庞加莱对偶、希尔伯特复形及其几何应用
Poincaré duality, Hilbert complexes and geometric applications
摘要：
设$(M,g)$是开的定向黎曼流形.本文研究了$L^2$上同调群的两个序列：$H^i_{2,m\rightarrow M}(M,g)$（定义为象$\im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g))$）和$\bar{H}^i_{2,m\rightarrow M}(M,g)$（定义为$\im(\bar{H}^i_{2,min}(M,g)\rightarrow \bar{H}^i_{2,max}(M,g))$）的一些性质.在一定的假设下，我们证明了第一序列是一个合适的Hilbert复形的上同调，它包含最小的一个，并包含在最大的一个。我们还表明，当第二个序列是有限维的，然后庞加莱对偶持有它，在相同的假设下，当$dim(M)\equiv0\ mod\ 4$，我们可以用它来定义一个$L^2$签名$M$。此外，我们还讨论了紧致光滑分层伪流形的交上同调的一些应用，得到了关于$Δ_{i}$的Friedrichs扩张$Δ_{i}^\mathcal{F}$的一些结果.
英文摘要：
Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\rightarrow M}(M,g)$ defined as the image $\im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g))$ and $\bar{H}^i_{2,m\rightarrow M}(M,g)$ defined as $\im(\bar{H}^i_{2,min}(M,g)\rightarrow \bar{H}^i_{2,max}(M,g))$. We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence is finite dimensional then Poincaré duality holds for it and that, in the same assumptions, when $dim(M)\equiv0\ mod\ 4$ we can use it to define a $L^2$ signature on $M$. Moreover we show several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension $Δ_{i}^\mathcal{F}$ of $Δ_{i}$.
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