Massey产品在映射环中
Massey products in mapping tori
摘要：
令$ϕ: M\to M$是$C^\infty$紧连通流形的微分同胚，而$X$是其映射环面。有一个天然的纤维化$p:X\to S^1$，用$ξ\in H^1(X, \mathbb{Z})$表示相应的上同调类。令$λ\in \mathbb{Z}^*$。考虑$ϕ$在次数为$k$的$M$的上同调中诱导的自同态$ϕ_k^*$，并用$J_k(λ)$表示其特征值为$λ$的Jordan块的最大尺寸.用$ρ_λ(g) = λ^{p_*(g)}.$定义一个表示$ρ_λ: π_1(X)\to\mathbb{C}^*$令$H^*(X,ρ_λ)$是$X$的相应扭曲上同调。我们证明了$J_k(λ)$等于$\langle ξ, \ldots , ξ, a\rangle$形式的非零Massey乘积的最大长度，其中$a\in H^k(X,ρ_λ)$（这里的长度表示$ξ$的条目数）。特别地，如果$X$是一个强形式空间（例如Kähler流形），那么$ϕ_k^*$的所有Jordan块的大小都是1。如果$X$是一个形式空间，那么所有特征值为1的Jordan块的大小都为1。这导致了一个简单的建设正式的，但不是强正式的映射环面。主要定理的证明是基于这样一个事实，即上述形式的Massey乘积可以用Massey谱序列中的微分来识别，而Massey谱序列又可以用$ϕ^*$的Jordan标准形来显式计算。
英文摘要：
Let $ϕ: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $ξ\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $λ\in \mathbb{Z}^*$. Consider the endomorphism $ϕ_k^*$ induced by $ϕ$ in the cohomology of $M$ of degree $k$, and denote by $J_k(λ)$ the maximal size of its Jordan block of eigenvalue $λ$. Define a representation $ρ_λ: π_1(X)\to\mathbb{C}^*$ by $ρ_λ(g) = λ^{p_*(g)}.$ Let $H^*(X,ρ_λ)$ be the corresponding twisted cohomology of $X$. We prove that $J_k(λ)$ is equal to the maximal length of a non-zero Massey product of the form $\langle ξ, \ldots , ξ, a\rangle$ where $a\in H^k(X,ρ_λ)$ (here the length means the number of entries of $ξ$). In particular, if $X$ is a strongly formal space (e.g. a Kähler manifold) then all the Jordan blocks of $ϕ_k^*$ are of size 1. If $X$ is a formal space, then all the Jordan blocks of eigenvalue 1 are of size 1. This leads to a simple construction of formal but not strongly formal mapping tori. The proof of the main theorem is based on the fact that the Massey products of the above form can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of $ϕ^*$.
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