手术和旋量τ不变量
Surgery and the spinorial tau-invariant
摘要：
我们将紧致自旋流形M与实值不变量τ（M）联系起来，方法是取所有共形类上的上确界和第一个正狄拉克特征值的每个共形类内的下确界，归一化到体积1。这个不变量是Schoen的$σ$ -常数的旋量类似物，也被称为光滑Yamabe数。我们证明了如果N是通过余维数至少为2的运算从M得到的，则$τ(N) \geq \min\{τ(M),Λ_n\}$与$Λ_n>0$。可以得出各种拓扑结论，特别是τ是低于$Λ_n$的自旋边界不变量。在$Λ_n$以下，当在固定维度的所有流形上变化时，$τ$的值不能从上面累积。
英文摘要：
We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $σ$-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least 2, then $τ(N) \geq \min\{τ(M),Λ_n\}$ with $Λ_n>0$. Various topological conclusions can be drawn, in particular that τis a spin-bordism invariant below $Λ_n$. Below $Λ_n$, the values of $τ$ cannot accumulate from above when varied over all manifolds of a fixed dimension.
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