Dissolving knot surgered 4-manifolds by classical cobordism arguments

TL;DR

This paper demonstrates that classical cobordism techniques can quickly show that knot surgered 4-manifolds become standard after stabilization, and that many such manifolds are nearly decomposable, simplifying their classification.

Contribution

It introduces a unified cobordism-based approach to prove stabilization and decomposability results for knot surgered 4-manifolds, extending classical methods to modern problems.

Findings

Knot surgered 4-manifolds become diffeomorphic to original after stabilization.

Classical cobordism arguments provide quick proofs of stabilization.

Many almost completely decomposable 4-manifolds are shown to be nearly decomposable.

Abstract

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 x S^2 or CP^2 # -CP^2, and almost complete decomposability of X_K for many almost completely decomposable X, such as the elliptic surfaces.

Full text

In the late 1970s, Moishezon and Mandelbaum discovered that many algebraic surfaces, such as the elliptic surfaces $E(n)$, are almost completely decomposable, i.e. after a single connected sum with the complex projective plane ${\mathbb{CP}}{}^{2}$, they dissolve –become diffeomorphic– to a connected sum of ${\mathbb{CP}}{}^{2}$s and $\overline{\mathbb{CP}}{}^{2}$s; see for example [9, 10, 11, 12, 13, 14]. Most recently, Choi, Park and Yun considered knot surgered elliptic surfaces $E(n)_{K}$ of Fintushel and Stern [5], which (for each $n$) constitute a large family of irreducible $4$–manifolds that contain infinitely many symplectic but not algebraic, and even non-symplectic $4$–manifolds that are homeomorphic but not pairwise diffeomorphic [5]. Using handle diagrams and Kirby calculus, they proved that, after connected summing with a single ${\mathbb{CP}}{}^{2}$, all $E(n)_{K}$ become diffeomorphic to each other (in particular to $E(n)$ when $K$ is the unknot), and concluded that $E(n)_{K}$ are ACD [4]. Here we give an alternate proof using $5$–dimensional cobordism arguments, which follow the classical framework of Mandelbaum and Moishezon —with further contributions from Gompf [7]. These general arguments show that many other knot surgered $4$–manifolds are almost completely decomposable (e.g. many irreducible $4$–manifolds in [15]). These are knot surgery $4$–manifolds that can be seen as fiber sums of almost completely decomposable $4$-manifolds and knot surgered $E(n)$).) A simpler version of our proof provides general results on the stable equivalence of knot surgered $4$–manifolds after a single stabilization by connected summing with $S^{2}\times S^{2}$ or ${\mathbb{CP}}{}^{2}\#\overline{\mathbb{CP}}{}^{2}$.

The first part of our theorem recaptures somewhat more specific results of Auckly and Akbulut, who used handle diagrams and Kirby calculus to show that knot surgered simply-connected $4$–manifolds $X_{K}$ become diffeomorphic to $X$ after a stabilization by connected summing with ${\mathbb{CP}}{}^{2}\#\overline{\mathbb{CP}}{}^{2}$ or with $S^{2}\times S^{2}$, provided the torus fiber is contained in a Gompf nucleus [2], or in a cusp neighborhood [1], respectively. Other general results on the sufficiency of single stabilizations for homeomorphic simply-connected $4$–manifolds to become diffeomorphic for various constructions of exotic $4$–manifolds were given by the author and Sunukjian in [3] using $5$–dimensional round handle cobordisms.

Below $(X_{1},F_{1})\#_{\phi}(X_{2},F_{2}):=(X_{1}\setminus\nu F_{1})\cup_{\phi}(X_{2}\setminus\nu F_{2}$) denotes the (generalized) fiber sum [8] of $X_{i}$, $i=1,2$, along square zero closed orientable surfaces $F_{i}\subset X_{i}$ of the same genera, with the gluing map $\phi$. Let $K$ be a knot in $S^{3}$. We can recast the construction of $X_{K}$ as a fiber sum: for $T$ a square zero torus with simply-connected complement in $X$, we have $X_{K}=(X,T)\#_{\phi}(S^{1}\times S^{3},T_{K})$, where $T_{K}=S^{1}\times K$. The gluing map $\phi$ identifies the two circles $a$ and $b$, which frame the $T$ factor of $T\times D^{2}\cong\nu T$, with $S^{1}\times\text{pt}$ and $\text{pt}\times K$, which frame the $T_{K}$ factor of $T_{K}\times D^{2}\cong\nu\,T_{K}=S^{1}\times\nu K$. (Here $\phi$ identifies the meridian of $T$, which is null-homotopic in $X\setminus\,\nu T$ with $\text{pt}\times\mu$, where $\mu$ is the meridian of $K$, normally generating $\pi_{1}(S^{3}\setminus\nu K)$.) When $X$ is the elliptic surface $E(n)$, $T$ is a regular elliptic fiber [5].

Now for knot surgered $4$–manifolds, let us first consider the most specific case of $X_{1}=E(1)$ and $Z=E(1)_{K}$. We note that the following is essentially a generalization of the classical proof of the almost complete decomposability for $E(n)$, since $E(n)=E(n)_{K}$, for $K$ the unknot. Because $E(n)_{K}\cong(E(1)_{K},T)\#(E(n-1),T)$, the arguments in the previous paragraph show that it suffices to prove the almost complete decomposability for $E(1)_{K}$. Observe that $(E(1),F)$ is obtained from $({\mathbb{CP}}{}^{2}\#\,8\overline{\mathbb{CP}}{}^{2},F^{\prime})$, where $F^{\prime}$ is an $8$ times blown-up cubic in ${\mathbb{CP}}{}^{2}$. Now for $E(1)_{K}=(E(1),T)\#_{\phi}(S^{1}\times S^{3},T_{K})$, the only difference is that $S^{1}\times S^{3}$ is not simply-connected. In this case, we attach two $5$–dimensional $2$–handles $H_{1}$ and $H_{2}$ to ${\mathbb{CP}}{}^{2}\#\,8\overline{\mathbb{CP}}{}^{2}\#\,S^{1}\times S^{3}$ to arrive at $E(1)_{K}\#\,{\mathbb{CP}}{}^{2}$. By the chosen gluing map $\phi$, the attaching circle of $H_{1}$ (resp. $H_{2}$) is the connected sum of $a$ (resp. $b$) in $T^{\prime}\subset{\mathbb{CP}}{}^{2}\#\,8\overline{\mathbb{CP}}{}^{2}$ and $S^{1}\times\text{pt}$ (resp. $\text{pt}\times K$) in $T_{K}\subset S^{1}\times S^{3}$. As $a$ and $b$ are null-homotopic in ${\mathbb{CP}}{}^{2}\#\,8\overline{\mathbb{CP}}{}^{2}\setminus\nu T^{\prime}$, the attaching circles of $H_{1},H_{2}$ are isotopic to the latter summands. (Homotopy implies isotopy for $1$–manifolds in a $4$–manifold.) Attaching $H_{1}$ along $S^{1}\times\text{pt}$ of $S^{1}\times S^{3}$ yields $S^{4}$. (To see this, consider for example the standard handle decomposition of $S^{1}\times S^{3}$ with a single handle for every index but $2$. We then trade the $1$–handle with a [math]–framed $2$–handle attached along an unknot, which cancels against the $3$–handle.) Now, attaching $H_{2}$ along —the image of— $\text{pt}\times K$ in $S^{4}$ amounts to a connected sum with $S^{2}\times S^{2}$ or ${\mathbb{CP}}{}^{2}\#\overline{\mathbb{CP}}{}^{2}$ (which doesn’t matter here, since the other summand ${\mathbb{CP}}{}^{2}\#\,8\overline{\mathbb{CP}}{}^{2}$ is non-spin). Hence, the second to last level of the cobordism reads from the bottom as ${\mathbb{CP}}{}^{2}\#8\overline{\mathbb{CP}}{}^{2}\#S^{2}\times S^{2}\cong 2{\mathbb{CP}}{}^{2}\#9\overline{\mathbb{CP}}{}^{2}$. From the top, it reads as $E(1)_{K}\#{\mathbb{CP}}{}^{2}$, showing that $E(1)_{K}$ is ACD. This in turn concludes that $E(n)_{K}$ is ACD.

At this point it should be clear that running the above arguments for arbitrary $X=X_{1}$ instead of $E(1)$, under respective assumptions on $X$, provides a proof of the second part of the theorem. For the stabilization results in the first part, we run the same arguments simply without eliminating the extra ${\mathbb{CP}}{}^{2}$, as we discussed in the very first paragraph of our proof. ∎

With these in mind, we can run similar arguments as in the proof of Theorem 1, now for a cobordism $W$ from the disjoint union $X\sqcup\,S^{1}\times Y_{K}$ to $X_{K}$. In fact, we can make it very comparable to our earlier cobordism if we regard $S^{1}\times Y_{K}$ as the $4$–manifold obtained from $S^{1}\times S^{3}$ by attaching a $5$–dimensional round $2$–handle along $S^{1}\times K$, which consists of attaching a $5$–dimensional $2$–handle $H^{2}$ (along $\text{pt}\times K$ in $S^{1}\times S^{3}$) and a $5$–dimensional $3$–handle $H^{3}$ that goes over $H^{2}$ geometrically twice but algebraically zero times; see e.g. [3]. This way, we can extend the cobordism $W$ on one end to once again consider a cobordism $W^{\prime}$ from $X\sqcup\,S^{1}\times S^{3})$ to $X_{K}$. Now, the $3$–handle ($D^{1}\times h^{2}\times D^{2}$) of the original cobordism $W$ is attached along a $2$–sphere that is the union of a normal disk to $T$, a normal disk to $\mu$ (which is a $2$–disk capping $K$ in $Y_{K}$), and a cylinder between the meridian $\mu_{T}$ of $T$ and $K$. So it goes over the $2$–handle $H^{2}$ of the round $2$–handle once, and we can cancel them. Attaching the $1$–handle $D^{1}\times h^{0}\times D^{2}$, and the $2$–handles $H_{1}$ and $H_{2}$ to $X\sqcup\,S^{1}\times S^{3}$, we get at a connected sum of $X$ with $S^{2}\times S^{2}$ or ${\mathbb{CP}}{}^{2}\#\overline{\mathbb{CP}}{}^{2}$ as before. Attaching the $3$–handle $H^{3}$ of the round $2$–handle to the other end $X_{K}$ as a $2$–handle, we arrive at the same conclusions.

Acknowledgments. Many thanks to Bob Gompf and Danny Ruberman for helpful comments. The author is supported by the NSF Grant DMS-$1510395$.