[Paper Review] Exotic Smooth Structures on Small 4-Manifolds with Odd Signatures

Anar Akhmedov, B. Doug Park|arXiv (Cornell University)|Jan 29, 2007

Geometric and Algebraic Topology11 references4 citations

TL;DR

This paper constructs exotic smooth structures on small 4-manifolds with odd signatures, specifically on ${\mathbb{CP}}^2\#m\overline{\mathbb{CP}}^2$ and $3{\mathbb{CP}}^2\#k\overline{\mathbb{CP}}^2$ for certain $m,k$, using Luttinger and torus surgeries. It proves the existence of an irreducible symplectic 4-manifold and an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to each such manifold, filling 52 open points in the 4-manifold geography plane.

ABSTRACT

Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer $n\geq 3$. We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic smooth structures when $M$ is $\CP#4\CPb$ or $3\CP# k \CPb$ for $k=6,8,10$.

Motivation & Objective

• To construct exotic smooth structures on small 4-manifolds with odd signatures, particularly those homeomorphic to $m{\mathbb{CP}}^2\#n\overline{\mathbb{CP}}^2$ with $m$ odd and $m < n \leq 5m+4$.
• To extend the known range of 4-manifolds with multiple smooth structures by constructing such structures on $m{\mathbb{CP}}^2\#n\overline{\mathbb{CP}}^2$ for $m=1,3$ and specific $n$.
• To resolve open cases in the 4-manifold geography problem by realizing new $(\chi_h, c_1^2)$ coordinates in the symplectic and non-symplectic regions.
• To prove the existence of an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to specified small 4-manifolds with odd signatures.
• To establish a general construction for symplectic 4-manifolds with fundamental group isomorphic to that of a given symplectic 4-manifold, using Lagrangian torus surgeries.

Proposed method

• The construction uses $2n+4$ surgeries—$2n+3$ Luttinger surgeries and one torus surgery—on the 4-manifold $\Sigma_2 \times \Sigma_n$ to produce families of 4-manifolds with the same cohomology ring as $(2n-3)(S^2 \times S^2)$.
• The fundamental group of the resulting 4-manifolds is computed via van Kampen's theorem, using relations from the surgery curves and their intersections.
• Exotic smooth structures are obtained by performing Luttinger and torus surgeries on symplectic normal connected sums of building blocks, preserving the homeomorphism type but altering the smooth structure.
• The construction leverages the fact that certain Lagrangian tori in symplectic 4-manifolds can be used to perform surgeries that preserve symplectic or non-symplectic structures while altering the diffeomorphism type.
• For Theorem 2, the method involves skipping a single Luttinger surgery in a symplectic normal connected sum to produce a non-simply-connected symplectic 4-manifold $N$ with a Lagrangian torus $T'$, ensuring the meridian of $T'$ is trivial in $\pi_1(N \setminus \nu T')$.
• The resulting 4-manifold $Y$ is constructed by surgering $N$ along $T'$, and its fundamental group is shown to be isomorphic to $\pi_1(X)$, with $\chi_h(Y) = \chi_h(X) + \chi$ and $c_1^2(Y) = c_1^2(X) + c$ for specified $\chi, c$.

Experimental results

Research questions

• RQ1 Can exotic smooth structures be constructed on small 4-manifolds with odd signatures, such as ${\mathbb{CP}}^2\#2\overline{\mathbb{CP}}^2$ and $3{\mathbb{CP}}^2\#4\overline{\mathbb{CP}}^2$?
• RQ2 Do there exist irreducible symplectic 4-manifolds homeomorphic to $m{\mathbb{CP}}^2\#n\overline{\mathbb{CP}}^2$ for $m$ odd and $m < n \leq 5m+4$?
• RQ3 Can an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds be constructed on such manifolds?
• RQ4 Is it possible to realize new points in the 4-manifold geography plane by constructing such exotic structures?
• RQ5 Can the fundamental group of a symplectic 4-manifold be preserved under surgery while increasing $\chi_h$ and $c_1^2$ by specified amounts?

Key findings

• An irreducible symplectic 4-manifold homeomorphic to ${\mathbb{CP}}^2\#2\overline{\mathbb{CP}}^2$ is constructed, and this is the smallest known Euler characteristic among simply-connected 4-manifolds with multiple smooth structures.
• An infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to ${\mathbb{CP}}^2\#2\overline{\mathbb{CP}}^2$ is constructed.
• For $3{\mathbb{CP}}^2\#k\overline{\mathbb{CP}}^2$ with $k=4,6,8,10$, the paper constructs both an irreducible symplectic 4-manifold and an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to the same manifold.
• For $(2n-1){\mathbb{CP}}^2\#2n\overline{\mathbb{CP}}^2$ with $n \geq 3$, the paper constructs an irreducible symplectic 4-manifold and an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to the manifold.
• The results fill in 52 new points in the 4-manifold geography plane, specifically in the region where $2e + 3\sigma \geq 0$, $e + \sigma \equiv 0 \pmod{4}$, and $\sigma \leq -1$, or in $(\chi_h, c_1^2)$ coordinates with $0 \leq c_1^2 \leq 8\chi_h - 1$.
• Theorem 2 establishes that for any closed symplectic 4-manifold $X$ containing a symplectic torus of self-intersection 0 with trivial fundamental group image, and for any $(\chi, c)$ with $0 \leq c \leq 8\chi - 1$, there exists a symplectic 4-manifold $Y$ with $\pi_1(Y) \cong \pi_1(X)$, $\chi_h(Y) = \chi_h(X) + \chi$, and $c_1^2(Y) = c_1^2(X) + c$, with $Y$ having an odd indefinite intersection form.