Relative $K$-cycles and elliptic boundary conditions

Guihua Gong

Relative $K$-cycles and elliptic boundary conditions

Analysis of PDEs 2008-02-03 v1 Differential Geometry K-Theory and Homology

Authors: Guihua Gong

Abstract

In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator $D$ on smooth manifold $M$ with boundary $\p M$ , $D$ possesses an elliptic boundary condition if and only if $\partial [D]$ = 0 in $K_1(\partial M)$ , where $[D]$ is the relative $K$ -cycle in $K_0(M, \partial M)$ corresponding to $D$ . We prove the ``if'' part of this conjecture for $\dim(M)$ $\not=$ 4, 5, 6, 7 and the ``only if'' part of the conjecture for arbitrary dimension.

Keywords

elliptic partial differential equations boundary value problems operator theory and elliptic operators

Cite

@article{arxiv.math/9301213,
  title  = {Relative $K$-cycles and elliptic boundary conditions},
  author = {Guihua Gong},
  journal= {arXiv preprint arXiv:math/9301213},
  year   = {2008}
}