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Reverse engineering small 4-manifolds. (English) Zbl 1142.57018
The authors developed a technique named Reverse Engineering which allows the construction of infinite families of distinct smooth structures on many smoothable 4-manifolds. On a simply connected 4-manifold \(X\), reverse engineering is a three step process:
(1)
Find a 4-manifold \(M_{X}\) (called the model manifold) having the same Euler number and signature as \(X\), but with \(b_{1}>0\), and whose Seiberg-Witten invariant is non-trivial.
(2)
Find \(b_{1}\) essential tori that carry generators of \(H_{1}(M_{X})\) and apply some \(p/q\)-surgery on each of these tori in order to kill \(H_{1}(M_{X})\). Thereafter, the killing of \(\pi_{1}(M_{X})\) must be checked, and this may not be a simple computation due to the presentation of the group.
(3)
Compute the Seiberg-Witten invariants.
When the model manifold \(M_{X}\) for \(X\) is symplectic and \(b_{1}-1\) of the above tori are Lagrangian, the condition in (1) that the Seiberg-Witten invariants be non-trivial will be satisfied. In this case, each time a Luttinger surgery is performed to reduce \(b_{1}\), the resulting manifold is symplectic. If the resulting manifold after \(b_{1}\) surgeries is simply connected, then one can show that there are infinitely many distinct smooth structures on \(X\).
The \(p/q\)-surgery used is defined as follows: Suppose \(T\) is a torus with tubular neighborhood \(N_{T}\simeq T\times D^{2}\). Let \(\alpha\) and \(\beta\) be generators of \(\pi_{1}(T)\) and let \(S^{1}_{\alpha}\) and \(S^{1}_{\beta}\) be loops in \(T^{3}=\partial N_{T}\) homologous in \(N_{T}\) to \(\alpha\) and \(\beta\), respectively. Let \(\mu_{T}=\partial D^{2}\) denote the meridional circle to \(T\) in \(X\). By a \(p/q\)-surgery on \(T\) with respect to \(\beta\) the authors mean
\[ X_{T;\beta}(p/q)=(X\backslash N_{T})\bigcup_{\phi}(S^{1}\times S^{1}\times D^{2}), \]
where the gluing map \(\phi:S^{1}\times S^{1}\times\partial D^{2}\rightarrow \partial(X\backslash N_{T})\) satisfies
\[ \phi_{*}([\partial D^{2}])=q[S^{1}_{\beta}]+p[\mu_{T}]\in H_{1}(\partial(X\backslash N_{T});\mathbb{Z}). \]
Let \(T_{p/q}\) be the core torus \(S^{1}\times S^{1}\times\{0\}\subset X_{T;\beta}(p/q)\). For \(n\in\mathbb{N}\), let \(X_{1/n}=X_{T,\beta}(1/n)\) and \(X_{0}=X_{T,\beta}(0)\).
The authors apply reverse engineering to produce infinitely many smooth structures on \(X=\mathbb{C}P^{2}\# 3\overline{\mathbb{C}P}^{2}\). (This provides a streamlined construction of the examples originally discovered by S. Baldridge and P. Kirk [Geom. Topol. 12, No. 2, 919–940 (2008; Zbl 1152.57026)] and A. Akhmedov and B. Park [Invent. Math. 173, No. 1, 209–223 (2008; Zbl 1144.57026)].) In this case, the model manifold used by the authors is the 2-fold symmetric product \(M_{X}=\text{Sym}^{2}(\Sigma_{3})\) of a genus 3 surface \(\Sigma_{3}\), which is a symplectic manifold with \(H_{1}(\text{Sym}^{2}(\Sigma_{3}),\mathbb{Z})=\mathbb{Z}^{6}\), Euler number \(e(M)=e(\mathbb{C}P^{2}\# 3\overline{\mathbb{C}P}^{2})=6\) and signature \(\text{sign}(M_{X})=\text{sign}(\mathbb{C}P^{2}\# 3\overline{\mathbb{C}P}^{2})=-2\). According to the surgery notation above, consider \(X_{0}\), the manifold obtained after performing five Luttinger surgeries on the model \(M\) to kill five of the generators of \(H_{1}(M,\mathbb Z)\). One more surgery and step (2) is completed. In this case, the authors managed to prove that the manifold \(X\) obtained is simply connected, hence homeomorphic to \(\mathbb{C}P^{2}\# 3\overline{\mathbb{C}P}^{2}\). Along the surgeries all manifolds obtained are symplectic, so their Seiberg-Witten invariant is non zero. By the theorem below, it follows that \(\mathbb{C}P^{2}\# 3\overline{\mathbb{C}P}^{2}\) has an infinite number of inequivalent smooth structures.
Reverse engineering has so far not been successful in producing infinitely many smooth structures on \(X=S^{2}\times S^{2}\). In this case, the model manifold used is the Kähler manifold \(M_{X}=\Sigma_{2}\times \Sigma_{2}\) given by the product of two closed surfaces with genus 2, because \(e=4\), \(\text{sign}=0\) and \(H_{1}=\mathbb{Z}^{8}\). Although the manifold \(X\) obtained after performing the eight Luttinger surgeries has the homology type of \(S^{2}\times S^{2}\), so far it is unknown whether or not \(\pi_{1}(X)\) is trivial. Therefore, in this case, this technique proves the existence of an infinite family of fake homology \(S^{2}\times S^{2}\)’s.
The authors consider the modified Seiberg-Witten invariant \(SW_{X}':\{k\in H_{2}(X;\mathbb{Z})\mid \hat{k}\equiv w_{2}(X)\bmod 2\}\rightarrow \mathbb{Z}\), \(SW_{X}'(k)=\sum_{c(s)=k}SW_{X}(s)\), where \(c(s)\in H_{2}(X,\mathbb{Z})\) is the Poincaré dual of \(c_{1}(W^{+}_{s})\), and \(W^{+}_{s}\) is the positive spinor bundle associated to the spin\(^{c}\) structure \(s\). This is used to prove the following theorem upon which their surgery algorithm is developed;
Theorem: Let \(X\) be a smooth closed oriented 4-manifold which contains a null-homologous torus \(\Lambda\), and let \(\lambda\) be a simple loop on \(\Lambda\) so that \(S^{1}_{\lambda}\) is nullhomologous in \(X\backslash N_{\lambda}\). If the Seiberg-Witten invariant of \(X_{\Lambda;\lambda}(0)\) is non-trivial in the sense that for some basic class \(\kappa_{0}\), \(\sum_{i}SW_{X_{\Lambda;\lambda}(0)}'(\kappa_{0}+2i[\Lambda_{0}])\neq 0\), then among the manifolds \(\{X_{\Lambda;\lambda}(1/n)\}\), infinitely many are pairwise nondiffeomorphic.
Corollary: Suppose \(X_{0}=X_{T,\beta}(0)\) has, up to sign, exactly one Seiberg-Witten basic class. Then the manifolds \(X_{n}=X_{T,\beta}(1/n)\), \(n\in\mathbb{N}^{*}\), are pairwise nondiffeomorphic.

MSC:
57R55 Differentiable structures in differential topology
57R57 Applications of global analysis to structures on manifolds
14J26 Rational and ruled surfaces
53D05 Symplectic manifolds (general theory)
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References:
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