The relative Whitney trick and its applications

Abstract

We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary.Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological 4-manifold. We further prove that any link in a homology sphere is order k Whitney tower concordant to a link in $S^3$ for all k. Finally, we explore the minimum Gordian distance from a link in $S^3$ to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.

Appendix A: Freely slicing boundary links

Cha-Kim-Powell [5] describe a set of conditions on a link in $S^3$ that ensure the link is freely slice. In Sect. 4, we generalized these conditions to links in a general homology 3-sphere Y and claimed in Theorem 4.4 that our conditions guaranteed the link was freely slice in the contractible 4-manifold X bounded by Y. The proof of this is a close imitation of the argument from Cha-Kim-Powell [5, Sects. 4 & 5] and, as such, we only sketch the argument below. An attempt has been made to include enough detail to follow the argument, but without repeating too much of what already appears in [5].

We begin by recalling some terminology and a theorem from Freedman-Quinn [12]. A transverse pair is two copies of $S^2\times D^2$ plumbed together at one point. This model is a neighbourhood of the pair of spheres

$$\begin{aligned}(S^2\times \{pt\})\cup (\{pt\}\times S^2)\subseteq S^2\times S^2.\end{aligned}$$

Take the disjoint union $N_1, \dots , N_\ell$ of copies of the transverse pair and perform further plumbings between the copies, possibly including self-plumbings, then map the result into a topological 4-manifold W via a continuous map that is a homeomorphism to its image. The result of this process is a map $f:\coprod _i N_i\rightarrow W$ which is called an immersion of a union of transverse pairs.

An immersion of a union of transverse pairs is said to have algebraically trivial intersections if the images of the further plumbings we performed can be arranged in pairs by Whitney disks in W (that may a priori meet $\coprod _i N_i$). Such a map f is called $\pi _1$-null if the inclusion induced map $\pi _1\left( f\left( \coprod _i N_i\right) \right) \rightarrow \pi _1(W)$ is trivial.

If middle-dimensional homology classes can be represented in this arrangement, then one is able to use a result of Freedman-Quinn [12, Theorem 6.1] to conclude that f is s-cobordant rel. boundary to an embedding. In the particular case of interest to us, this gives the following.

Theorem A.1

Suppose W is a compact topological 4-manifold, bounded by $M_L$, with $\pi _1(W)$ free and generated by the meridians of L. Let $f:\coprod _i N_i\rightarrow W$ be a $\pi _1$-null immersion of a union of transverse pairs with algebraically trivial intersections, and inducing an isomorphism $f_*:H_2(\coprod _i N_i)\rightarrow H_2(W)$. Then there exists a compact topological 4-manifold $W'$, bounded by $M_L$, with $\pi _1(W')$ free and generated by the meridians of L, and a locally flat embedding $f':\coprod _i N_i\hookrightarrow W'$ inducing an isomorphism $f'_*:H_2(\coprod _i N_i)\rightarrow H_2(W')$.

We now follow the standard surgery-theoretic approach to slice L, sketched in the introduction. Recall that the 0-surgery on L is denoted by $M_L$.

Proposition A.2

Let L be a boundary link with a good disky basis in a homology sphere Y, then $M_L$ bounds a compact oriented 4-manifold W such that

• (1) $\pi _1(W)$ is free and generated by the meridians of L, and

• (2) $H_2(W;\mathbb {Z})$ is free and represented by a $\pi _1$-null immersion of a union of transverse pairs with algebraically trivial intersections.

The argument we now use is almost identical to that appearing in [5, Sect. 5].

Summary of the proof of Proposition A.2

Let $F=F_1\cup \dots \cup F_n$ be a boundary link Seifert surface for L and let $\{a_i, b_i\}_{1\le i \le g}$ be a good disky basis, with

$$\begin{aligned} \left\{ \Delta _j^+,\Delta _i \mid 1\le j \le 2g, 1\le i \le g\right\} \end{aligned}$$

the immersed disks as in Definition 4.2. Recalling these conditions, for each i, $\partial \Delta _{i}^+ = a_i$, $\partial \Delta ^+_{g+i} = (b_i')^+$, and $\partial \Delta _i = b_i'$, where $b_i'$ is the result of pushing $b_i$ off F such that it has zero linking with $a_i$, and $(b_i')^+$ is a zero linking parallel copy of $b_i'$. These disks are all disjoint except that the disks $\{\Delta _j^+\}_{1\le j \le 2g}$ might intersect each other. Write X for the contractible 4-manifold bounded by Y.

Fig. 13

Curves $a_i$, $\beta _i$, $\gamma _i$ and $\delta _i$ sitting in a produce neighbourhood of a Seifert surface for L. Attaching a 1-handle using the dotted $\beta _i$ curves and attaching 2-handles along the 0-framings of $a_i$, $\gamma _i$, and $\delta _i$

For each i, let $\beta _i\cup \gamma _i$ be the Bing double of $b_i$ appearing in Fig. 13. Attach 1-handles to X along $\beta _i$ and 2-handles to X along the 0-framings of $a_i$, $\gamma _i$, and $\delta _i$ to get a 4-manifold W. A straightforward argument shows that W has boundary $M_L$ and has fundamental group freely generated by the meridians of L; see [5, Claim A] for details. Clearly $H_2(W;\mathbb {Z})\cong \mathbb {Z}^{2g}$. This basis is generated by framed immersed spheres $\Sigma _1,\dots ,\Sigma _{2g}$ described as follows. For each i, take $\Sigma _{2i-1}$ to be the union of $\Delta _i^+$ and the core of the 2-handle attached to $a_{i}$. For each i, we can and will assume $b_i'$ and $(b'_i)^+$ lie on the gray surface bounded by $\gamma _i$ depicted in Fig. 14 (left). We use this to define a planar surface $P_i$ bounded by $\gamma _i$, $b_i'$ and $(b_i')^+$ as in Fig. 14 (right). Take $\Sigma _{2i}$ to be the union of $\Delta _{g+i}^+$, $\Delta _i$, the core of the handle attached to $\gamma _i$, and $P_i$; cf. [5, Claim B].

Fig. 14

Left: A section of the surface F containing $\{a_i,b_i\}$ ($b_i$ not depicted). The curves $\beta _i$ and $\gamma _i$ form a bing double of the curve $b_i$ in a neighbourhood of F. A gray genus one surface disjoint from $\beta _i$ with boundary $\gamma _i$ is also depicted. Right: A close-up of the gray surface. The annulus $A_i$ on the gray surface with boundary $b_i$ and $b'_i$ is depicted. The complement of the interior of the annulus in the gray surface is the planar surface $P_i$

For each i, a regular neighbourhood of $\Sigma _{2i-1}\cup \Sigma _{2i}$ can now be viewed as an immersed transverse pair. The same arguments from [5, Claim C] and [5, Claim D] now reveal that $\cup _{i=1}^{2g} \Sigma _i$ has algebraically trivial intersections and is $\pi _1$-null. $\square$

Finally, we can confirm that Cha-Kim-Powell [5, Theorem A] generalizes as claimed.

Proof of Theorem 4.4

Let W be the 4-manifold and $f:\coprod _i N_i\rightarrow W$ the immersion of a union of transverse pairs representing $H_2(W;\mathbb {Z})$ described in Proposition A.2. Applying Theorem A.1, we obtain $W'$ and $f'$. Note that the image of $f'$ consists of a tubular neighbourhood of locally flat embedded 2-spheres representing generators for $H_2(W';\mathbb {Z})\cong H_2(W;\mathbb {Z})$. These embedded 2-spheres come in transverse pairs and we now perform surgery on one sphere from each transverse pair. Since the second sphere from each transverse pair intersected the surgered sphere geometrically once, these surgeries preserve $\pi _1(W')$. Thus, we obtain $W''$ with boundary $M_L$, with $H_2(W'';\mathbb {Z})=0$, and with $\pi _1(W'')$ freely generated by the meridians of L. Now attach 2-handles to $M_L$ along the meridians of the link components, with framing so that the 0-surgery is reversed. This has Y as the effect of surgery, and by glueing across meridians we ensure that $\pi _1(W'')=0$. The resultant 4-manifold is contractible and has boundary Y. The link L has slice disks given by the cocores of the 2-handles we have just attached so it is slice and moreover freely slice as $\pi _1(W'')$ is free. $\square$