Surfaces in the complex projective plane and their mapping class groups

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。

Smooth, nonsymplectic embeddings of rational balls in the complex projective plane

We exhibit an infinite family of rational homology balls which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson's diagonalisation theorem, and use this to show that no two of our examples may be embedded disjointly.

Immersed projective planes, Arf invariants and even 4-manifolds

In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.

Topological structure of Morse functions on the projective plane

To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We prove recurent formulas for the number of Reeb graphs with the given number of sadles (vertex).

The mapping class group of manifolds which are like projective planes

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 $\dim M \ne 4$.

映射类群的极限集

Limit sets of mapping class groups

该研究在射影测度叶状结构空间中为映射类群子群引入两类极限点概念，将Teichmüller测地线射线分类，且证明分类可由定向叶状结构的拓扑性质确定。

A Projective Representation of the Modular Group

Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest non-commutative torus which remains fixed by all elements of the mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.

实射影平面的辫群、不可定向曲面的映射类群及其群环的代数K理论

Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings

该研究刻画了3弦实射影平面辫群及3标记点实射影平面映射类群整群环的低次代数K理论，并给出了k≥3个标记点的不可定向曲面映射类群群环的代数K理论群一般公式。

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.

The octonionic projective plane

This small note, without claim of originality, constructs the projective plane over the octonionic numbers and recalls how this can be used to rule out the existence of higher-dimensional real division algebras, using Adams' solution of the Hopf invariant $1$ problem.

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 $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.

Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane

We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves on results of Berrick and Matthey in the case of S^2, and extends them in the case of RP^2. We begin by recalling the situation for the Artin braid groups. As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show th…

Extensions of maps to the projective plane

It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.

关于32维Rosenfeld射影平面

On the 32-dimensional Rosenfeld projective plane

本文在已有代数拓扑相关研究基础上，对属于嘉当紧对称空间列表中EIII类的32维Rosenfeld射影平面进行了详细研究。

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 …

On the cohomology of the mapping class group of the punctured projective plane

The mapping class group $Γ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/Σ_k \to X_k \to BSO(3)$ where $X_k$ is a $K(Γ^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/Σ_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. As a consequence, we express the mod-2 cohomol…

具有小指数商奇点的有理同调射影平面研究

On rational homology projective planes with quotient singularities of small indices

本文研究拓扑与光滑阻碍对带小指数商奇点的有理同调复射影平面存在性的影响，完成了指数≤3且光滑轨迹一阶整同调平凡的这类曲面的商奇点类型分类。

射影平面中凸区域的极限多边形

Limit polygons of convex domains in the projective plane

本文研究了实射影平面中真凸区域序列在Pick微分零点重数有界且间距不断增大的条件下，经射影变换规范化后的所有Hausdorff极限区域，并证明该结果适用于射影三角反射群生成的凸区域。

A diffeomorphism classification of manifolds which are like projective planes

We give a complete diffeomorphism classification of 1-connected manifolds (of dimension different from 4) whose integral homology is H(M)=Z+Z+Z.

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.

The Toledo invariant, and Seshadri constants of fake projective planes

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\C$-Fuchsian curves in complex hyperbolic spaces.

Hochschild上同调、模张量范畴与映射类群

Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups

本文针对有限模张量范畴，将带边界紧曲面关联至上链复形，证明曲面映射类群在其上同调群上射影作用，零度时与手征共形块空间已知作用一致，并在环面与可因子化带状Hopf代数情形下恢复先前结果。