Narrow Results

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$-manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) ${\pi }_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2$, $3$ or $4$, or (b) $p=2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2$, $3$ or $4$, then $\mathrm{\text{dim}}{H}_1(M;F_p)<157.763\cdot \mathrm{\text{vol}}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that ${\pi }_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g=2,\dots ,8$. These results should be compared with those of our previous paper “The ratio of homology rank to hyperbolic volume, I,” in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by this paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of ${\pi }_1(M)$ in terms of $\mathrm{\text{vol}}M$, assuming that either ${\pi }_1(M)$ is $9$-free, or $M$ is closed and ${\pi }_1(M)$ is $5$-free.

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree $d$ and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces ${{𝒫}}_d^{c\mathrm{\Theta }}$ of real monic univariate polynomials of degree $d$ whose real divisors avoid sequences of root multiplicities, taken from a given poset $\mathrm{\Theta }$ of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of ${{𝒫}}_d^{c\mathrm{\Theta }}$ and of some related topological spaces. We find explicit presentations for the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in $d$. We also show that ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ stabilizes for $d$ large. The mechanism that generates ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree $d$ polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder $ℝ\times S^1$, whose images avoid the tangency patterns from $\mathrm{\Theta }$ with respect to the generators of the cylinder.

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin^c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

We study fundamental groups related with log-terminal singularities, and show that fundamental groups are preserved by a resolution of singularities. As corollaries, we show that fundamental groups are invariant under various fundamental operations in the Minimal Model Program, for example, by contractions of extremal rays, by flips, by pluricanonical morphisms of minimal varieties of general type.

In this note, we will show that the fundamental group of any negatively δ-pinched manifold cannot be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F_4(-20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for δ-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in SO(n,1)(n≥3) is the fundamental group of a quasi-compact Kähler manifold [21].

We consider the class of quasiprojective varieties admitting a dominant morphism onto a curve with negative Euler characteristic. The existence of such a morphism is a property of the fundamental group. We show that for a variety in this class the number of maps onto a hyperbolic curve or surfaces can be estimated in terms of the numerical invariants of the fundamental group. We use this estimates to find the number of biholomorphic automorphisms of complements to some arrangements of lines.

We construct numerous continuous families of irreducible subfactors of the hyperfinite II₁ factor which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II₁ factor with Jones index 6, which have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II₁ factor.

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link L_p/q of two components modulo the (A + 1)-torsion.

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

We will prove that given a genus-2 fibration f : X → C on a smooth projective surface X such that b₁(X) = b₁(C) + 2, the fundamental group of X is almost isomorphic to π₁(C) × π₁(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X → C such that b₁(X) > b₁(C).

We establish a relative volume comparison theorem for minimal volume form of Finsler manifolds under integral Ricci curvature bound. As its applications, we obtain some results on integral Ricci curvature and topology of Finsler manifolds. These results generalize the corresponding properties with pointwise Ricci curvature bound in the literatures.

We construct a group measure space II₁ factor that has two nonconjugate Cartan subalgebras. We show that the fundamental group of the II₁ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is nontrivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.

Let be a von Neumann algebra of type II₁ which is also a complemented subspace of . We establish an algebraic criterion, which ensures that is an injective von Neumann algebra. As a corollary we show that if is a complemented factor of type II₁ on a Hilbert space , then is injective if its fundamental group is nontrivial.

We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski–Van Kampen algorithm and exploit the Cremona transformations used in the construction of these curves. We simplify and study the group presentations so obtained and determine if they are abelian, finite or big, i.e. if they contain free non-abelian subgroups. We also study the quotients of these groups to some extent.

Mehta and Seshadri proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with the set of equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this paper, we discuss the Mehta–Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.

Studying self-similar solutions of geometric flows on manifolds plays an important role in understanding geometrical and topological properties of underlying manifolds. In this paper, we prove that a complete shrinking Ricci–Bourguignon harmonic flow soliton $((M,g),(N,h),\varphi ,X,\rho ,\lambda )$ is compact if and only if $\left|\right|X\left|\right|$ is bounded on $M$. Also, we show that a complete shrinking Ricci–Bourguignon harmonic flow soliton has finite fundamental group.

Our main purpose in this paper is to give a counterexample to some interesting conjectures on a rational fibred surface with multiple fibers proposed by R. V. Gurjar and D.-Q. Zhang.

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let $S$ be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane ${\mathbb{P}}^2$. Such $S$ has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of $X={\mathbb{P}}^2\setminus S$ [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. $X$ has a 9-fold Galois covering space $\tilde{X}$ given by the complement of the Hesse configuration of 12 lines in ${\mathbb{P}}^2$. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of ${\pi }_1(\tilde{X})$, which we carry out in this note.

A topological invariant, analogous to the linking number as defined in knot theory, is defined for pairs of digital closed paths of ℤ³. This kind of invariant is very useful for proofs which involve homotopy classes of digital paths. Indeed, it can be used, for example, in order to state the connection between the tunnels in an object and the ones in its complement. Even if its definition is not as immediate as in the continuous case it has the good property that it is immediately computable from the coordinates of the voxels of the paths with no need of a regular projection. The aim of this paper is to state and prove that the linking number has the same property as its continuous analogue: it is invariant under any homotopic deformation of one of the two paths in the complement of the other.

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods.