Torsion In Homology Research Articles

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

Homology torsion growth of finitely presented pro-p groups

We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-[Formula: see text] groups can grow arbitrarily fast. By way of contrast in [Formula: see text]-adic analytic groups the torsion growth is at most polynomial.

On torsion in homology of finite connected quandles

It is known that the order of a finite quandle annihilates its reduced quandle homology and the torsion subgroup of its rack homology if the quandle is quasigroup. However, this does not hold in general if a quandle is connected. In this paper, we prove that under a certain condition, the reduced quandle homology and the torsion subgroup of the rack homology of a connected quandle are annihilated by the order of the inner automorphism group of the quandle.

Torsion in Khovanov homology of homologically thin knots

We prove that every [Formula: see text]H-thin link has no [Formula: see text]-torsion for [Formula: see text] in its Khovanov homology. Together with previous results by Lee [The support of the Khovanov’s invariants for alternating knots, preprint (2002), arXiv:math.GT/0201105; An endomorphism of the Khovanov invariant, Adv. Math. 197(2) (2005) 554–586, arXiv:math.GT/0210213] and the author [Torsion of Khovanov homology, Fund. Math. 225 (2014) 343–364, arXiv:math.GT/0405474], this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over [Formula: see text]. We conjecture that a similar relation exists between the corresponding spectral sequences.

Mod p and torsion homology growth in nonpositive curvature

We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and L\"uck's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.

Examples of Homological Torsion and Volume Growth

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular planar lattice is given by the volume of a hyperbolic polyhedron.

Torsion in the magnitude homology of graphs

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.

Torsion in thin regions of Khovanov homology

AbstractIn the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is, links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only $\mathbb {Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported on two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb {Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of three-braids, strictly containing all three-strand torus links, thus giving a partial answer to Sazdanović and Przytycki’s conjecture that three-braids have only $\mathbb {Z}_2$ torsion in Khovanov homology. We use these computations and our main theorem to obtain the integral Khovanov homology for all links in this family.

Knot cobordisms, bridge index, and torsion in Floer homology

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for T p , q and T p , q # T ¯ p , q , respectively. We also show that the bridge index of T p , q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show that knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

Analytic, Reidemeister and homological torsion for congruence three–manifolds

Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplicity results for the irreducible components of the space of cuspidal forms.

Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic $3$-manifolds

In this paper, it is shown that every closed hyperbolic $3$-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2, \mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth. However, the constructed sublattices are not normal in general.

Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes

ABSTRACTWe study torsion in homology of the random d-complex Y ∼ Yd(n, p) experimentally. Our experiments suggest that there is almost always a moment in the process, where there is an enormous burst of torsion in homology Hd − 1(Y). This moment seems to coincide with the phase transition studied in [Aronshtam and Linial 13, Linial and Peled 16, Linial and Peled 17], where cycles in Hd(Y) first appear with high probability.Our main study is the limiting distribution on the q-part of the torsion subgroup of Hd − 1(Y) for small primes q. We find strong evidence for a limiting Cohen–Lenstra distribution, where the probability that the q-part is isomorphic to a given q-group H is inversely proportional to the order of the automorphism group |Aut(H)|.We also study the torsion in homology of the uniform random -acyclic 2-complex. This model is analogous to a uniform spanning tree on a complete graph, but more complicated topologically since Kalai showed that the expected order of the torsion group is exponentially large in n2 [Kalai 83]. We give experimental evidence that in this model also, the torsion is Cohen–Lenstra distributed in the limit.

Equivalence of two definitions of set-theoretic Yang–Baxter homology and general Yang–Baxter homology

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

Small Simplicial Complexes with Prescribed Torsion in Homology

For $$d \ge 2$$ and G a finite abelian group, define $$T_d(G)$$ to be the minimum number of vertices n so that there exists a simplicial complex X on n vertices which has the torsion part of $$H_{d - 1}(X)$$ isomorphic to G. Here we use the probabilistic method, in particular the Lovasz Local Lemma, to establish an upper bound on $$T_d(G)$$ which matches the known lower bound up to a constant factor. That is, we prove that for every $$d \ge 2$$ there exist constants $$c_d$$ and $$C_d$$ so that for any finite abelian group $$\begin{aligned} c_d(\log |G|)^{1/d} \le T_d(G) \le C_d(\log |G|)^{1/d}. \end{aligned}$$

Samelson products in quasi-$p$-regular exceptional Lie groups

There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samelson products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular.

Growth of homology torsion in finite coverings and hyperbolic volume

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

Rank, combinatorial cost, and homology torsion growth in higher rank lattices

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in SL(n,R), n≥3, and SO(p,q) for some values of p,q. This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.

Chromatic homology, Khovanov homology, and torsion

In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that all torsion in the categorification of the chromatic polynomial is of order two, and hence all torsion in Khovanov homology in the gradings where the isomorphism is defined is of order two. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.

A functorial control of integral torsion in homology

We study the integral torsion of the values of strict polynomial functors defined over the integers. We interpret some classical homological invariants as values of strict polynomial functors, and thus obtain estimates of the integral torsion of these inv