Narrow Results

Given a compact geodesic space $X$, we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.

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.

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.

We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.

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.

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

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.

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 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.

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.

In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The difference in the topology of their complements can only be detected via finer invariants or techniques. In our case the generic braid monodromies, the fundamental groups, the characteristic varieties and the existence of dihedral coverings of ℙ² ramified along them can be used to distinguish the two sextics. Our intention is not only to use different methods and give a general description of them but also to bring together different perspectives of the same problem.

Suppose $\mathrm{\Gamma }$ is a discrete group, and $\alpha \in Z^3(B\mathrm{\Gamma };A)$, with $A$ an abelian group. Given a representation $\rho :{\pi }_1(M)\to \mathrm{\Gamma }$, with $M$ a closed 3-manifold, put $F(M,\rho )=〈{(B\rho )}^{\ast }[\alpha ],[M]〉$, where $B\rho :M\to B\mathrm{\Gamma }$ is a continuous map inducing $\rho$ which is unique up to homotopy, and $〈-,-〉:H^3(M;A)\times H_3(M;\mathbb{Z})\to A$ is the pairing. We extend the definition of $F(M,\rho )$ to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing $F(M,\rho )$ when $M$ is given by a surgery along a link $L\subset S^3$. In particular, the Chern–Simons invariant can be computed this way.

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank $2$ are $2$-bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered.

Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π₁(M) ≅ π₁(M/G); if additionally ϕ is proper, then π₁(M) ≅ π₁ (ϕ^-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π₁(M) ≅ π₁(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π₁(M) ≅ π₁(ϕ^-1(G⋅a)) ≅ π₁(ϕ^-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then for all a ∈ ϕ(M), where is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π₁(M/G) ≅ π₁(ϕ^-1(G⋅a)/G) for all a ∈ ϕ(M).

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e. small loops at $p\in X$ generate a subgroup of the fundamental group of the unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results:

(1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence.

(2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup $\mu$ of $P$ and ${\beta }^{*}$, the fundamental relation on a given polygroup $P$, we construct an Engel fuzzy subgroup ${\mu }_{{\beta }^{*}}$. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group $P$/$\sim$, the group of equivalence classes derived from a fuzzy subpolygroup of $P$. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

In this paper, our prime aim is to develop the concept of fundamental group structure in soft set theoretic approach. To execute this, first, we have defined soft homotopy of maps, soft path, soft loop, soft path homotopy, soft loop homotopy, product of soft loops, soft homotopy class, etc., at a soft element using generalized soft mappings and their important behaviors are studied. Finally, we have introduced the notion of fundamental group whose members are soft homotopy classes. Some examples are also discussed in different soft topological spaces.