相关论文: Exact triangles in Seiberg-Witten Floer theory. Pa…
We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut-Lott. Then we consider the direct image of a…
We adapt and generalise results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this…
We give a geometric setup in which the connecting homomorphism in the localization long exact sequence for Witt groups decomposes as the pull-back to the exceptional fiber of a suitable blow-up followed by a push-forward.
We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology…
Extriangulated categories axiomatize extension-closed subcategories of triangulated categories. We show that the homotopy category of an exact quasi-category can be equipped with a natural extriangulated structure.
Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex $M_n$, which is the simplicial complex of matchings in the complete graph $K_n$.…
We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres.…
We introduce and study an abstract class of semirings, which we call exact semirings, defined by a Hahn-Banach-type separation property on modules. Our motivation comes from the tropical semiring, and in particular a desire to understand…
The aim of the current paper is to introduce a new class of contractive mappings, which are contracting (a feature of) triangles. We prove that maps contracting triangles are continuous and give the fixed point result for such mappings. We…
Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and…
We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant…
Let $f \colon X \to Y$ be a morphism of concentrated schemes. We characterize $f$-perfect complexes $\mathcal{E}$ as those such that the functor $\mathcal{E} \otimes^{\mathbf{L}}_X \mathbf{L} f^*-$ preserves bounded complexes. We prove, as…
We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3-manifolds and answer questions of Reid-Wang-Zhou and Agol-Liu.
We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the…
For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological…
This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification…
Given an abelian category, we characterize the long exact sequences of length six which can be obtained from the snake lemma. Equivalently, these are the long exact sequences which arise as the homology of a triangle in the corresponding…
In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer…
In this paper, we introduce the first and third cohomology groups on Leibniz triple systems, which can be applied to extension theory and $1$-parameter formal deformation theory. Specifically, we investigate the central extension theory for…
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…