Authors:Yunhyung Cho; Min Kyu Kim; Dong Youp Suh Pages: 1197 - 1212 Abstract: The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if (M, ω) admits a Hamiltonian S 1-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c 1(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S 1-invariant symplectic 2-torus T in (M, ω) such that ‹c 1(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c 1(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ < 0, then G must be trivial, (2) if λ = 0, then the G-action is non-Hamiltonian, and (3) if λ > 0, then the G-action is Hamiltonian. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1031-7 Issue No:Vol. 38, No. 6 (2017)

Authors:Suyoung Choi; Boram Park; Hanchul Park Pages: 1213 - 1222 Abstract: The authors compute the (rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1032-6 Issue No:Vol. 38, No. 6 (2017)

Authors:Daciberg Lima Gonçalves; John Guaschi Pages: 1223 - 1246 Abstract: Let X be a topological space. In this survey the authors consider several types of configuration spaces, namely, the classical (usual) configuration spaces F n (X) and D n (X), the orbit configuration spaces F G n (X) and F G n (X)/S n with respect to a free action of a group G on X, and the graph configuration spaces F Г n (X) and F Г n (X)/H, where Г is a graph and H is a suitable subgroup of the symmetric group S n . The ordered configuration spaces F n (X), F G n (X), F Г n (X) are all subsets of the n-fold Cartesian product n Π 1 X of X with itself, and satisfy F G n (X) → F n (X) → F Г n (X) → n Π 1 X. If A denotes one of these configuration spaces, the authors analyse the difference between A and n Π 1 X from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion ι: A → n Π 1 X, the homotopy type of the homotopy fibre I ι of the map ι via certain constructions on various spaces that depend on X, and the long exact sequence in homotopy of the fibration involving I ι and arising from the inclusion ι. In this respect, if X is either a surface without boundary, in particular if X is the 2-sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space S k /G of the k-dimensional sphere by a free action of a Lie group G, the authors present recent results obtained by themselves for the first case, and in collaboration with Golasiński for the second and third cases. The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, a few of their basic properties are described and proved. A list of open questions and problems is given at the end of the paper. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1033-5 Issue No:Vol. 38, No. 6 (2017)

Authors:Hideya Kuwata; Mikiya Masuda; Haozhi Zeng Pages: 1247 - 1268 Abstract: The authors study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number p, the authors find a necessary condition for the integral cohomology of X to have no p-torsion. Then it is proved that the necessary condition is sufficient in some cases. The authors also give an example of X which shows that the necessary condition is not sufficient in general. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1034-4 Issue No:Vol. 38, No. 6 (2017)

Authors:Pascal Lambrechts; Jeremy Lane; Donald Stanley Pages: 1269 - 1274 Abstract: A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S 4n−1 → S 2n ×S m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1035-3 Issue No:Vol. 38, No. 6 (2017)

Authors:Zhiguo Li; Fengchun Lei; Jingyan Li Pages: 1275 - 1286 Abstract: The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones’s original construction. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1036-2 Issue No:Vol. 38, No. 6 (2017)

Authors:Ivan Limonchenko Pages: 1287 - 1302 Abstract: The author constructs a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology for any given n. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus Tm-action called moment-angle manifolds Z P , whose orbit spaces are simple n-dimensional polytopes P obtained from an n-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes P are flag nestohedra but not graph-associahedra. The author also describes the numbers β −i,2(i+1)(Q) for an associahedron Q in terms of its graph structure and relates it to the structure of the loop homology (Pontryagin algebra) H *(ΩZ Q ), and then studies higher Massey products in H *(Z Q ) for a graph-associahedron Q. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1037-1 Issue No:Vol. 38, No. 6 (2017)

Authors:Ximin Liu; Changtao Xue Pages: 1303 - 1310 Abstract: Let X be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to 2k(−E s ) ⊕ lH, where H is the hyperbolic form. In this paper, the authors prove that if there exists a locally linear pseudofree ℤ3-action on X, then Sign(g,X) ≡ −k mod 3. They also investigate the smoothability of locally linear ℤ3-action satisfying above congruence. In particular, it is proved that there exist some nonsmoothable locally linear ℤ3-actions on certain elliptic surfaces. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1038-0 Issue No:Vol. 38, No. 6 (2017)

Authors:Jiming Ma; Fangting Zheng Pages: 1311 - 1320 Abstract: In this paper, it is shown that for a 3-dimensional small cover M over a polytope P, there are only 2-torsions in H 1(M; Z). Moreover, the mod 2 Betti number growth of finite covers of M is studied. PubDate: 2017-11-01 DOI: 10.1007/s11401-007-1039-5 Issue No:Vol. 38, No. 6 (2017)

Authors:Hanchul Park Pages: 1321 - 1334 Abstract: This paper deals with two things. First, the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G. Second, the author focuses on a specific canonical extensions called doublings and presents their various properties. They include existence of infinitely many real topological toric manifolds admitting complex structures, and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups. Moreover, some questions about real topological toric manifolds related to Halperin’s toral rank conjecture are presented. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1040-6 Issue No:Vol. 38, No. 6 (2017)

Authors:Yi Sun Pages: 1335 - 1344 Abstract: Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes, which now can be seen as “colorings”. The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K, then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex. It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3. This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1041-5 Issue No:Vol. 38, No. 6 (2017)

Authors:Yusuke Suyama Pages: 1345 - 1352 Abstract: The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1042-4 Issue No:Vol. 38, No. 6 (2017)

Authors:Wei Wang Pages: 1353 - 1364 Abstract: This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1043-3 Issue No:Vol. 38, No. 6 (2017)

Authors:Baoqun Zhang; Xuezhi Zhao Pages: 1365 - 1372 Abstract: The authors consider the difference of Reidemeister traces and difference cochain of given two self-maps, and find out a relation involving these two invariants. As an application, an inductive formula of the Reidemeister traces for self-maps on a kind of CW-complex, including spherical manifolds is obtained. PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1044-2 Issue No:Vol. 38, No. 6 (2017)

Authors:Yan Zhao; Fengchun Lei; Fengling Li Pages: 1373 - 1380 Abstract: Let M be a compact connected 3-submanifold of the 3-sphere S 3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S 1, · · ·, S n } properly embedded in M, ∂S = {∂S 1, · · ·, ∂S n } is a complete curve system on F. We call S a complete surface system for M, and ∂S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup H n . PubDate: 2017-11-01 DOI: 10.1007/s11401-017-1045-1 Issue No:Vol. 38, No. 6 (2017)

Authors:Chao Gong; Yong Lin Pages: 1059 - 1070 Abstract: The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n). PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1022-8 Issue No:Vol. 38, No. 5 (2017)

Authors:Ghulam Mustafa Pages: 1077 - 1092 Abstract: The objective of this article is to introduce a generalized algorithm to produce the m-point n-ary approximating subdivision schemes (for any integer m, n ≥ 2). The proposed algorithm has been derived from uniform B-spline blending functions. In particular, we study statistical and geometrical/traditional methods for the model selection and assessment for selecting a subdivision curve from the proposed family of schemes to model noisy and noisy free data. Moreover, we also discuss the deviation of subdivision curves generated by proposed family of schemes from convex polygonal curve. Furthermore, visual performances of the schemes have been presented to compare numerically the Gibbs oscillations with the existing family of schemes. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1024-6 Issue No:Vol. 38, No. 5 (2017)

Authors:Zhihong Wen; Guantie Deng; Cuiqiao Wang; Feifei Qu Pages: 1093 - 1110 Abstract: This paper deals with two topics mentioned in the title. First, it is proved that function f in L p (∂D a ) can be decomposed into a sum g + h, where D a is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H p (D a ) and \({H^p}\left( {\overline D _a^c} \right)\) in the sense of L p (D a ), respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1025-5 Issue No:Vol. 38, No. 5 (2017)

Authors:Heman Fu; Feng Tan Pages: 1119 - 1130 Abstract: For each real number λ ∈ 2 [0, 1], λ-power distributional chaos has been introduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as λ varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos, λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally (n + 1)-scrambled tuples. For each λ ∈ 2 [0, 1], λ-power distributional n-chaos can still appear in minimal systems with zero topological entropy. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1027-3 Issue No:Vol. 38, No. 5 (2017)

Authors:Tongzhu Li Pages: 1131 - 1144 Abstract: Let x: M n → S n+1 be an immersed hypersurface in the (n + 1)-dimensional sphere S n+1. If, for any points p, q ∈ M n , there exists a Möbius transformation ϕ: S n+1 → S n+1 such that ϕ ○ x(M n ) = x(M n ) and ϕ ○ x(p) = x(q), then the hypersurface is called a Möbius homogeneous hypersurface. In this paper, the Möbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Möbius transformation. PubDate: 2017-09-01 DOI: 10.1007/s11401-017-1028-2 Issue No:Vol. 38, No. 5 (2017)