Topic
Stable map
About: Stable map is a research topic. Over the lifetime, 98 publications have been published within this topic receiving 1330 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, the authors define a Deligne-Mumford stack Xp r which depends on a scheme X, an effective Carrier divisor D C X, and a positive integer r. This construction was known to several mathematicians before they became aware of it, but very little appeared in the literature aside from a brief mention in (AGV, 3.5.
Abstract: We define a Deligne-Mumford stack Xp r which depends on a scheme X, an effective Carrier divisor D C X, and a positive integer r. Then we show that the Abramovich-Vistoli mod- uli stack of stable maps into Xq r provides compactifications of the locally closed substacks of Mgtn(X, (3) corresponding to relative stable maps. 1. Introduction. This paper has two main purposes. The first is to introduce the rth root construction, which is a way of creating stack structure along a divisor. Given a scheme X, an effective Cartier divisor D C X, and a positive integer r which is invertible on X, it produces a Deligne-Mumford stack Xo,r- When r is not invertible, the construction still makes sense and produces an Artin stack which has finite diagonal over X. Since the output is in general an Artin stack, we give the definition and basic properties when X is an Artin stack. This simplifies the exposition, since we need to apply the construction multiple times. This construction was known to several mathematicians before we became aware of it, but very little appeared in the literature aside from a brief mention in (AGV, 3.5.3). Secondly, we compare twisted stable maps into Xp,r with ordinary stable maps into X. The theory of twisted stable maps was developed by Chen and Ruan (CR) in the symplectic category and by Abramovich and Vistoli (AV) in the algebraic category. The latter showed that there is a Deligne-Mumford stack Xg,n(X,(3) parametrizing morphisms/: £ - ► X from twisted rc-marked genus g nodal curves £ into a Deligne-Mumford stack X such that/*(£) = (3. In the case where X is a scheme, this is just the space of Kontsevich stable maps Mg,n(X, (3). The theory of relative stable maps was developed by several symplectic ge- ometers, and the algebraic definition was worked out partially by Andreas Gath- mann (Ga) and in general by Jun Li (Li). Let X be a smooth complex projective veriety, D C X a smooth divisor, and (3 G B\(X) such that D • (3 > 0. Here B\(X) denotes the group of 1 -cycles modulo algebraic equivalence. Choose a partition D - (3 = q\ + - - + Qn, where each qi is a nonnegative integer. Then a stable map /: C - ► X from a smooth curve C having marked points x\, . . . ,xn such that f*(C) = (3 and/*D = Y.Qixi ls a relative stable map. The space of such mor- phisms was compactified by (Ga) and (Li) and they used this to define relative Gromov-Witten invariants.
313 citations
TL;DR: In this article, it was shown that every 3-manifold can be constructed from a 3-Manifold M 3 of complexity n to a 4-manivold M 4 with shadow complexity n, where the complexity of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation.
Abstract: It has been known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, but little attention has been paid to the complexity of the 4-manifold produced. Given a 3-manifold M 3 of complexity n, we construct a 4-manifold bounded by M of complexity , where the ‘complexity’ of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation.The proof goes through the notion of ‘shadow complexity’ of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We further prove that, for a manifold M satisfying the geometrization conjecture with Gromov norm G and shadow complexity S, we have , for suitable constants , . In particular, the manifolds with shadow complexity 0 are the graph manifolds.In addition, we give an bound for the complexity of a spin 4-manifold bounding a given spin 3-manifold. We also show that every stable map from a 3-manifold M with Gromov norm G to has at least crossing singularities, and if M is hyperbolic there is a map with at most crossing singularities.
105 citations
1 Jan 1985
TL;DR: In this paper, the authors present a method to induce coincidence between one map and another, regardless of their respective projections, by transforming the map to a rubber sheet stretched to coincide with a stable base map at an average of 15 control points.
Abstract: We present a computationally simple and inexpensive method to induce coincidence between one map and another, regardless of their respective projections. We imagine the map to be transformed on a rubber sheet stretched to coincide with a stable base map at an average of 15 control points; for more precise coincidence, we add more control points. We compute triangles on the rubber-sheet map, using control points as vertices, and with a procedure called piecewise linear homeomorphism, map these triangles linearly onto corresponding triangles on the stable map.
59 citations
TL;DR: In this article, the authors define a global surgery operation on an n-dimensional manifold, which simplifies the configuration of the critical value set and does not change the diffeomorphism type of the source manifold.
Abstract: Let f: M -R2 be a C? stable map of an n-dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on f which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold M. For this purpose, we also study the quotient space Wf of f, which is the space of the connected components of the fibers of f, and we completely determine its local structure for arbitrary dimension n of the source manifold M. This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuya for orientable manifolds of dimension 4. We also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.
54 citations
TL;DR: In this paper, the authors studied various topological properties of generic smooth maps between manifolds whose regular fibers are disjoint unions of homotopy spheres and showed that if a closed 4-manifold admits such a generic map into a surface, then it bounds a 5-Manifold with nice properties.
Abstract: In this paper, we study various topological properties of generic smooth maps between manifolds whose regular fibers are disjoint unions of homotopy spheres. In particular, we show that if a closed 4 -manifold admits such a generic map into a surface,then it bounds a 5 -manifold with nice properties. As a corollary, we show that each regular fiber of such a generic map of the 4 -sphere into the plane is a homotopy ribbon 2 -link and that any spun 2 -knot of a classical knot can be realized as a component of a regular fiber of such a map.
52 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 2 |
| 2019 | 2 |
| 2018 | 1 |
| 2017 | 4 |
| 2016 | 1 |