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arXiv 2001.05186math.AG
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次解析奇异空间上$L^p$上同调的Poincaré对偶

Poincaré duality for $L^p$ cohomology on subanalytic singular spaces

Guillaume Valette

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摘要：

研究了$\mathbb{R}^n$（不一定是紧的）的有界次解析子流形上的$L^p$微分形式的Poincaré对偶问题.我们证明了，当$p$足够接近$1$，那么这样一个子流形的$L^p$上同调同构于它的奇异同调。在$p$是大的情况下，我们表明，$L^p$上同调是对偶的交叉同源。因此，如果$p$和$q$彼此是Hölder共轭的，并且$p$足够大，我们可以推出$L^p$上同调是$L^q$上同调的Poincaré对偶.

英文摘要：

We investigate the problem of Poincaré duality for $L^p$ differential forms on bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarily compact). We show that, when $p$ is sufficiently close to $1$ then the $L^p$ cohomology of such a submanifold is isomorphic to its singular homology. In the case where $p$ is large, we show that $L^p$ cohomology is dual to intersection homology. As a consequence, we can deduce that the $L^p$ cohomology is Poincaré dual to $L^q$ cohomology, if $p$ and $q$ are Hölder conjugate to each other and $p$ is sufficiently large.

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