Homology Spheres Research Articles

Niemeier lattices, smooth 4-manifolds and instantons

We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.

More Brieskorn spheres bounding rational balls

We call an integral homology sphere non-trivially bounds a rational homology ball if it is obstructed from bounding an integral homology ball. After Fintushel and Stern's well-known example Σ(2,3,7), Akbulut and Larson recently provided the first infinite families of Brieskorn spheres non-trivially bounding rational homology balls: Σ(2,4n+1,12n+5) and Σ(3,3n+1,12n+5) for odd n. Using their technique, we present new such families: Σ(2,4n+3,12n+7) and Σ(3,3n+2,12n+7) for even n. Also manipulating their main argument, we simply recover some classical results of Akbulut and Kirby, Fickle, Casson and Harer, and Stern about Brieskorn spheres bounding integral homology balls.

Vector Fields and Genus in Dimension 3

Abstract Given a vector field on a 3D rational homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. This provides a formula for the genus of a transverse surface, and in particular, of a Birkhoff section. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to helicity.

New curvature conditions for the Bochner Technique

We show that manifolds with $$ \lceil \frac{n}{2} \rceil $$ -positive curvature operators are rational homology spheres. This follows from a more general vanishing and estimation theorem for the pth Betti number of closed n-dimensional Riemannian manifolds with a lower bound on the average of the lowest $$n-p$$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot–Meyer, and Gallot.

Spherical 3-Manifolds Bounding Rational Homology Balls

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology balls induced from Donaldson’s diagonalization theorem and Heegaard Floer correction terms.

Monopole Floer homology and SOLV geometry

We study the monopole Floer homology of a SOLV rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable SOLV metrics on Y, small regular perturbations of the Seiberg-Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that Y is an L-space.

The rigidity of the graphs of homology spheres minus one edge

We prove that for d−1≥3, the graph of any homology (d−1)-sphere Δ with g2(Δ)≥1 and without missing (d−1)-faces is generically redundantly d-rigid. This confirms a conjecture of Nevo and Novinsky.

$g$-vectors of manifolds with boundary

We extend several g-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain Kühnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion Δ ^ of a manifold with boundary Δ; it is obtained from Δ by coning off the boundary of Δ with a single new vertex. We show that despite the fact that Δ ^ has a singular vertex, its Stanley–Reisner ring shares a few properties with the Stanley–Reisner rings of homology spheres. We close with a discussion of a connection between three lower bound theorems for manifolds, PL-handle decompositions, and surgery.

AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

AbstractWe introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

Cosmetic two‐strand twists on fibered knots

Let $K$ be a knot in a rational homology sphere $M$. This paper investigates the question of when modifying $K$ by adding $m>0$ half-twists to two oppositely-oriented strands, while keeping the rest of $K$ fixed, produces a knot isotopic to $K$. Such a two-strand twist of order $m$, as we define it, is a generalized crossing change when $m$ is even and a non-coherent band surgery when $m=1$. A cosmetic two-strand twist on $K$ is a non-nugatory one that produces an isotopic knot. We prove that fibered knots in $M$ admit no cosmetic generalized crossing changes. Further, we show that if $K$ is fibered, then a two-strand twist of odd order $m$ that is determined by a separating arc in a fiber surface for $K$ can only be cosmetic if $m=\pm 1$. After proving these theorems, we further investigate cosmetic two-strand twists of odd orders. Through two examples, we find that the second theorem above becomes false if `separating' is removed, and that a key technical proposition fails when the order equals 1. A closer look at an order-one example, an instance of cosmetic band surgery on the unknot, reveals it to be nearly trivial in a sense that we name `weakly nugatory'. We correct the technical proposition to obtain a means of using double branched covers to show that certain band surgeries are weakly nugatory. As an application, we prove that every cosmetic band surgery on the unknot is of this type.

τ–invariants for knots in rational homology spheres

Ozsv\'ath and Szab\'o used the knot filtration on $\widehat{CF}(S^3)$ to define the $\tau$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $\tau$-invariants associated to a knot $K$ in a rational homology sphere $Y$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative definite 4-manifold with boundary $Y$..

Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action

In this paper, we study the action of $\text{Homeo}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an $\mathbb{F}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an $\mathbb{F}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of $\text{Homeo}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the "coning-off" construction.

CYCLIC BRANCHED COVERINGS OF BRIESKORN SPHERES BOUNDING ACYCLIC 4-MANIFOLDS

AbstractWe show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic 4-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well-known invariants of homology spheres that obstruct acyclic bounding 4-manifolds and then use a method based on equivariant Yang–Mills moduli spaces to rule out extensions of the actions.

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.The first part contains the definition of ‘generic structure’ based on the work of Laufer [14]. The second technical ingredient is the Abel map developed in [21].The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves (see e.g. [1] and [6]), though the tools and machineries are very different.

Induced equators in flag spheres

We propose a combinatorial approach to the following strengthening of Gal's conjecture: γ(Δ)≥γ(E) coefficientwise, where Δ is a flag homology sphere and E⊆Δ an induced homology sphere of codimension 1. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag d-polytopes: h1(Δ)hi(Δ)≥(d−i+1)hi−1(Δ)+(i+1)hi+1(Δ) for all 0≤i≤d.

Torelli group, Johnson kernel, and invariants of homology spheres

In the late 1980s, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and its core part was identified with the secondary characteristic class d_1 associated with the fact that the first \mathrm{MMM} class vanishes on the Torelli group (however it turned out that Johnson proved the former part highly likely prior to the above, see Remark 1.1). This secondary class d_1 is a rational generator of H^1(\mathcal{K}_g;\mathbb Z)^{\mathcal{M}_g}\cong\mathbb Z where \mathcal{K}_g denotes the Johnson subgroup of the mapping class group \mathcal{M}_g . Hain proved, as a particular case of his fundamental result, that this is the only difference in degree 2 . In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree 6 . We apply this to determine H_1(\mathcal{K}_g;\mathbb Q) explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology 3 -spheres of degrees up to 6 , including the second and the third Ohtsuki invariants, can be expressed by d_1 and lifts of Johnson homomorphisms.

Entropy in uniformly quasiregular dynamics

AbstractLet$M$be a closed, oriented, and connected Riemannian$n$-manifold, for$n\geq 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map$f:M\rightarrow M$, the topological entropy$h(f)$is$\log \deg f$. This proves Shub’s entropy conjecture in this case.

E8 spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

Rational homology 3–spheres and simply connected definite bounding

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,q$, we obtain an infinite family of irreducible rational homology 3-spheres which are homology cobordant to the lens space $L(p,q)$ but cannot obtained by a knot surgery.

Trisections, intersection forms and the Torelli group

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of 4-manifolds. Specifically, if $X$ and $Y$ admit handle decompositions without 1- or 3-handles and have isomorphic intersection forms, then a trisection of $Y$ can be obtained from a trisection of $X$ by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology 3-spheres can be applied, via this result, to obstruct intersection forms of smooth 4-manifolds. As an application, we use the Casson invariant to recover Rohlin's Theorem on the signature of spin 4-manifolds.