关于3维球面中与混合多项式相关的实代数链环

Algebraic links in lens spaces

The lens space $L_{p,q}$ is the orbit space of a $\mathbb{Z}_{p}$-action on the three sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in $L_{p,q}$. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in lens spaces, whose lift in the $3$-sphere is a torus link.

A survey of surface braid groups and the lower algebraic K-theory of their group rings

We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the homotopy groups of the 2-sphere. The braid groups of the 2-sphere and the real projective plane are of particular interest because they possess elements of finite order, and we discuss in detail their torsion and the classification of their finite and virtually cyclic subgroups. F…

局部像矩映射那样重构具有指定像、且与到1维实仿射空间的典范投射复合的实代数映射

Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space

本研究提出一类局部类似矩映射、原像为单点集或球面乘积的非正余维数显式实代数映射，补充了Nash-Tognoli理论下显式实代数映射实例不足的问题，并给出其与典范投射复合的显式函数族。

The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B\_4(\mathbb{S}^2)]$

We study $K$-theoretical aspects of the braid groups $B\_n(\mathbb{S}^{2})$ on $n$ strings of the $2$-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\cite{JM}. In light of this, in order to determine the algebraic $K$-theory of the group ring $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon{\c c}alves and the first author. We calculate the Whitehead and $K\_{-1}$-groups of the group rings of the finite subgroups (dicyclic and binary po…

关于 codimension $-1$ 的 round fold 映射的实代数实现

On real algebraic realization of round fold maps of codimension $-1$

本文研究了 codimension $-1$ 的 round fold 映射的实代数实现，探讨其在实代数几何中的基础和重要性。

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a…

SO(3) invariants of Seifert manifolds and their algebraic integrality

For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), τ^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is proved that $(\sqrt {\dfrac{4}{r}}\sin \dfracπ{r})^ντ^{'}_r(M)$ is an algebraic integer in the r-th cyclotomic field, where $ν$ is the first Betti number of $M$. For the torus bundle obtained from trefoil knot with framing 0, i.e. $X_{tref}(0)=X(-2/_{\f{1}},3/_{\f{1}},6/_{\f{1}}), τ^{'}_r$ is obtained in a simple form if $3\mid\llap /r$, which shows in some sen…