arXiv research

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many…

Study homology growth in nonpositive curvature spaces, finding examples of torsion.

We give a complete calculation of the infinity flavor of Heegaard Floer homology with mod 2 coefficients for all three-manifolds and torsion Spin^c structures. The computation agrees with the conjectured calculation of Ozsvath and Szabo. This therefore establishes an isomorphism with Mark's cup homology mod 2.

New operator reveals unique features of sl(N) link homology.

We show that the mod 2 Seiberg-Witten invariant can be determined for a spin manifold X which has the same homology groups as the 4-torus. The value depends on the structure of the cohomology ring of X, and in particular on the 4-fold cup product on H^1(X). We also consider some examples of homology tori.

We define Khovanov homology mod 2 for graph-links.

Smooth approximation of integral cycles mod 2 in Riemannian manifolds.

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

Ozsváth, Rasmussen and Szabó constructed odd Khovanov homology. It is a link invariant which has the same reduction modulo 2 as (even) Khovanov homology. Szabó introduced a spectral sequence with mod 2 coefficients from mod 2 Khovanov homology to another link homology. He got his spectral sequence from a chain complex …

New results show area-minimizing surfaces have fewer singularities than expected.

This paper introduces a new method to create mod m almost classical links from virtual knots.

Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group of automorphisms on the first homology group with $\mathbb Z_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\c…

Ozsvath, Rasmussen and Szabo constructed odd Khovanov homology. It is a link invariant which has the same reduction modulo 2 as (even) Khovanov homology. Szabo introduced a spectral sequence with mod 2 coefficients from mod 2 Khovanov homology to another link homology. He got his spectral sequence from a chain complex …

Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.

Aramayona and Leininger have provided a "finite rigid subset" $\mathfrak{X}(Σ)$ of the curve complex $\mathscr{C}(Σ)$ of a surface $Σ= Σ^n_g$, characterized by the fact that any simplicial injection $\mathfrak{X}(Σ) \to \mathscr{C}(Σ)$ is induced by a unique element of the mapping class group $\mathrm{Mod}(Σ)$. In this…

We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorp…

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes…

The study explores how the mapping class group acts on the homology of surface covers.

We construct an invariant $z (M) =1+a_1(A^4-1)+ a_2(A^4-1)^2+a_3(A^4-1)^3 + \cdots \in \mathbb{Q} [[A^4-1]]= \mathbb{Q} [[A+1]]$ for an integral homology $3$-sphere $M$ using a completed skein algebra and a Heegaard splitting. The invariant $z(M)\mathrm{mod} ((A+1)^{n+1})$ is a finite type invariant of order $n$. In p…

Hasse principle applied to area-minimizing submanifolds across different homology types.

This paper defines a functor for $L_\infty$-modules and applies it to Khovanov homology.

We show that the rational Novikov conjecture for a group $Γ$ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an E$Γ$. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjectu…

We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We a…

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".

To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the sp…

Study on torsion in homology of Torelli group for surfaces.

Let $\text{Man}_{*}$ denote the category of closed, connected, oriented and based $3$-manifolds, with basepoint preserving diffeomorphisms between them. Juhász, Thurston and Zemke showed that the Heegaard Floer invariants are natural with respect to diffeomorphisms, in the sense that there are functors $HF^{\circ}: \te…

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure…

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of a 3-manifold with the integral homology of a torus, given by counting projectively flat connections. We show that its mod 2 evaluation is given by the triple cup prod…

Homological model for quantum representations of mapping class groups.

Torsion found in knot homology, challenging augmentation theories.

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, incl…

Any continuous action of SL(n,Z), where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n - 1. In particular, any continuous action of SL(n+2,Z) on the n-dimensional sphere factors through a finite group action.

We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of…

The paper identifies manifolds with free circle actions.

We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces sug…

The paper studies the structure of a specific homology group related to mapping class groups.

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the act…

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces…

New homology $q_2$ from 3D cobordism to integers.

Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $Σ_g$ on the lower central series of $π_1(Σ_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a …

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there…

In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are $(1, 1)$ knots and, therefore, admit a genus one Heeg…

The paper finds a new lower bound on the genus of surfaces in indefinite 4-manifolds.

We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V = V_{2,1}$, we can deduce that t…

Paper extends gauge theory results to homology tori, preserving spin structure obstructions.

By calculating the Fermat limit of certain q-Fermat functions, we get explicit surgery formulae for the second and third Ohtsuki invariants for homology 3-spheres. The surgery formula of the second Ohtsuki invariant λ_2, combined with an argument using the general theory of finite type invariants of homology 3-spheres,…