相对$K$ -圈与椭圆边界条件
Relative $K$-cycles and elliptic boundary conditions
摘要：
本文讨论了Baum-Douglas提出的猜想：对光滑流形$\p M$上的任何一阶椭圆微分算子$D$，$D$具有椭圆边界条件当且仅当$\partial [D]$ = 0 in $K_1(\partial M)$，其中$[D]$是$K_0(M, \partial M)$中对应于$D$的相对$K$ -圈.我们证明了对于$\dim(M)$ $\not=$ 4，5，6，7猜想的“如果”部分和对于任意维数猜想的“仅当”部分.
英文摘要：
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.
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