Fiberings with Fiber RP（κ）

Let κ be non-negative integer. The unoriented bordism classes, which can be represented as [RP（ξ^κ）] where ξ^κ is a k-plane bundle, form an ideal of the unoriented bordism ring MO.. A group of generators of this ideal expressed by a base of MO. and a necessary and sufficient condition for a bordism class to belong to this ideal are given.     

Bordism invariants of the mapping class group

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin bordism, and we prove that this single representation contains all of the information given by the Johnson homomorphism, the Birman-Craggs homomorphism, and the Morita homomorphism.     

不动点集为RP（1）×CP（N）的带有对合的流形

令 (M,T)是一个在光滑闭流形上的光滑对合 ,它的不动点集为 F ,本文确定了 F=RP(1 )× CP(N )的对合的协边分类     

The cobordism classification of involution on DOLD manifoldP(1,2l)

Supposing the smooth involution of the DOLD manifoldP(1,2l) satisfies the following condition: the fiberation π:P(1,2l)× _T×(−1)S^∞→RP(∞) is totally nonhomologous to zero (cf. [1, p373]), this paper determines the classification of smooth involution on the DOLD manifoldP(1,2l) totally. Supported by the Foundation of Tian Yuan and the Natural Science Foundation of Hebei province.     

Involution Fixing∪from (i=1) to r(RP(1))~m_i×(CP(1))~s_i*)

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Bordism theory and the Kervaire semi-characteristic

By using the bordism group, this paper provides an alternative proof of Weiping Zhangs’ theorem on counting Kervaire semi-characteristic.     

几乎自由的Z_2~m作用的Z_2~-不动点集

本文回答了“Z2m-1自由作用的未定向协边群里,哪些类可以作为带有几乎自由的Z2m作用的流形的Z2-不动点集被实现,且Z2-不动点集具有常余维数k？”的问题     

Pontryagin–Thom construction for approximation of mappings by embeddings

Let n3 and be positive integers, f :Sⁿ→Sⁿ be a C⁰-mapping, and denote the standard embedding. As an application of the Pontryagin–Thom construction in the special case of the two-point configuration space, we construct complete algebraic obstructions O(f) and to discrete and isotopic realizability (realizability as an embedding) of the mapping Jf. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y)Sⁿ×Sⁿf(x)=f(y), x≠y}.

A standard method of solving problems in differential topology is to translate them into homotopy theory by means of bordism theory and Pontryagin–Thom construction. By this method we give a generalization of the van-Kampen–Skopenkov obstruction to discrete realizability of f and the van-Kampen–Melikhov obstruction to isotopic realizability of f. The latter are complete only in the case d=0 and are the images of our obstructions under a Hurewicz homomorphism.

We consider several examples of computation of the obstructions.     

Fiberings with Fiber RP(k)

Let k be non–negative integer. The unoriented bordism classes, which can be represented as [RP(ξ ^k )] where ξ ^k is a k–plane bundle, form an ideal of the unoriented bordism ring MO_*. A group of generators of this ideal expressed by a base of MO_* and a necessary and sufficient condition for a bordism class to belong to this ideal are given. This work is supported by HNSF (Grant No: 103144) and NNSF of China (10371029)     

不动点集为RP(2m)(∪) P(2m,2n-1)的对合

Let (M2m+4n+k-2, T) be a smooth closed manifold with a smooth involution T whose fixed point set is RP(2m)(∪) P(2m, 2n - 1) (m > 3, n > 0). For 2n≥2m, (M2m+4n+k-2, T) is bordant to (P(2m, RP(2n)), To).