Signature via Novikov numbers

Von Neumann Betti numbers and Novikov type inequalities

It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for $L^2$ cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.

On the Conical Novikov Homology

Let $ω$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([ω])=0$. Let $ξ:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat Λ}_ξ$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $ω$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat Λ}_ξ$ by the set of zeroes of $ω$. In this paper we introduce a refinement of this construction. We define a subring $\hatΛ_Γ$ of ${\hat Λ}_ξ$ and show that the Novikov compl…

Morse-Novikov同调与$β$-临界点

Morse-Novikov homology and $β$-critical points

本文针对流形上的$β$-临界点问题，基于Morse-Novikov同调给出$β$-Morse函数$i$指标$β$-临界点数量的下界并推广到无穷远二次生成函数，还应用于检测本质Liouville弦。

Conformal symplectic geometry of cotangent bundles

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $β$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.

χ_y-示性类、陈数不等式与签名

The $χ_y$-genus, Chern number inequalities and signature

本文研究了几乎复流形上修改的χ_y-示性类的两种正性条件，推导出最优陈数不等式，并探讨了辛流形在辛圆作用下的签名应用。

A note on the geometry of linear Hamiltonian systems of signature 0 in R^4

It is shown that a linear Hamiltonian system of signature zero in 4 dimensions is elliptic or hyperbolic according to the number of Lagrangian planes in the null-cone $H^{-1}(0)$, or equivalently the number of invariant Lagrangian planes. An extension to higher dimensions is described.