Maps to the projective plane

自对偶映射 III：射影链环

Self-dual Maps III: projective links

本文给出了链环为射影的充分必要条件，并引入对称圈概念，提供防止射影链环成为交错链环的简单条件。

Symplectic maps to projective spaces and symplectic invariants

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

Indecomposability of branched coverings of even degree on the projective plane

In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.

Rational homology of configuration spaces of the Moebius strip and projective plane

All rational homology groups of unordered configuration spaces of the Moebius strip and the projective plane are calculated

Graphs immersions to the projective plane

Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.

Toward a geometric construction of fake projective planes

We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a $(2,3)$-elliptic surface that is the minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of ${\mathbb Q}$-homology projective planes with cusps only.

射影平面上最小亏格分支覆盖的可分解性

Decomposability of minimal defect branched coverings over the projective plane

本文刻画了射影平面上最小亏格原始分支覆盖的可分解性，扩展了实现族，并推广了原始置换群的相关结果。

On 2-knots and connected sums with projective planes

In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh's proof of the case that $K$ is a 2-bridge knot.

穿孔射影平面上极大曲线系统的唯一性

Uniqueness of Maximal Curve Systems on Punctured Projective Planes

本文研究穿孔射影平面上极大1-系统的唯一性，证明其唯一性成立当且仅当穿孔数不超过五。

Projective planes and their look-alikes

We classify all closed 1-connected manifolds $M$ which look like projective planes, i.e. with integral homology $H_*(M)=Z^3$. Furthermore, we give an explicit construction of these manifolds as Thom spaces of open disk bundles.

Exceptional collections and the bicanonical map of Keum's fake projective planes

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.

Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curves

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study …

Real aspects of the moduli space of genus zero stable maps

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

The topology of projective codes and the distribution of zeros of odd maps

We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean sp…

An inequality for length and volume in the complex projective plane

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.