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For a Legendrian link $\Lambda \subset J^1M$ with $M = \mathbb{R}$ or $S^1$, immersed exact Lagrangian fillings $L \subset \mbox{Symp}(J^1M) \cong T^*(\mathbb{R}_{>0} \times M)$ of $\Lambda$ can be lifted to conical Legendrian fillings $\Sigma \subset J^1(\mathbb{R}_{>0} \times M)$ of $\Lambda$. When $\Sigma$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation $\alpha: \mathcal{A}(\Sigma) \rightarrow \mathbb{Z}/2$ of the LCH algebra of $\Sigma$, there is an induced augmentation $\epsilon_{(\Sigma,\alpha)}: \mathcal{A}(\Lambda) \rightarrow \mathbb{Z}/2$. With $\Sigma$ fixed, the set of homotopy classes of all such induced augmentations, $I_\Sigma \subset \mathit{Aug}(\Lambda)/{\sim}$, is a Legendrian isotopy invariant of $\Sigma$. We establish methods to compute $I_\Sigma$ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary $n \geq 1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $\rho \neq 1$ and $\Lambda \subset J^1\mathbb{R}$ every $\rho$-graded augmentation of $\Lambda$ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of $\rho$-graded augmented Legendrian cobordism., Comment: 51 pages, 22 figures. Accepted version to appear in Journal of Topology. Version 2 is shorter than Version 1 with more efficient exposition. In places, readers desiring more details are referred to Version 1

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold ( M n , g ) (M^n,g) is always pluriclosed, and when the manifold is compact and g g is not Kähler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2 n=2 , the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n ≥ 3 n\geq 3 , and the manifold does not admit any Hermitian symplectic metric.

The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones., 63 pages; v3: perturbative version of the Viterbo conjecture now proven for arbitrary symplectic capacities, added more properties to the normal form, added a statement on the local rigidity of Zoll contact forms

We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group diagram of $M$ such that the $T$-action on $M$ is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples., 18 pages, 6 figures, 3 tables. Minor changes, in particular additional explanations were added in the proof of Proposition 2.3. Proposition 3.6 added. Results are unchanged

In our previous work, we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko’s results on permutation representations.

We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak T(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HFK^-(T_{(p,q)})$ can be realized. Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots., Comment: 17 pages, 3 figures. To appear in Quantum Topol

In this note, we give a short proof of the localization formula for the loop space Chern character of a compact Riemannian spin manifold M, using the rescaled spinor bundle on the tangent groupoid associated to M., 19 pages

We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular we show that in low dimensions $d \leq 4$ a homologically regular point on a PL $d$-manifold is always strongly regular. Examples show that this fails to hold in higher dimensions $d \geq 5$. One of our constructions involves an 8-vertex embedding of the dunce hat into a polytopal 4-sphere with 8 vertices such that a regular neighborhood is Mazur's contractible 4-manifold., 24 pages, 3 figures

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur., Comment: v3: final version

We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6., To appear in Comm. Math. Helv

We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition., 27 pages

Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is well-defined up to quasi-isomorphism., Comment: 31 pages

Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots., 29 pages, 9 figures

A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural non-associative algebra $\mathfrak{m}$ associated to the affine structure of the Stiefel manifold, computing explicitly the automorphism group $Aut(\mathfrak{m})$ and inducing this computation to the Isometry group.

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperk\"ahler variety as a Fano visitor., Comment: 34 pages, nearly identical to published version

Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a network category whose isomorphism classes of objects correspond to the orbits of quiver representations, in which neural networks map input data. We then prove that the output of a neural network depends only on the corresponding point in the moduli space. Finally, we present a different perspective on mapping neural networks with a specific activation function, called ReLU, to a moduli space using the symplectic reduction approach to quiver moduli., 27 pages

In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_\infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_\infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_\infty$-morphism. Finally, we perfom a comparison with other reduction procedures., 30 pages, comments are welcome!

We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, in particular giving the first Floer-theoretic proof that Legendrian (n,m) torus links have infinitely many Lagrangian fillings, if n is greater than 3 and m greater than 6, or (n,m)=(4,4),(4,5). In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients., Comment: 82 pages; version accepted for publication in the Journal of Topology

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$, are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained., Comment: 18 pages, 1 diagram

Perelman has proved that there cannot exist a nontrivial breather for Ricci flow on a closed manifold. Here we construct nontrivial expanding breathers for Ricci flow in all dimensions when the underlying manifold is allowed to be noncompact., Comment: Version 2 contains new remark pointing out that the same technique works for other geometric flows such as mean curvature flow

We prove that the quotient of the integer homology cobordism group by the subgroup generated by the Seifert fibered spaces is infinitely generated., Comment: 22 pages, 11 figures. v4: Minor expositional improvements. This version to be published in Transactions of the American Mathematical Society

We give a topological explanation of the main results of V.Shchigolev, Categories of Bott-Samelson Varieties, Algebras and Representation Theory, 23 (2), 349-391, 2020. To this end, we consider some subspaces of Bott-Samelson varieties invariant under the action of the maximal compact torus $K$ and study their topological and homological properties. Moreover, we describe multiplicative generators of the equivariant cohomologies of Bott-Samelson varieties.

In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapani to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive., 22 pages, final version

We set up a topological framework for degenerations of symplectic manifolds into singular spaces paying a special attention to the behavior of Lagrangian manifolds and their (holomorphic) membranes. We show that degenerations into singular toric varieties provide a source of exotic Lagrangian tori., Comment: accepted version

In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M \times [0, \infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $\pi_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $M$, and the fiber is a closed cone in the space of cubic differentials. For 3-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus., Comment: 42 pages, 1 figure, comments welcome

In this paper, we establish an equivariant version of Dai-Zhang's Toeplitz index theorem for compact odd-dimensional spin manifolds with even-dimensional boundary., Comment: 26 pages, 1 figure. 2 appendices. Published version

We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem., Comment: 36 pages, 2 figures. In v3, theorems and definitions were renumbered to align with journal style. To appear in Commentarii Mathematici Helvetici

In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two., 14 pages

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of $b$-symplectic manifolds started in [12], we prove a slice theorem for Lie group actions on $b$-symplectic manifolds., Comment: 18 pages, 2 figure, final form accepted at Bulletin of the London Mathematical Society

We consider sums of squares operators globally defined on the torus. We show that if some assumptions are satisfied the operators are globally analytic hypoelliptic. The purpose of the assumptions is to rule out the existence of a Hamilton leaf on the characteristic variety lying along the fiber of the cotangent bundle, i.e. the case of the (global) M\'etivier operator.

Cross-manifold clustering is a hard topic and many traditional clustering methods fail because of the cross-manifold structures. In this paper, we propose a Multiple Flat Projections Clustering (MFPC) to deal with cross-manifold clustering problems. In our MFPC, the given samples are projected into multiple subspaces to discover the global structures of the implicit manifolds. Thus, the cross-manifold clusters are distinguished from the various projections. Further, our MFPC is extended to nonlinear manifold clustering via kernel tricks to deal with more complex cross-manifold clustering. A series of non-convex matrix optimization problems in MFPC are solved by a proposed recursive algorithm. The synthetic tests show that our MFPC works on the cross-manifold structures well. Moreover, experimental results on the benchmark datasets show the excellent performance of our MFPC compared with some state-of-the-art clustering methods., Comment: 12 pages, 58 figures

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula.

In this paper, we consider double ramification cycles with orbifold targets. An explicit formula for double ramification cycles with orbifold targets, which is parallel to and generalizes the one known for the smooth case, is provided. Some applications for orbifold Gromov--Witten theory are also included., Comment: 49 pages; typo fixed; overall expression improved; comments are welcome!

We prove that if the moduli $\mathbb Q$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb Q$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve., Comment: 18 pages, v2: minor revisions following the referee's comments, v3: very minor modifications, v4: very minor modifications

Bivariant theory is a unified framework for cohomology and Borel-Moore homology theories. In this paper, we extract an $\infty$-enhanced bivariant homology theory from Gaitsgory-Rozenblyum's six functor formalism.

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting we show that under certain conditions -- which in particular hold for a compact orientable manifold with discrete fixed point set -- the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold., 16 pages

We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the "Dirac-Nijenhuis" structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case., Comment: 35 pages

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $\mathbb S^1$-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the $n$-torus has a unique symplectically aspherical strong filling up to diffeomorphism., Comment: v5: Final version of the paper. To appear in Inventiones Mathematicae. arXiv admin note: text overlap with arXiv:1903.11866

We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the $C^0$-metric and the Hofer metric. This answers a variant of a question of Entov--Polterovich--Py which is one of the open problems listed in the monograph of McDuff--Salamon. Throughout the proof, we make extensive use of the idea of working with different coefficient fields in quantum cohomology rings. As a by-product of the arguments in the paper, we answer a question of Polterovich--Wu regarding quasimorphisms on the group of Hamiltonian diffeomorphisms of the complex projective plane and prove some intersection results about Lagrangians in the four dimensional quadric hypersurface., v2: 40 pages, minor corrections, v3: 45 pages, accepted version, to appear in Commentarii Mathematici Helvetici

In the present paper, we introduce the notion of $\ast$-Miao-Tam critical equation on almost contact metric manifolds and studied on a class of almost Kenmotsu manifold. It is shown that if the metric of a $(2n + 1)$-dimensional $(k,\mu)'$-almost Kenmotsu manifold $(M,g)$ satisfies the $\ast$-Miao-Tam critical equation, then the manifold $(M,g)$ is $\ast$-Ricci flat and locally isometric to the Riemannian product of a $(n + 1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold. Finally, an illustrative example is presented to support the main theorem., Comment: 10 pages

We consider a family of smooth del Pezzo surfaces of degree four and study the geometry and arithmetic of a genus one fibration with two reducible fibres for which a Brauer element is vertical., Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:2002.11539

We prove that Morse local-to-global groups grow exponentially faster than their infinite index stable subgroups. This generalizes a result of Dahmani, Futer, and Wise in the context of quasi-convex subgroups of hyperbolic groups to a broad class of groups that contains the mapping class group, CAT(0) groups, and the fundamental groups of closed 3-manifolds. To accomplish this, we develop a theory of automatic structures on Morse geodesics in Morse local-to-global groups. Other applications of these automatic structures include a description of stable subgroups in terms of regular languages, rationality of the growth of stable subgroups, density in the Morse boundary of the attracting fixed points of Morse elements, and containment of the Morse boundary inside the limit set of any infinite normal subgroup., Comment: Version to appear in Journal of Topology. Minor updates, including a new corollary on the non-existance of a Cannon--Thurston map from the boundary of a hyperbolic normal subgroup to the Morse boundary of the ambient Morse local-to-global group

In 1985 Diederich and Ohsawa proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains, we show that it is hyperconvex.

Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode the groupoid cardinalities of their graded components. As suggested in the work of Baez and Dolan, we use analytic continuation of such generating functions to define a complex-valued cardinality for groupoids whose usual groupoid cardinality diverges. The complex nature of such a cardinality invariant is shown to reflect a recursion of structure which we refer to as `nested equivalence'., No figures

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog can be interpreted as a quantitatve statement on the limitation for Gromov's Oka principle. For any finite open Riemann surface $X$ (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from $X$ to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface. If $X_{\sigma}$ is the $\sigma$-neighbourhood of a skeleton of an open Riemann surface with finitely generated fundamental group, then the number of irreducible holomorphic mappings up to homotopy from $X_{\sigma}$ to the twice punctured complex plane grows exponentially in $\frac{1}{\sigma}$., Comment: 51 pages, 4 figures

Kreck's modified surgery theory reduces the classification of closed, connected 4-manifolds, up to connect sum with some number of copies of $S^2\times S^2$, to a series of bordism questions. We implement this in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, the computations simplify considerably. We use this to solve some cases in which $\pi_1(M)$ is finite of order 2 mod 4: under an assumption on cohomology, there are nine stable diffeomorphism classes for which M is pin$^+$, one stable diffeomorphism class for which M is pin$^-$, and four stable diffeomorphism classes for which M is neither. We also determine the corresponding stable homeomorphism classes., Comment: 10 pages. Comments welcome! v2: fixed an error in an important lemma

This paper seeks to demonstrate the relation between homology group and homotopy group. The result in this paper is a construction of the homology of a complex torus.

Suppose $(M, \gamma)$ is a balanced sutured manifold and $K$ is a rationally null-homologous knot in $M$. It is known that the rank of the sutured Floer homology of $M\backslash N(K)$ is at least twice the rank of the sutured Floer homology of $M$. This paper studies the properties of $K$ when the equality is achieved for instanton homology. As an application, we show that if $L\subset S^3$ is a fixed link and $K$ is a knot in the complement of $L$, then the instanton link Floer homology of $L\cup K$ achieves the minimum rank if and only if $K$ is the unknot in $S^3\backslash L$., Comment: 18 pages, 1 figures, comments are welcome!