The automorphism group of a compact
complex manifold acts on its cohomology ring. Such action is faithful for
Riemann surfaces with genus bigger than one, but not always faithful on
algebraic surfaces or algebraic varieties of higher dimension. In [Cai-Liu-Zhang, Compos. Math. (2013)], Jin-Xing Cai and his collaborators showed the
action on a complex minimal surface of general type with irregularity bigger
than 2 is faithful. In [Cai, J. Algebra (2012)], Cai gave a complete classification and explicit construction of
surfaces of general type with automorphism group of order 4, geometric genus at
least 35 and irregularity at least 3.
3.
Contact Topology (DING Fan)
In [Ding-Geiges, J. Topol. (2009)] Fai Ding and his coauthor constructed the contact analogue of Kirby
calculus, namely, they gave the contact versions of various topological handle
moves, including handle cancellation and the first and second Kirby moves. This
continues their previous work on the surgery description of contact
three-manifolds from the standard contact three-sphere.
In [Ding-Geiges-van Koert, J. Lond. Math. Soc. (2) (2012)], Ding et al. gave
representations of a certain class of contact 5-manifolds via 2-dimensional
diagrams, and described moves on such diagrams that do not change the contact
5-manifold. As an application, they classified Stein fillable contact 5-manifolds
for which the 6-dimensional Stein filling has no 3-handles, up to connect sums.
In [Ding-Geiges, Compos. Math. (2010)], the authors computed the mapping class groups of S1×S2
using contact geometry, originally due to H. Gluck.
4.
Knot Theory (JIANG Boju, WANG Jiajun, ZHENG Hao)
In [Jiang, Acta Math. Sin. (Engl. Ser.) (2016)], Boju Jiang gave a new skein
characterization of the Conway potential function for colored links. As a
special case, they gave a new skein characterization of the Alexander-Conway
polynomial of knots. In [Jiang- Wang-Zheng, J.
Knot Theory Ramifications (2017)], they gave a skein characterization of the
HOMFLY polynomial. These characterizations do not use the smoothing of
crossings.
5.
Three-dimensional Topology (WANG Shicheng)
1) In [Derbez-Liu-Wang, J. Topol. (2015)], Shicheng Wang et al. considered the set
of volumes of all representations from the fundamental group of a
three-manifold to the isometry group of the hyperbolic three-space or the
universal cover of SL(2,R). They showed that a three-manifold with positive Gromov
norm or Seifert fibered piece in its JSJ decomposition has a finite cover with
nontrivial representation volume to one of the two groups. They also gave
examples that there are manifolds with trivial representation volumes in both
cases.
2) A three-manifold M is said to dominate
N if there is a nonzero degree map from M to N. In [Boileau-Rubinstein-Wang, Comment. Math. Helv. (2014)], Wang et al. showed
that a closed orientable 3-manifold dominates only finitely many integral
homology 3-spheres, and an integral homology 3-sphere 1-dominates at most
finitely many closed 3-manifolds.
3) In [Wang-Wang-Zhang-Zimmermann, Groups Geom. Dyn. (2015); Wang-Wang-
Zhang, Acta Math. Sin. (Engl. Ser.) (2016); Wang-Wang-Zhang, J. Knot Theory
Ramifications (2017)], they considered the finite group action on surfaces
which can extend to a group action on S3 for some embedding of the
surface in S3. For each genus, Wang et al. obtained a complete list of the
maximal order of such groups (finite and cyclic) and concretely constructed
these actions.
