Mutative hyperbolic homology 3-spheres with the same Floer homology

We construct a large class of finitely many hyperbolic homology 3-spheres making the following invariants equal, simultaneously, the integral homology, the quantum SU(2) invariants, the hyperbolic volume, the hyperbolic isometry group, the -invariant, the Chern-Simons invariant, and the Floer homology.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

Rohlin’s invariant and gauge theory,I. Homology 3-tori

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H ¹ (Y;₂) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Cassons homology sphere invariant reduces mod 2 to the Rohlin invariant.     

Reidemeister torsion and lens surgeries on knots in homology 3-spheres II

We introduce the norm and the order of a polynomial and of a homology lens space. We calculate the norm of the cyclotomic polynomials, and apply it to lens surgery problem for a knot whose Alexander polynomial is the same as an iterated torus knot.     

A spanning tree cohomology theory for links

In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard–Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the _E3$E_3$-term of their spectral sequence. The main purpose of the present paper is to prove directly that this _E3$E_3$-term is a link invariant. We also give some concrete examples of computation of the invariant.     

Branched spines and Heegaard genus of 3-manifolds

We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S ³ and S ² × S ¹. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.     

High distance Heegaard splittings from involutions

Fixed an oriented handlebody H=H₊ with boundary F, let η(H₊)=H₋ be the mirror image of H₊ along F, so η(F) is the boundary of H₋, for a map f:F→F, we have a 3-manifold by gluing H₊ and H₋ along F with attaching map f, and denote it by Mf=H_+∪f:F→FH₋. In this note, we show that there are involutions f:F→F which are also reducible, such that Mf have arbitrarily high Heegaard distances.     

Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

In Ozsváth and Szabó (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsváth and Szabó (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsváth and Szabó (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF⁺ for a variety of three-manifolds. Moreover, we show how the structure of HF⁺ constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for .     

On the Heegaard Floer homology of branched double-covers

Let be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S³, branched along L. When L is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E² term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ(L).     

The homology of path spaces and Floer homology with conormal boundary conditions

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions. Dedicated to Felix E. Browder     

Refined Kirby calculus for three-manifolds of first homology groups of odd prime orders

Every integral homology 3-sphere is presented by a framed link with framing ±1 and without linking numbers. Restricting such presentations, Habiro arranged Kirby calculus so that it preserves framings and linkings and moreover showed that his calculus suffices to relate all links with the same results. This paper provides an extension of his result for manifolds of first homology groups of odd prime orders. After defining our set of links, we establish Habiro calculus over it, and show that, for many orders, it works on those manifolds. We further give the existence of the Casson-Walker invariant for them.     

Rational homology spheres and the four-ball genus of knots

Using the Heegaard Floer homology of Ozsváth and Szabó we investigate obstructions to a rational homology sphere bounding a four-manifold with a definite intersection pairing. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.     

Heegaard splittings of twisted torus knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain “twisted torus knots” originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.     

Knots with unknotting number one and Heegaard Floer homology

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with 10 crossings and unknotting number equal to one.     

Quadratic functions and complex spin structures on three-manifolds

We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex spin structure. Our main result states that two closed oriented three-manifolds endowed with a complex spin structure are undistinguishable by complex spin invariants of degree zero if, and only if, their associated quadratic functions are isomorphic.     

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.     

The Heegaard genera of surface sums

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M₁ and M₂ (resp. a manifold M₁). Then M is called the surface sum (resp. self surface sum) of M₁ and M₂ (resp. M₁) along F, denoted by M=M₁F_∪M₂ (resp. M=M₁F_∪). In this paper, we will study how g(M) is related to χ(F), g(M₁) and g(M₂) when both M₁ and M₂ have high distance Heegaard splittings.     

Equivariant Floer groups for binary polyhedral spaces

This work was partially supported by a grant from the National Sciences and Engineering Research Council     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written