Shadows of 4-manifolds with complexity zero and polyhedral collapsing

@article{Naoe2016ShadowsO4,
  title={Shadows of 4-manifolds with complexity zero and polyhedral collapsing},
  author={Hironobu Naoe},
  journal={arXiv: Geometric Topology},
  year={2016}
}
  • Hironobu Naoe
  • Published 1 May 2016
  • Mathematics
  • arXiv: Geometric Topology
Our purpose is to classify acyclic 4-manifolds having shadow complexity zero. In this paper, we focus on simple polyhedra and discuss this problem combinatorially. We consider a shadowed polyhedron $X$ and a simple polyhedron $X_0$ that is obtained by collapsing from $X$. Then we prove that there exists a canonical way to equip internal regions of $X_0$ with gleams so that two 4-manifolds reconstructed from $X_0$ and $X$ are diffeomorphic. We also show that any acyclic simple polyhedron whose… 
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References

SHOWING 1-10 OF 13 REFERENCES
Stein domains and branched shadows of 4-manifolds
We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact four-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds…
Four-manifolds with shadow-complexity one
We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev shadows. This complexity roughly measures how complicated the…
Stable maps and branched shadows of 3-manifolds
In the early 1990s, Turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3-manifolds. Later, Costantino–Thurston revealed a strong relation between the Stein…
Pants Decompositions of Surfaces
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices…
BRANCHED SHADOWS AND COMPLEX STRUCTURES ON 4-MANIFOLDS
We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped…
Shadows, ribbon surfaces, and quantum invariants
Eisermann has shown that the Jones polynomial of a $n$-component ribbon link $L\subset S^3$ is divided by the Jones polynomial of the trivial $n$-component link. We improve this theorem by extending…
Complexity of 4-Manifolds
We define and study a notion of complexity for smooth, closed and orientable 4-manifolds. This notion, based on the theory of Turaev shadows, represents the 4-dimensional analogue of Matveev's…
Smoothings of piecewise linear manifolds
The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy…
3‐manifolds efficiently bound 4‐manifolds
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…
Introduction to Piecewise-Linear Topology
1. Polyhedra and P.L. Maps.- Basic Notation.- Joins and Cones.- Polyhedra.- Piecewise-Linear Maps.- The Standard Mistake.- P. L. Embeddings.- Manifolds.- Balls and Spheres.- The Poincare Conjecture…
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