Characteristic Class of Isotopy for Surfaces

It is an important problem in topology to verify whether two embeddings are isotopic.This work proposes an algorithm for computing Haefliger-Wu invariants for isotopy based on algebraic topological methods.Given a simplicial complex embedded in the Euclidean space,the deleted product of it is the direct product with diagonal removed.The Gauss map transforms the deleted product to the unit sphere.The pull-back of the generator of the cohomology group of the sphere defines characteristic class of the isotopy of the embedding.By using Mayer Vietoris sequence and Ku¨nneth theorem,the computational algorithm can be greatly simplified.The authors prove the ranks of homology groups of the deleted product of a closed surface and give explicit construction of the generators of the homology groups of the deleted product.Numerical experimental results show the efficiency and efficacy of the proposed method.

Bordism Classes of Involutions on RP(2n)、CP(2n) and HP(2n)

In this paper we discuss all possible bordism classes of involutions on RP(2n),CP(2n) and HP(2n).

TOPOLOGY OF CONFIGURATION SPACE FOR A THREE-BODY PROBLEM

A mechanics system consisting of three mass points on sphere S 2 is considered. The configuration space of the system is a fibre bundle over S 2 . It is proved that first Chern class of the bundle is -2 c 1(γ) where γ is the canonical line bundle over the complex projective space CP 1=S 2 , which shows the bundle is non trivial. The information about the first Chern class makes the cohomology groups and homotopy groups of the configuration space worked out. In addition the effects of these topolo gical properties of the configuration space on the behavior in large scale of the system, as the number of equilibrium positions, periodic orbits and reduced phase space, are discussed.

INVOLUTIONS WITH FIXED POINT SET RP(2m)∪P(2m, 2n + 1)

Let （M, T） be a smooth closed manifold with a smooth involution T whose fixed point set is a disjoint union of an even-dimensional real projective space and a Dold manifold. In some cases, the equivariant bordism classes of （M, T） are determined.

Historical development of the Gauss-Bonnet theorem

A historical survey of the Gauss-Bonnet theorem from Gauss to Chern.

On the Relative Twist Formula of l-adic Sheaves

We propose a conjecture on the relative twist formula of l-adic sheaves,which can be viewed as a generalization of Kato-Saito’s conjecture.We verify this conjecture under some transversal assumptions.We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward.A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.

ETA INVARIANTS, DIFFERENTIAL CHARACTERS AND FLAT VECTOR BUNDLES

The purpose of this paper is to give a refinement of the Atiyah-Singer families index theorem at the level of differential characters. Also a Riemann-Roch-Grothendieck theorem for the direct image of flat vector bundles by proper submersions is proved,with Chern classes with coefficients in C/Q. These results are much related to prior work of Gillet-Soule, Bismut-Lott and Lott.

Involutions Fixing RP（2^m） U P（2^m, 2n- 1）

Let （M^2m＋4n＋k-2, T） be a smooth closed manifold with a smooth involution T whose fixed point set is RP（2^m） ∪ P（2^m, 2n - 1） （m 〉 3, n 〉 0）. For 2n ≥ 2^m, （M^2m＋4n＋k-2, T） is bordant to （P（2^m, RP（2n））, To）.