Symplectized Torelli mapping tori

Abstract

Examples of aspherical closed symplectic 4-manifolds are presented whose Sullivan minimal models are (1,n)-formal for any n, without being formal. They have as cohomology algebra, signature, canonical class, those of a product of a closed Riemann surface of genus g>1 and an elliptic curve, and the fundamental group, resp. $A_{\infty}$ category defined by their de Rham complex, is isomorphic to that of the product of surfaces up to brackets of order n+1, resp. products of order n+1. Nevertheless, the manifolds do not admit any holomorphic structure. These examples are derived from the fact that the Torelli groups are pro-unipotent. The mapping tori for representatives of suitable classes in the Torelli group are considered, and their product with $S^1$ is symplectized a la Thurston.

Figures (3)

• Figure 1: Homotopy Picard--Lefschetz formula: up to homotopy, the effect of a Dehn twist along a simple loop $C$ on a path $\gamma$ intersecting $C$ transversely is to insert a copy of $C$ at each intersection point with it of $\gamma$.
• Figure 2: Presentation of $\pi_1(S_g,*)$ adapted to a decomposition by a simple closed curve $C$ in 2 connected components with genus $g_1,g_2>0$.
• Figure 3: Decomposition of a genus 4 surface $S_4$ in 4 handles of genus 1 by two simple closed curves $C_1,C_2$ meeting transversely at 2 points, and an adapted presentation for $\pi_1(S_4,*)$, in which $C_1=(a_1,b_1)(a_2,b_2), C_2=(a_1,b_1)(a_4,b_4)$. Note that $\tau_{C_1}, \tau_{C_2}$ both fix the classes $a_1,b_1$ of the handle containing the base point $*$, and that for every other handle with generators $a_i,b_i$ the effect of $\tau_{C_j}$ is the identity if the path $\gamma_i$ connecting the handle to the base point $*$ does not cross $C_j$, and conjugation by $C_j$ if the path $\gamma_i$ does cross $C_j$.

Theorems & Definitions (16)

• Definition 1
• Theorem 2
• Corollary 3
• Lemma 4
• Example 5
• Proposition 6
• Proposition 7
• proof
• Proposition 8
• proof