On the Homotopy Classification of Elliptic Operators on Manifolds with Edges

On the homotopy classification of elliptic operators on stratified manifolds

We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the $K$-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.

Elliptic Theory on Manifolds with Corners: II. Homotopy classification and $K$-Homology

We establish the stable homotopy classification of elliptic pseudodifferential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.

The homotopy classification and the index of boundary value problems for general elliptic operators

We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.

Elliptic Operators and Higher Signatures

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; - the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

指标为1的真Fredholm映射的同伦分类

The homotopy classification of proper Fredholm maps of index one

本研究在前期已完成非正指标真Fredholm映射同伦分类的基础上，引入新的数值不变量，完成了单连通希尔伯特流形到其模型空间的指标为1的真Fredholm映射的同伦类计算。