In this paper we compute the mapping class group of closed simply-connected 6-manifolds $M$ which look like complete intersections, i.~e.~ $H_2(M;\mathbb Z) \cong \mathbb Z $ and $x^3 \ne 0$ where $x \in H^2(M; \mathbb Z)$ is a generator. We determine some algebraic properties of the mapping class group; for example we compute its abelianization and its center. We show that modulo the center the mapping class group is residually finite and virtually torsion-free. We also study low dimensional homology groups. The results are very similar to the computation of the mapping class group of Riemann surfaces. We give generators of the mapping class group, and generators and relations for the subgroup acting trivially on $π_{3}(M)$.
We extend Donaldson's diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.
We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson homomorphism.
We determine the contributions of isolated singularities of spin V 4-manifolds to the index of the Dirac operator over them. From these data we derive certain constraints on the intersection forms of spin 4-manifolds bounded by spherical 3-manifolds, and also on the embeddings of the real projective planes into 4-manifolds.
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from being homology cobordant to certain graph manifolds.
We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem of detecting the geometric intersection between separating simple closed curves in a representation.
In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we also consider the homology cobordism monoid, and give a necessary condition using w-invariants for the homology 3-spheres to belong to the inertia group associated to some homology 3-spheres.
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity. As a corollary of our methods, we obtain that most curves that are homotopic are als…
In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.
In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known re…
The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this work we show the intersection matrix is enough to determine the triangulation of a 3-manifold up to isomorphism.
In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold $X$ with second positive Betti number $b_2^+(X)=1$. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this…
Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$-spheres. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $\pmΣ(2,3,6k\pm1)$. Along the way, we also give an alternative proof of Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total spaces of orientable vector bundles over M.
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local coefficient version of Furuta's 10/8-inequality. As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.
We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn, but our proofs are fundamentally different.
We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional cycles are developed. By means of these algorithms it is possible to construct a basis of cohomology group from the homology group of two cycles of complementary dimensions.
The second Betti number of a smooth, closed, connected and simply connected, four-dimensional spin manifold is greater or equal 11/8 times the abolute value of its signature.
We give some constraints on intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres in terms of the $\barμ$ invariant and compare them with those in terms of the $κ$ invariant. We also show that the difference between $κ$ and $-\barμ$ for a Seifert rational homology 3-sphere is at most 2, and under certain extra conditions the difference is 0.
We show that the action of the mapping class group on the space of closed curves of a closed surface effectively tracks the corresponding action on Teichmüller space in the following sense: for all but quantitatively few mapping classes, the information of how a mapping class moves a given point of Teichmüller space determines, up to a power saving error term, how it changes the geometric intersection numbers of a given closed curve with respect to arbitrary geodesic currents. Applications include an effective estimate describing the speed of convergence of Teichmüller geodesic rays to the bou…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class groups of punctured surfaces is proposed, with efficiency understood in a more restrictive way than the most common one.
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer K_G-split homology sphere; this concept may be useful in an approach to the 11/8 conjecture.
In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $\cN \in \N$ we show that the level $\cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {\bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $\cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely enco…
In this note, we present a proof of Donaldson's Diagonalization Theorem via an abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple spinors. Like the other proof of Donaldson's theorem using the standard Seiberg-Witten theory, Elkies' theorem also plays a key role in our argument.
We give a simple criterion when a Gluck twisting an odd smooth 4-manifold along a 2-sphere $S\subset X$ does not change its diffeomorphism type. We obtain this by handlebody techniques and plug twisting operation, getting a slightly stronger version of the known fact that Gluck twisting of a 2-sphere $S\subset X$ of a compact smooth 4-manifold with an odd spherical class, in the complement of S, does not change the diffeomorphism type of X. This is the best possible result on Gluck twisting manifolds with odd homology classes.
Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(Σ;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing on the diagram surface $Σ$. This includes explicit formulas for the second and third homology groups of $X$ as well an algorithm to compute the intersection form. Moreover, we show that all $(g;k,0,0)$-trisections admit "algebraically trivial" diagrams.
We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds…
We construct infinitely many non-diffeomorphic examples of $5$-dimensional contact manifolds which are tight, admit no strong fillings, and do not have Giroux torsion. We obtain obstruction results for symplectic cobordisms, for which we give a proof not relying on the polyfold abstract perturbation scheme for SFT. These results are part of the author's PhD thesis, and are the first applications of higher-dimensional Siefring intersection theory for holomorphic curves and hypersurfaces.
We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for $J$-holomorphic curves in almost complex $4$-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce $J$-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset o…
We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Frøyshov, and Ozsváth and Szabó. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $Σ$, guiding the gluing of the handlebodies. Any morphism $φ$ from $π_1(X)$ to a finitely generated free abelian group induces a morphism on $π_1(Σ)$. We express the twisted homology and Reidemeister torsion of $(X;φ)$ in terms of the first homology of $(Σ;φ)$ and the three subspaces generated by the collections of curves. We also express the intersec…
Let M be a compact oriented PL manifold and let C_*M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex G of C_*M\otimes C_*M. We prove that G is a "full" subcomplex, that is, the inclusion of G in C_*M \otimes C_*M is a quasi-isomorphism. An analogous result is true for the domain of the iterated intersection pairing. Using this, we show that the intersection pairing gives C_*M a structure of partially defined commutative DGA, which in particular implies that C_*M is canonically quasi-isomorphic to an E_\infty chain algebra. An erratum is attached whic…
In an earlier paper (math.SG/0110169), we introduced absolute gradings on the three-manifold invariants developed in math.SG/0101206 and math.SG/0105202. 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'' (from math.SG/0101206), restrictions on knots whose surgeries give rise to lens spaces, and calculations of $\HFp$ for a variety of three-manifolds. Moreover, we show how the structure of $\HFp$ constrains the exoticness of definite intersection forms for s…
Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. When M is the product of a Riemann surface with a circle the 4-dimensional class does not enter and the path integral takes the form of a Riemann-Roch formula albeit in infinite dimensions. The discussion is generalised to include Wilson lines along the fibre direction in M.
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 f…
An obstruction theory for representing homotopy classes of surfaces in 4-manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor's link-homotopy invariants, and can give the complete obstruction to pulling apart 2-spheres in certain families of 4-manifolds. It is also shown that in an arbitrary simply connected 4-manifold any number of parallel copies of an immersed surface with vanishing self-intersection number can be pulled apart, a…
Let $X$ be a smooth simply connected closed 4-manifold with definite intersection form. We show that any automorphism of the intersection form of $X$ is realized by a diffeomorphism of $X \mathbin{\#} S^2 \times S^2$. This extends and completes Wall's foundational result from 1964.
This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed primarily at students and require only a minimal background in holomorphic curve theory, as the emphasis is on topological rather than analytical issues. Some of the gaps in the analysis are then filled in by the appendices, which include self-contained proofs of the similarity principle and positivity of intersections, and conclude with a "quick reference" fo…
In this paper we determined all of the possible self mapping degrees of the manifolds with $S^3$-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self mapping degrees of all closed orientable 3-manifold in Thurston's picture.
By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any closed orientable 3-manifold $M$?
Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather's result. Namely, we show that the set of stable mappings in the set of all mappings is never dense if the source manifold is non-compact. Moreover, as a corollary of Mather's result and the main theorem of th…
This article is the author's contribution to the volume "Problems on mapping class groups and related topics" which will be published in December 2006, with Benson Farb as Editor. Various individuals were invited by Farb to submit open problems which seemed interesting to them about surface mapping class groups. We singled out a particular recent development, Hempel's concept of the `Heegaard distance' of an element in the mapping class group, and used it to describe a number of open questions which relate to the correspondence between Heegaard surfaces in 3-manifolds and the mapping class gro…
The topology of the intersection of two real homogeneous coaxial quadrics was studied by the second author who showed that its intersection with the unit sphere is in most cases diffeomorphic to a connected sum of sphere products. Combining that approach with a recent one (due to Antony Bahri, Martin Bendersky, Fred Cohen and the first author) we study here the intersections of k>2 quadrics and we identify very general families of such manifolds that are diffeomorphic to connected sums of sphere products. These include those moment-angle manifolds for which the result was conjectured by Freder…
In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.
We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. As an example we show that small positive surgeries on torus knots do not bound negative-definite four-manifolds.
We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.
