Cyclic Group Action of Composite Order on Indefinite 4-manifolds

In this paper, we study the possibilities for several kinds of topological, locally linear cyclic group actions of non-prime order on some closed, simply connected 4-manifolds with indefinite intersection form. Especially, we discuss the existence of locally linear pseudofree C9 action on this kind of 4-manifolds.

NORMAL EULER NUMBERS OF EMBEDDING NONORIENTABLE SURFACES IN 4-MANIFOLDS

ＮＯＲＭＡＬＥＵＬＥＲＮＵＭＢＥＲＳＯＦＥＭＢＥＤＤＩＮＧＮＯＮＯＲＩＥＮＴＡＢＬＥＳＵＲＦＡＣＥＳＩＮ４－ＭＡＮＩＦＯＬＤＳＧＡＯＨｏｎｇｚｈｕ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＢｅｉｊｉｎｇＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉ...

Branched Coverings and Embedded Surfaces in Four-manifolds

In this paper, we get a lower bound on the genera of surfaces re, presenting certain divisible classes in some 4-manifold X with Hi（X; Z） finite.

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).

不可定向四维流形的一些例子(英文)

利用 Ｔａｕｂｅｓ的结果，给出了一些紧致带边不可定向四维拓扑流形，它们的 Ｋｉｒｂｙ－Ｓｅｉｂｅｎ－ｍａｎｎ不变量平凡。

A Note on Donaldson's “Tamed to Compatible” Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson＇s ＂tamed to compatible＂ question （Draghici T, Zhang W. A note on exact forms on almost complex manifolds, arXiv： 1111. 7287vl [math. SC]. Submitted on 30 Nov. 2011）. That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure？ They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson＇s question in dimension 4. In this note, we give another simpler proof of their result.

ON NORMAL EULER NUMBERS OF EMBEDDING RP^2 IN INDEFINITE 4-MANIFOLDS

Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle vf. The normal Euler class e（vf） of f is an element in H2（N,）, where is the local coefficient determined by w1（vf） =w1（N）. It is very important to determine e（vf）[N] for all embeddings. This problem is closely related to whether a two-dimensional homology class can be represented by a smooth embedded sphere. In this note, we determine all the possible normal Euler numbers of embedding real projective plane into indefinite 4-manifolds.

Two-dimensional submanifolds in non-simply connected 4-manifolds

A foundamental problem in 4-dimensional differential topology iS to find a surface with minimal genus that can represent a given two-dimensional homology class. Hsiang, Rohlin have got some results about the homological l-connected manifolds. This note will discuss this problem under the condition that H1（N4） is finite. Under this assumption, the possible normal Euler numbers of embedding nonorientable surfaces in 4-manifolds are also determined.

ON NORMAL EULER NUMBERS OF EMBEDDING SURFACES INTO 4-MANIFOLDS

Let N be a closed,orientable 4-manifold satisfying H1（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H2（M,（?））,where （?） denotes the twisted integer coefficients determined byw1（v）=w1（M）.We study the possible values of e（v）[M],and prove H1（N-M）=Z2 or 0.Underthe condition of H1（N-M,Z）=Z2,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β2）,2σ（N）-2（n+β2-2）,2σ（N）-2（n+β2-4）,…,2σ（N）+2（n+β2）,where σ（N） is the usual index of N,n the nonorientable genus of M and β2 the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S4.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.

Rational-slice Knots via Strongly Negative-amphicheiral Knots

We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.

GENUS MINIMIZING IN SYMPLECTIC 4-MANIFOLDS

The authors show that a symplectically embedded surface in a symplectic 4-manifold with hi greater than one minimizes genus in its homology class.

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori,Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.