Homology Spheres Research Articles

A combinatorial approach to surgery formulas in Heegaard Floer homology

Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.

Some combinatorial properties of flag simplicial pseudomanifolds and spheres

A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $\Delta$ of dimension $d-1$ is minimized when $\Delta$ is the boundary complex of the $d$-dimensional cross-polytope.

Cyclotomy and endomotives

We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expansions. Both have compatible endomorphisms actions of the multiplicative semigroup of positive integers. As a higher dimensional generalization, we consider a crossed product ring obtained using Manin’s multivariable generalizations of the Habiro functions and an action by endomorphisms of the semigroup of integer matrices with positive determinant. We then construct a corresponding class of multivariable BC endomotives, which are obtained geometrically from self maps of higher dimensional algebraic tori, and we discuss some of their quantum statistical mechanical properties. These multivariable BC endomotives are universal for (torsion free) Λ-rings, compatibly with the Frobenius action. Finally, we discuss briefly how Habiro’s universal Witten-Reshetikhin-Turaev invariant of integral homology 3-spheres may relate invariants of 3-manifolds to gadgets over \( \mathbb{F}_1 \) and semigroup actions on homology 3-spheres to endomotives.

Universal abelian covers of superisolated singularities

AbstractWe give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Lefschetz properties and basic constructions on simplicial spheres

The well known g-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the g-conjecture for piecewise-linear spheres.

Face enumeration—from spheres to manifolds

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h -vector of balanced semi-Eulerian complexes and the toric h -vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3-dimensional constructions [47], allow us to give a complete characterization of the f -vectors of arbitrary simplicial triangulations of S^1 \times S^3 , ℂP^2, K3 surfaces, and (S^2 \times S^2) \# (S^2 \times S^2) . We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the g -conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.

The Lefschetz property for barycentric subdivisions of shellable complexes

We show that an ‘almost strong Lefschetz’ property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of $1$.

On the Casson invariant of homology 3-spheres of Mazur type

We compute the Casson invariant for some integral homology 3-spheres. We also show that for every integer n, there exists an integral homology 3-sphere of Mazur type with the Casson invariant 2 n.

A SIMPLE FORMULA FOR THE CASSON–WALKER INVARIANT

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case — the celebrated Casson–Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson–Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson–Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations.

The residual gravity acceleration effect in the Poincaré dodecahedral space

In a flat space, the global topology of comoving space can induce a weak acceleration effect similar to dark energy. Does a similar effect occur in the case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish the Poincare space from other well-proportioned spaces? The residual acceleration effect in the Poincare space is studied here using a massive particle and a nearby test particle of negligible mass, in S^3 embedded in R^4. The weak limit gravitational attraction on a test particle at distance r is set \propto [r_C \sin(r/r_C)]^{-2}, where r_C = curvature radius, in order to satisfy Stokes' theorem. A finite particle horizon large enough to include the adjacent topological images of the massive particle is assumed. The regular, flat, 3-torus T^3 is re-examined, and two other well-proportioned spaces, S^3/T^* and S^3/O^*, are also studied. The residual gravity effect occurs in all four cases. In a perfectly regular 3-torus of side length L_a, and in S^3/T^* and S^3/O^*, the highest order term in the residual acceleration is the third order term in the Taylor expansion of r/L_a (3-torus), or r/r_C, respectively. However, the Poincare dodecahedral space is unique among the four spaces. The third order cancels, leaving the fifth order term \sim \pm 300 (r/r_C)^5 as the most significant. Not only are three of the four perfectly regular well-proportioned spaces better balanced than most other multiply connected spaces in terms of the residual acceleration effect by a factor of about a million (setting r/L_a = r/r_C \sim 10^{-3}), but the fourth of these spaces is about 10^4 times better balanced than the other three. This is the Poincare dodecahedral space. Is this unique dynamical property of the Poincare space a clue towards a theory of cosmic topology?

Ohtsuki's invariant cannot determine the full so (3) quantum invariants

Two lens spaces are given to show, that Ohtsuki's τ for rational homology spheres does not determine Kirby-Melvin's {τ′r odd ≥ 3}.

Surgery formulae for finite type invariants of rational homology 3–spheres

We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form $$\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$$ where $N$ is the set of components of a framed algebraically split link $L$ in a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the Dehn surgeries on the components of $I$. The first formula treats the case when $L$ is a boundary link with $n$ components, while the second one is for $3n$--component algebraically split links. In the third formula, the link $L$ has $2n$ components and the Milnor triple linking numbers of its 3--component sublinks vanish. The presented formulae are then applied to the study of the variation of $Z_n$ under a $p/q$-surgery on a knot $K$. This variation is a degree $n$ polynomial in $q/p$ when the class of $q/p$ in $\QQ/\ZZ$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.

Orders and actions of branched coverings of hyperbolic links

Let M, M ′ be compact oriented 3-manifolds and L ′ a link in M ′ whose exterior has positive Gromov norm. We prove that the topological types of M and ( M ′ , L ′ ) determine the degree of a strongly cyclic covering p : M → M ′ branched over L ′ . Moreover, if M ′ is a homology sphere then these topological types determine also the covering up to conjugacy.

OHTSUKI INVARIANTS FOR INTEGRAL HOMOLOGY SPHERES AND HABIRO'S CYCLOTOMIC EXPANSION

We give surgery formulas for the Ohtsuki invariants λ1, λ2and λ3of an integral homology sphere obtained by surgery from S3along a knot, related to Habiro's cyclotomic expansion of the colored Jones polynomial of the knot. As an application, we prove that the Ohtsuki invariants λ1, λ2and λ3separate integral homology 3-spheres obtained from S3by surgery along the Borromean rings.

On Kontsevich's characteristic classes for higher-dimensional sphere bundles II: Higher classes

This paper studies Kontsevich’s characteristic classes of smooth bundles with fibre in a ‘singularly framed’ odd-dimensional homology sphere, which are defined through his graph complex and configuration space integral. We will give a systematic construction of smooth bundles parameterized by trivalent graphs and will show that our smooth bundles are non-trivially detected by Kontsevich’s characteristic classes. It turns out that there are surprisingly many non-trivial elements of the rational homotopy groups of the diffeomorphism groups of spheres that are in some ‘non-stable’ range. In particular, the homotopy groups of the diffeomorphism groups in some ‘non-stable’ range are not finite.

Trivial cocycles and invariants of homology 3-spheres

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We apply this study to give a new purely algebraic construction of the Casson invariant. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

Virtually fibered Montesinos links

We prove that all generalized Montesinos links in S3 which are not classic S L ˜ 2 -type are virtually fibred except the trivial link of two components. We also prove the virtually fibred property for a family of infinitely many classic Montesinos links of type S L ˜ 2 . As a byproduct we find the first family of infinitely many virtually fibred hyperbolic rational homology 3-spheres.

Heegaard Floer homology and genus one, one-boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, ϕ), where T is a genus one surface with one-boundary component. In addition, we compute the Heegaard Floer homology of every T2-bundle over S1 with first Betti number equal to 1, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-fillings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.

On the set of the logarithm of the LMO invariant for integral homology 3-spheres

AbstractThe LMO invariant is a very strong invariant such that it is expected to classify integral homology 3-spheres. In this paper we identify the set of the degree ≤ 6 parts of the logarithm of the LMO invariant for integral homology 3-spheres. As an application, we obtain a complete set of relations which characterize the set of Ohtsuki's invariants {λi(M)} fori≤ 6. For any simple Lie algebra$\frak{g}$, we also obtain a complete set of relations which characterize the set of perturbativeP$\frak{g}$invariants {$\lambda^{\frakg}_i$(M)} fori≤ 3.