Homology Spheres Research Articles

Inflation and Topological Phase Transition Driven by Exotic Smoothness

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an exotic smoothness structure for the space-time. Here we consider a space-time with topologyS3×ℝ. In case of an exoticS3×ℝ, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski orR2model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated.

A result on Ricci curvature and the second Betti number

We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction. As consequences we obtain an interesting curved restriction for compact Kahler-Einstein manifolds and a homology sphere theorem in ${\rm dim}=4,5$.

An Alternative Manifold for Cosmology Using Seifert Fibered and Hyperbolic Spaces

We propose a model with 3-dimensional spatial sections, constructed from hyperbolic cusp space glued to Seifert manifolds which are in this case homology spheres. The topological part of this research is based on Thurston’s conjecture which states that any 3-dimensional manifold has a canonical decomposition into parts, each of which has a particular geometric structure. In our case, each part is either a Seifert fibered or a cusp hyperbolic space. In our construction we remove tubular neighbourhoods of singular orbits in areas of Seifert fibered manifolds using a splice operation and replace each with a cusp hyperbolic space. We thus achieve elimination of all singularities, which appear in the standard-like cosmological models, replacing them by “a torus to infinity”. From this construction, we propose an alternative manifold for cosmology with finite volume and without Friedmann-like singularities. This manifold was used for calculating coupling constants. Obtaining in this way a theoretical explanation for fundamental forces is at least in the sense of the hierarchy.

The Poincaré homology sphere and almost-simple knots in lens spaces

Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S 3 S^3 . Rasmussen showed that when the ‘left-handed’ one is in the homology class of the dual to a Berge knot of type VII, it admits an L-space homology sphere surgery. In this note we give a simple proof that these L-space homology spheres are always the Poincaré homology sphere.

DECOHERENCE IN QUANTUM COSMOLOGY AND THE COSMOLOGICAL CONSTANT

We discuss a spacetime having the topology of S3×ℝ but with a different smoothness structure. The initial state of the cosmos in our model is identified with a wildly embedded 3-sphere (or a fractal space). In previous work we showed that a wild embedding is obtained by a quantization of a usual (or tame) embedding. Then a wild embedding can be identified with a (geometrical) quantum state. During a decoherence process this wild 3-sphere is changed to a homology 3-sphere. We are able to calculate the decoherence time for this process. After the formation of the homology 3-sphere, we obtain a spacetime with an accelerated expansion enforced by a cosmological constant. The calculation of this cosmological constant gives a qualitative agreement with the current measured value.

Heegaard Floer genus bounds for Dehn surgeries on knots

We provide a new obstruction for a rational homology 3-sphere to arise by Dehn surgery on a given knot in the 3-sphere. The obstruction takes the form of an inequality involving the genus of the knot, the surgery coefficient and a count of L-structures on the 3-manifold, that is spinc-structures with the simplest possible associated Heegaard Floer group. Applications include an obstruction for two framed knots to yield the same 3-manifold, an obstruction that is particularly effective when working with families of framed knots. We introduce the rational and integral Dehn surgery genera for a rational homology 3-sphere, and use our inequality to provide bounds, and in some cases exact values, for these genera. We also demonstrate that the difference between the integral and rational Dehn surgery genera can be arbitrarily large.

METABELIAN SL(N, C) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is conjugate to an SU(n) representation. In this case we show it determines a smooth point x in the SL(n,C) character variety, and we use a deformation argument to establish the existence of a smooth (n-1)-dimensional family of characters of irreducible SL(n,C) representations near x. Combining this with our previous existence results, we deduce the existence of large families of irreducible SU(n) and SL(n,C) non-metabelian representation for knots K in homology 3-spheres S with nontrivial Alexander polynomial. We then relate the condition on twisted cohomology to a more accessible condition on untwisted cohomology of a certain metabelian branched cover of S branched along K.

Asymptotics of the Self-Dual Deformation Complex

We analyze the indicial roots of the self-dual deformation complex on a cylinder $(\mathbb{R}\times Y^{3}, dt^{2} + g_{Y})$ , where Y 3 is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section Y 3, which is crucial in gluing results for orbifolds in the case of cross-section Y 3=S 3/Γ. We also resolve a conjecture of Kovalev–Singer in the case where Y 3 is a hyperbolic rational homology 3-sphere, and show that there are infinitely many examples for which the conjecture is true, and infinitely many examples for which the conjecture is false.

Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary \(p\)-groups in the groups concerned. In some cases, including \({\mathrm{Sp}}(2n,\mathbb Z )\), the bounds we obtain are sharp: if \(X\) is a generalized \(\mathbb Z /3\)-homology sphere of dimension less than \(2n-1\) or a \(\mathbb Z /3\)-acyclic \(\mathbb Z /3\)-homology manifold of dimension less than \(2n\), and if \(n\ge 3\), then any action of \({\mathrm{Sp}}(2n,\mathbb Z )\) by homeomorphisms on \(X\) is trivial; if \(n=2\), then every action of \({\mathrm{Sp}}(2n,\mathbb Z )\) on \(X\) factors through the abelianization of \({\mathrm{Sp}}(4,\mathbb Z )\), which is \(\mathbb Z /2\).

Homology equivalences of manifolds and zero-in-the-spectrum examples

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-dimensional knots proved by Karvaire. We also use it to get counter-examples to the zero-in-the-spectrum conjecture found by Farber-Weinberger, and by Higson-Roe-Schick.

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.

Spheres are rare

We prove that triangulations of homology spheres in any dimension grow much slower than general triangulations. Our bound states in particular that the number of triangulations of homology spheres in 3 dimensions grows at most like the power 1/3 of the number of general triangulations.

The Lefschetz coincidence class of p maps

Abstract Let X be an arbitrary topological space and let Y be a closed connected oriented n-dimensional manifold. In this work we consider p maps f 1,...,fp : X → Y, p ≥ 2, define a Lefschetz class L(f 1,...,fp ) ∈ H n(p-1)(X;ℚ) and prove that L(f 1,...,fp ) ≠ 0 implies f 1(x) = f 2(x) = ⋯ = fp (x) for some x ∈ X. In the particular case where Y is a homology sphere there are presented some formulas to calculate L(f 1,...,fp ).

Plato, Poincaré, and the Enchanted Dodecahedron: Is the Universe Shaped Like the Poincaré Homology Sphere?

(2013). Plato, Poincare, and the Enchanted Dodecahedron: Is the Universe Shaped Like the Poincare Homology Sphere? Math Horizons: Vol. 20, No. 3, pp. 12-17.

On 3-manifolds with locally standard [formula omitted]-actions

We investigate what kind of closed 3-manifolds can admit locally standard (Z2)3-actions. In particular, we will prove that for a closed connected 3-manifold M with H1(M;Z2)=0, M admits a locally standard (Z2)3-action if and only if M is a connected sum of 8 copies of a Z2-homology sphere N. So if such an M is irreducible, it must be homeomorphic to S3. Moreover, the argument can be extended to study orientable rational homology 3-spheres M with H1(M;Z2)≅Z2 which admits locally standard (Z2)3-actions.

Heegaard Floer homology and several families of Brieskorn spheres

In Ozsváth and Szabó (2003) [15] the authors gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm in Némethi (2005) [12] for executing these computations for almost-rational plumbings, and his work in Némethi (2007) [13] gives a formula computing the invariants for the Brieskorn homology spheres −Σ(p,q,pqn+1). Here we give a formula for HF+(−Σ(p,q,pqn−1)), generalizing the one for the n=1 case given in Borodzik and Némethi (2011) [3]. We also compute HF+ for the families −Σ(2,5,k) and −Σ(2,7,k).

Small volume link orbifolds

This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the 3-sphere, or more generally in a Z_6 homology sphere. We also prove more general lower bounds under mild homological hypotheses.

Floer homology for 2-torsion instanton invariants

We construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with certain intersection forms.

From Flag Complexes to Banner Complexes

A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary dimension that are $(i+1)$-banner but not $i$-banner are constructed. It is shown that several theorems for flag complexes have appropriate $i$-banner analogues. Among them are (1) the codimension-$(i+j-1)$ skeleton of an $i$-banner homology sphere $\Delta$ is $2(i+j)$-Cohen--Macaulay for all $0\leq j\leq \dim\Delta+1-i$, and (2) for every $i$-banner simplicial complex $\Delta$ there exists a balanced complex $\Gamma$ with the same number of vertices as $\Delta$ whose face numbers of dimension $i-1$ and higher coincide with those of $\Delta$.

Finite type invariants of rational homology 3–spheres

We consider the rational vector space generated by all rational homology spheres up to orientation-preserving homeomorphism, and the filtration defined on this space by Lagrangian-preserving rational homology handlebody replacements. We identify the graded space associated with this filtration with a graded space of augmented Jacobi diagrams.