle the notion translates into a property of elements of the special linear group of integral matrices. We give a…
I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that…
We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U,…
The purpose of this article is to show that the bivariant algebraic $A$-cobordism groups considered previously by the author are independent of the chosen base ring $A$. This result is proven by analyzing the bivariant ideal generated by…
We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families…
This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(\pi_1(E))) and, provided that this obstruction vanishes and one…
Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group $\hat{\mathcal{C}}_{\mathbb{Z}}$ (resp. $\mathcal{C}_{\mathbb{Z}}$) was previously defined as the set of knots in homology spheres that…
If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.
Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted.…
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be…
In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The…
We study torus fibrations over the 2-sphere and Hurwitz equivalence of their monodromies. We show that, if two torus fibrations over $S^2$ have the same type of singularities, then their global monodromies are Hurwitz equivalent after…