不动点集为常余维数6的(Z2)2作用

记J_n,k^r为具有如下性质的n维未定向上协边类α构成的集合:存在α的一个代表元Mⁿ及(Z₂)^k在Mⁿ上的作用,其不动点集为常余维数r .记J_n,k^r=∑_nJ_n,k^r,则J_n,k^r为未定向上协边环MO_n=     

Two-weight estimates for singular and strongly singular integral operators

We give a Pontryagin-Thom type construction for Stein factorizations of fold maps of 3-manifolds into the plane. As an application, we compute the cobordism group of Stein factorizations of fold maps of oriented 3-manifolds into the plane and the oriented cobordism group of fold maps of 3-manifolds into the plane. It turns out that these two groups are isomorphic to Z ₂ ⊕ Z ₂. We have the analogous results about bordism groups as well.      

Oriented Cohomology Theories of Algebraic Varieties

This article contains proofs of the results announced by Panin and Smirnov (http://www. math.uiuc.edu/k-theory/0459/2000) in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples.     

Bak groups and equivariant surgery II

The previous paper showed that theG-surgery obstructions ofG-normal maps lie in the Bak groups. That paper remarked that in even-dimensional cases, theG-surgery obstruction is invariant under suitable cobordisms. This paper presents cobordism invariance theorems for theG-surgery obstruction not only in even-dimensional cases but also in odd-dimensional ones. We prove Theorems B-D by detaching equivariant issues from the singular sets and then by using arguments of C. T. C. Wall in ordinary surgery theory. We still need, however, to argue carefully, especially in the odd-dimensional cases. Actually, this paper contains details which are skipped over in Wall's work.Partially supported by Grant-in-aid for Development of Young Scientists.Dedicated to Kazutoshi Morioka on his 60th birthday     

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.     

A homomorphism between circuit cobordism groups and pseudohomology groups over configuration space <Emphasis Type="Italic">B</Emphasis>

We develop a new method to perturb singular circuits in configuration space B with trivial isotropy groups, and construct a homomorphism Φ* between its circuit cobordism groups and pseudohomology groups. This work can be viewed as a counterpart to Zinger’s construction.     

Torsion algebraic cycles and étale cobordism

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of ?-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.     

Cobordism invariance of the index of Callias-type operators

We introduce a notion of cobordism of Callias-type operators overcomplete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application, we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two noncompact but topologically simpler manifolds. As another application, we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.     

Commuting involutions with fixed point set of constant codimension

Special generators of the unoriented cobordism ringMO _* are constructed to determine the groupsJ _n,κ ^r ofn-dimensional cobordism classes inMO _n containing a representativeM ⁿ admitting a (Z ₂) ^k -action with fixed point set of constant codimension. This work is supported by HNSF     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类.利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数,推广了现有文献中的相关结果.     

Cobordism invariance of the family index

We give a K‐theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres $M\hookrightarrow \mbox{$\cal M$}\rightarrow B$, and define a notion of cobordant families using K¹‐groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push‐forward map in K‐theory to reduce it to families on $B\times \mathbb {R}^n$.     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类. 利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数, 推广了现有文献中的相关结果.     

Notes on homology cobordisms of plumbed homology 3-spheres

The gauge-theoretical invariants of Donaldson and Seiberg-Witten are used to detect some infinite order elements in the homology cobordism group of integral homology 3-spheres.

     

Generalized Elliptic Genus and Cobordism Class of Nonspin Real Grassmannians

We prove that Witten's generalized elliptic genusis rigid under certain group actions and constant(identically zero) on the real Grassmanniansr₄^2m+5, m 1, for appropriate vectorbundles.We also prove that all the Pontryagin numbers of theseGrassmannians are zero, i.e. they have cobordism class zero,which implies the vanishing of all genera.     

COBORDISM CLASSES OF FIBERINGS OVER PRODUCT OF PROJECTIVE SPACES

Let RP（k） denote the k-dimensional real projective space. This article determines which cobordism classes are represented by the total space of a fibering with prescribed base space RP（3）× RP（1）, RP（2） × RP（1）, RP（2）× RP（1）× RP（1） or RP（3）× RP（2）.     

COBORDISM AND ROKHLIN CONGRUENCES

In this paper, we give a cobordism proof of the higher dimensional Rokhlin congruences established in [8].     

棱柱上的小覆盖的等变配边分类

多面体上的小覆盖的等变配边类是由它的切表示集所决定的.本文通过将棱柱上的小覆盖的切表示集约化到一种素形式,来确定其等变配边分类.     

Grope cobordism and feynman diagrams

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in [CT]. We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the class is used to organize gropes. This implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3. We also show that the class is not a useful organizing complexity in 4 dimensions since only the Arf invariant survives. In contrast, measuring gropes according to height does lead to very interesting 4-dimensional information [COT]. Finally, several low degree calculations are explained, in particular we show that S-equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex. Mathematics Subject Classification (2000):57M27The first author was partially supported by NSF VIGRE grant DMS-9983660. The second author was partially supported by NSF grant DMS-0072775 and the Max-Planck Gesellschaft.     

The toric cobordisms

We introduce the notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori and compute the corresponding cobordism groups.

     

Commuting Involutions with Fixed Point Set of Variable Codimension

Special generators of the unoriented cobordism ring MO _* are constructed to determine some groups of cobordism classes in MO _n containing a representative M ⁿ admitting a (Z ₂) ^k -action with the fixed point set of (n−l _i )-dimensional submanifolds of M ⁿ . This work is supported by HNSF and BSHJJ