arXiv subjects
Publications and source records attributed to Zhi Lü.
The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of the paper [DKS] (arXiv:1809.03678) to compute its integral equivariant cohomology, in terms of generators and relations, coming from the corresponding orbifold torus graph.
In this paper we state a problem on rigidity of powers, which has a strong topological background for the rigid Hirzebruch genera and Kosniowski's conjecture of unitary circle actions. However, our statement of this problem is elementary enough and does not require any knowledge of algebraic topology. We shall give a solution of this problem for some particular cases. As a consequence, we obtain that Kosniowski's conjecture holds in the case of dimension less than 11 or equal to 14.
We explore the connection between simple polytopes and self-dual binary codes via the theory of small covers. We first show that a small cover $M^n$ over a simple $n$-polytope $P^n$ produces a self-dual code in the sense of Kreck-Puppe if and only if $P^n$ is $n$-colorable and $n$ is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorial information of $P^n$. Moreover, we can define a family of binary codes $\mathfrak{B}_k(P^n)$, $0\leq k\leq n$, from an arbitrary simple $n$-polytope $P^n$. We will give some necessary and sufficient conditions for $\mathfrak{B}_k(P^n)$ to be a self-dual code. A spinoff of our study of such binary codes gives some new ways to judge whether a simple $n$-polytope $P^n$ is $n$-colorable in terms of the associated binary codes $\mathfrak{B}_k(P^n)$. In addition, we prove that the minimum distance of the self-dual binary code obtained from a $3$-colorable simple $3$-polytope is always $4$.
Let $P$ be a simple polytope of dimension $n$ with $m$ facets. In this paper we pay our attention on those elementary symmetric polynomials in the Stanley--Reisner face ring of $P$ and study how the decomposability of the $n$-th elementary symmetric polynomial influences on the combinatorics of $P$ and the topology and geometry of toric spaces over $P$. We give algebraic criterions of detecting the decomposability of $P$ and determining when $P$ is $n$-colorable in terms of the $n$-th elementary symmetric polynomial. In addition, we define the Stanley--Reisner {\em exterior} face ring $\mathcal{E}(K_P)$ of $P$, which is non-commutative in the case of ${\Bbb Z}$ coefficients, where $K_P$ is the boundary complex of dual of $P$. Then we obtain a criterion for the (real) Buchstaber invariant of $P$ to be $m-n$ in terms of the $n$-th elementary symmetric polynomial in $\mathcal{E}(K_P)$. Our results as above can directly associate with the topology and geometry of toric spaces over $P$. In particular, we show that the decomposability of the $n$-th elementary symmetric polynomial in $\mathcal{E}(K_P)$ with ${\Bbb Z}$ coefficients can detect the existence of the almost complex structures of quasitoric manifolds over $P$, and if the (real) Buchstaber invariant of $P$ is $m-n$, then there exists an essential relation between the $n$-th equivariant characteristic class of the (real) moment-angle manifold over $P$ in $\mathcal{E}(K_P)$ and the characteristic functions of $P$.
This note shows that for each $n\geq 5$ with only $n\not= 6$, there exists a $2n$-dimensional specially omnioriented quasitoric manifold $M^{2n}$ which represents a nonzero element in $Ω_*^U$. This provides the counterexamples of Buchstaber--Panov--Ray conjecture.
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented bordism classification of $n$-dimensional 2-torus manifolds. We show that the $G_n$-equivariant unoriented bordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, solving the conjecture posed in [19]. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented bordism classes of 2-torus manifolds of all possible dimensions is generated by the classes of all generalized real Bott manifolds (as special small covers over the products of simplices). This gives a strong connection between the computation of $G_n$-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of small covers. As a computational application, with the help of computer, we completely determine the structure of the group formed by equivariant bordism classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs.
In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the $h$-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of Euler characteristic for the classical configuration space, which generalizes the Félix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retract of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which turns out that we are able to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.
Let $M^{2n}$ be a unitary torus $(2n)$-manifold, i.e., a $(2n)$-dimensional oriented stable complex connected closed $T^n$-manifold having a nonempty fixed set. In this paper we show that $M$ bounds equivariantly if and only if the equivariant Chern numbers $< (c_1^{T^n})^i(c_2^{T^n})^j, [M]>=0$ for all $i, j\in {\Bbb N}$, where $c_l^{T^n}$ denotes the $l$th equivariant Chern class of $M$. As a consequence, we also show that if $M$ does not bound equivariantly then the number of fixed points is at least $\lceil{n\over2}\rceil+1$.
In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) \cite[Chapter 8, §2 Gluing Manifolds Together]{h}, we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism $P^3(m)$ with $m\geq 3$. We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with ${\Bbb Z}_2$-coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism $P^3(m)$ (i.e., cohomology rings with ${\Bbb Z}_2$-coefficients of all small covers over a $P^3(m)$ determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over $P^3(m)$.
Let $\mathcal{A}_n^k$ denote all nonbounding effective smooth $({\Bbb Z}_2)^k$-actions on $n$-dimensional smooth closed connected manifolds, each of which is cobordant to one with finite fixed set. Motivated by GKM theory, one can associate to each action of $\mathcal{A}_n^k$ a $({\Bbb Z}_2)^k$-colored regular graph of valence $n$. Together with the combinatorics of colored graphs, equivariant cobordism and the tom Dieck-Kosniowski-Stong localization theorem, we give a lower bound for the number of fixed points of an action in $\mathcal{A}_n^k$, which can become the best possible in some cases; we determine the existence and the equivariant cobordism classification of all actions in $\mathcal{A}_n^k(h)$ with $h=3,4$, where $\mathcal{A}_n^k(h)$ is the subset of $\mathcal{A}_n^k$, each of which is equivariantly cobordant to an effective $({\Bbb Z}_2)^k$-action fixing just $h$ isolated points, and it is well-known that $\mathcal{A}_n^k(h)$ is empty if $h=1,2$; we characterize the explicit relationships among tangent representations at fixed points of each action in $\mathcal{A}_n^k(h)$ with $h=3,4$, which actually give the explicit solution of the Smith problem in such cases. As an application, we also study the minimum number of fixed points of all actions in $\mathcal{A}_n^k$.
In this paper, we give an algebra-combinatorics formula of the Möbius transform for an abstract simplicial complex $K$ on $[m]=\{1, ..., m\}$ in terms of the Betti numbers of the Stanley-Reisner face ring of $K$. Furthermore, we employ a way of compressing $K$ to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex $\mathcal{Z}_K$ (resp. real moment-angle complex ${\Bbb R}\mathcal{Z}_K$) of $K$, we show that the Halperin-Carlsson conjecture holds for $\mathcal{Z}_K$ (resp. ${\Bbb R}\mathcal{Z}_K$) under the restriction of the natural $T^m$-action on $\mathcal{Z}_K$ (resp. $({\Bbb Z}_2)^m$-action on ${\Bbb R}\mathcal{Z}_K$).
In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(\mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $\mathbb{R}P^3$ and $S^1\times\mathbb{R}P^2$ with certain $(\mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({\Bbb Z}_2)^3$-equivariant unoriented cobordism.
As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(\Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice manifolds with corners. Based upon this, we investigate what kinds of closed manifolds admit locally standard $(\Z_2)^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;\Z_2)=0$, it is shown that $M$ admits a locally standard $(\Z_2)^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some $\Z_2$-homology sphere $N$, and if further assuming $M$ is irreducible, then $M$ must be homeomorphic to $S^3$. In addition, the argument is extended to rational homology 3-sphere $M$ with $H_1(M;\Z_2) \cong \Z_2$ and an additional assumption that the $(\Z_2)^3$-action has a fixed point.
In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of the equivariant index, we give an explicit description of the equivariant cohomology of such a $G$-manifold in terms of algebra, so that we can obtain analytic descriptions of ring isomorphisms among equivariant cohomology rings of such $G$-manifolds, and a necessary and sufficient condition that the equivariant cohomology rings of such two $G$-manifolds are isomorphic. This also leads us to analyze how many there are equivariant cohomology rings up to isomorphism for such $G$-manifolds in 2- and 3-dimensional cases.
A 2-torus manifold is a closed smooth manifold of dimension $n$ with an effective action of a 2-torus group $(\Z_2)^n$ of rank $n$, and it is said to be locally standard if it is locally isomorphic to a faithful representation of $(\Z_2)^n$ on $\R^n$. This paper studies the equivariant classification of locally standard 2-torus manifolds.
In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\mathbb Z_2)^{n+1}$-manifold.
It has been known that an effective smooth $({\Bbb Z}_2)^k$-action on a smooth connected closed manifold $M^n$ fixing a finite set can be associated to a $({\Bbb Z}_2)^k$-colored regular graph. In this paper, we consider abstract graphs $(Γ,α)$ of $({\Bbb Z}_2)^k$-actions, called abstract 1-skeletons. We study when an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action. We also study the existence of faces of an abstract 1-skeleton (note that faces often have certain geometric meanings if an abstract 1-skeleton is a colored graph of some $({\Bbb Z}_2)^k$-action).
This paper establishes an equivalence between existence of free involutions on $H{\Bbb C}P^3$ and existence of involutions on $S^6$ with fixed point set an imbedded $S^3$, then a family of counterexamples of the Smith conjecture for imbeddings of $S^3$ in $S^6$ are given by known result on $H{\Bbb C}P^3$. In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smooth free involution, which answers an open problem [Pe]. Moreover, the study of existence problem for smooth orientation preserving involutions on $H{\Bbb C}P^3$ is completed.