[Paper Review] Geography of symplectic 4- and 6-manifolds

Rafael Torres, Jonathan Yazinski|arXiv (Cornell University)|Jul 20, 2011

Geometric and Algebraic Topology37 references3 citations

TL;DR

This paper establishes the existence of minimal symplectic 4- and 6-manifolds with prescribed fundamental groups and characteristic numbers, using symplectic sums and building blocks from known minimal symplectic 4-manifolds. It extends previous geography results by constructing examples with arbitrary fundamental groups and realizing all admissible Chern number triples in dimension 6.

ABSTRACT

The geography of minimal symplectic 4-manifolds with arbitrary fundamental group and symplectic 6-manifolds with abelian fundamental group of small rank, and with arbitrary fundamental group are addressed.

Motivation & Objective

• To extend the symplectic geography problem to non-simply connected 4-manifolds with arbitrary fundamental groups.
• To address the geography of symplectic 6-manifolds with abelian or arbitrary fundamental groups of small rank.
• To construct minimal symplectic 4-manifolds with specified characteristic numbers and fundamental group using symplectic sum techniques.
• To realize all admissible Chern number triples in symplectic 6-manifolds via blow-ups and symplectic sum constructions.
• To demonstrate the existence of exotic smooth structures on symplectic 4-manifolds with finite fundamental groups.

Proposed method

• Construct minimal symplectic 4-manifolds $M(G)$ with fundamental group $G$ using symplectic sums and building blocks with controlled Euler characteristic and signature.
• Utilize the minimal symplectic 4-manifold $BK$ with $c_2(BK) = 4(g+r)$, $ au(BK) = 0$, and $ ho_1(BK) o G$ as a base for constructing $X_1(G)$.
• Apply symplectic sum constructions along tori or surfaces of genus two to build 6-manifolds $W_i(G)$ from products of 4-manifolds with $S^2$, $T^2$, or $Σ_2$.
• Use blow-ups at points disjoint from symplectic surfaces to realize all possible values of $c_1c_2$ in 6-manifolds.
• Leverage known results on homeomorphisms and invariants (e.g., Boyer, Taubes, Kotschick) to ensure irreducibility and smooth structure distinctions.
• Verify Chern number computations via Lemma 22 and Lemma 24, relying on characteristic class additivity under symplectic sums.

Experimental results

Research questions

• RQ1 Can the symplectic geography problem be extended to minimal symplectic 4-manifolds with arbitrary fundamental groups?
• RQ2 What are the constraints on Chern numbers for symplectic 6-manifolds with abelian or arbitrary fundamental groups?
• RQ3 Can all admissible Chern number triples $(c_1^3, c_1c_2, c_3)$ be realized in symplectic 6-manifolds with a given fundamental group?
• RQ4 Do symplectic 4-manifolds with finite fundamental groups admit exotic smooth structures?
• RQ5 What is the role of symplectic sums and blow-ups in realizing the full range of characteristic numbers in higher-dimensional symplectic manifolds?

Key findings

• For any finitely presented group $G$ with $g$ generators and $r$ relations, there exists a minimal symplectic 4-manifold $M(G)$ with $ ho_1(M(G)) o G$, $c_1^2 = 2e+3 au+4(g+r)$, and $χ_h = \frac{1}{4}(e+\tau)+(g+r)$, satisfying $2e+3\tau \geq 0$, $e+\tau \equiv 0 \pmod{4}$, and $e+\tau \geq 8$.
• If $G$ is finite, $M(G)$ admits exotic smooth structures, as shown via homeomorphism results from Boyer and Kotschick.
• For any $n \in \mathbb{N}$ and $s \geq 1$, there exists a spin symplectic 4-manifold $X(G)$ with $(c_1^2, \chi_h) = (8n-8+8(g+r), 2s+n-1+(g+r))$, and an infinite family $\{X_m(G)\}$ of pairwise nondiffeomorphic manifolds homeomorphic to $X(G)$.
• Symplectic 6-manifolds $W_0(G), W_1(G), W_2(G)$ are constructed with fundamental group $G$ and prescribed Chern numbers via symplectic sums of 4-manifold products with $S^2$, $T^2$, or $Σ_2$.
• All triples $(a,b,c) \in \mathbb{Z}^3$ with $a \equiv c \equiv 0 \pmod{2}$, $b \equiv 0 \pmod{24}$, are realized as Chern numbers of symplectic 6-manifolds with any finitely presented fundamental group $G$.
• By blowing up points disjoint from symplectic genus-two surfaces, all possible values of $c_1c_2$ are realized, and the construction ensures that $c_1^3$ and $c_3$ span all even integers.