论文标题
多层诺维科夫同源性
Polytope Novikov Homology
论文作者
论文摘要
令$ m $为封闭的歧管，而$ \ Mathcal {a} \ subseteq H^1 _ {\ Mathrm {dr}}}}（m）$ a polytope。对于\ Mathcal {a} $中的每个$ a \，我们定义了一个novikov链复合体，其多重级$ \ MATHCAL {a} $编码的多个有限条件。由此产生的多层诺维科夫同源性概括了普通的Novikov同源性。我们证明，在规定的多面体中，任何两个共同体类别都会在与上述多层人士相关的novikov环上产生链同型等效的novikov复合物。作为应用，我们为（扭曲的）Novikov Morse同源性定理提出了一种新颖的方法，并证明了新的Polytope Novikov原理。后者概括了普通的诺维科夫原则，并且是阿贝尔案中帕吉特诺夫的最新结果。
Let $M$ be a closed manifold and $\mathcal{A} \subseteq H^1_{\mathrm{dR}}(M)$ a polytope. For each $a \in \mathcal{A}$ we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $\mathcal{A}$. The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated to said polytope. As applications we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.
