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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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