Pellegrini Alessio
Department of Mathematics, ETH Zürich, Zurich, Switzerland.
J Fixed Point Theory Appl. 2021;23(4):62. doi: 10.1007/s11784-021-00899-5. Epub 2021 Sep 24.
Let be a closed manifold and a polytope. For each , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope . 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 with 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.
设(M)是一个闭流形，(P)是一个多面体。对于每个(P)，我们定义一个具有由多面体(P)编码的多重有限性条件的诺维科夫链复形。由此产生的多面体诺维科夫同调推广了普通的诺维科夫同调。我们证明，在一个规定的多面体中的任意两个上同调类会产生在与所述多面体相关联的诺维科夫环上链同伦等价的多面体诺维科夫复形。作为应用，我们提出了一种处理（扭曲的）诺维科夫莫尔斯同调定理的新方法，并证明了一个新的多面体诺维科夫原理。后者推广了普通的诺维科夫原理以及帕伊特诺夫在阿贝尔情形下的一个近期结果。
