Surfaces in 4-manifolds

Choose an orientation for $\Sigma$, and let $N$ be the disk bundle over $\Sigma$ of Euler class $n$. Its boundary, a circle bundle over $\Sigma$, will be denoted by $Y$. We note that the for an embedding of $\Sigma$ with normal bundle $N$, the complement $X -\Sigma$ of $\Sigma$ deformation retracts onto the exterior of $\Sigma$, defined as $X - \mathrm{int}(N)$; it is more convenient henceforth to work with the exterior.  The idea of the proof of Theorem 1 is to find a suitable manifold $W$ with $\partial W = Y$ to serve as the exterior of $\Sigma$ in $X$. In fact we will take $X$ to be $N \cup W$, and the goal is to choose $W$ so that $X$ is simply connected. 

To this end, we recall that for a group $G$, the oriented bordism group $\Omega_k(G)$ consists of pairs $(M^k,\varphi)$ where $M$ is an oriented closed $k$-manifold and $\varphi:\pi_1(M) \to G$ is a homomorphism, up to an obvious cobordism relation. Equivalently, it is the bordism group $\Omega_k(BG)$. One defines, similarly, the spin bordism group $\Omega^{spin}_k(G)$ as spin cobordism classes of triples $(M,spin,\varphi)$ where $spin$ is a spin structure on $M$. One can prove that such bordism groups are functorial with respect to homomorphisms of $G$.

The following lemma is a straightforward consequence of the Atiyah-Hirzebruch spectral sequence for the generalised homology theory of oriented bordism or spin bordism. 

 Lemma 2: The map $\Omega_3(G) \to H_3(G;\mathbb{Z})$ that assigns $\varphi_*([M])$ to a pair $(M,\varphi)$ is an isomorphism.  If $H_1(G;\mathbb{Z}) = H_2(G;\mathbb{Z}) = 0$, then the same is true for $\Omega_3^{spin}(G)$.

Recall that the fundamental group of $Y$ is a central extension

\[   \{1\} \to \mathbb{Z} \to \pi_1(Y)  \to  \pi_{1}(\Sigma) \to \{1\}\]

and that the $\mathbb{Z}$ subgroup is the kernel of the map $\pi_1(Y) \to \pi_1(N)$. Denote by $\mu$ the generator of this $\mathbb{Z}$ subgroup corresponding to the oriented meridian of $\Sigma$.

 Lemma 3: Suppose that there is a finitely presented simple group $P$ and a homomorphism $\varphi: \pi_1(Y) \to P$ such that $\varphi(\mu)$ is non-trivial in $P$ and the bordism class of $(Y,\varphi)$ in $\Omega_3(P)$ is trivial.
Then there is an embedding of $\Sigma$ in a simply connected $4$-manifold $X$ such that the exterior $W =X - \mathrm{int}(N)$ has fundamental group $P$. If $n$ is even, choose a  spin structure on $N$ with restriction $\sigma_Y$ to $Y$. If the bordism class of  $(Y,\sigma_Y,\varphi)$ in $\Omega^{spin}_3(P)$ is trivial, then $X$ may be chosen to be a spin manifold.

Proof:    By hypothesis, there is a $4$-manifold $W_0$ and the portion connected by solid arrows in the diagram below.



We modify $W_0$ in two stages to fill in the dotted portion of the diagram so that $\Phi$ is an isomorphism. Since $P$ is finitely presented, there is a closed $4$-manifold $W_P$ with $\pi_1(W_P)\cong P$. Replace $W_0$ with $W_0 \# W_P$ and note that the homomorphism $\pi_1(W_0 \# W_P) \cong \pi_1(W_0) * P \to P$ given by $\Phi_0$ on the first factor and the identity on the second factor is a surjection. Choose finitely many generators $\{x_j\}$ for $\pi_1(W_0)$.  Now do surgery on circles in $W_0 \# W_P$ representing the elements $x_j^{-1}\Phi_0(x_j)$ to obtain a manifold $W_1$ with a surjection $\Phi_1$ as indicated. The homomorphism $i:\pi_1(W_0) \to \pi_1(W_1)$ is induced by the inclusion of the summand $W_0$ in $W_0 \# W_P$.  Do surgery on finitely many circles in $W_1$ to kill the kernel of $\Phi_1$, obtaining the manifold $W$ and isomorphism $\Phi$ as indicated.

The image of $\mu$ in $\pi_1(W)$ is taken to $\varphi(\mu)$, which is assumed to be non-trivial in $P$. Since $P$ is simple, $\mu$ normally generates $\pi_1(W)$. On the other hand, $\mu$ is trivial in $\pi_1(N)$, so van Kampen's theorem says that $X = W \cup N$ is simply connected.

The argument goes through in the spin case to produce a simply connected spin manifold $X$. By hypothesis, the initial manifold $W_0$ has a spin structure extending the given one on $Y$ (which by hypothesis extends over $N$). But it is standard that the framing of a surgery on a circle in a spin $4$-manifold can be chosen so that the new manifold inherits a spin structure. This choice of framing does not affect the fundamental group arguments. $\blacksquare$

Hence, to complete the proof of Theorem 1, it suffices to find groups and homomorphisms satisfying the hypotheses of Lemma 3. 

 As we all know, 4 is the bad number. Many of the tools we have for studying manifolds break or are much weaker specifically in dimension 4 because there are just too many pathologies in this dimension. One of the questions asked about this, that has led to spectacular developments in low dimensional topology, is the Thom conjecture. A smooth algebraic curve $C$ in the complex projective plane $\mathbb{CP}^2$ of degree $d$, has genus given by the genus–degree formula $g=(d-1)(d-2)/2$. The Thom conjecture asks whether this is the minimal genus of a smoothly embedded connected curve representing the same class in homology as $C$ in $H_2(\mathbb{CP}^2)$. Kronheimer and Mrowka, using and popularising Seiberg-Witten invariants, proved this, and the result has since been massively generalised to embedding in other 4-manifolds as well.

A basic invariant of an embedding of a surface $\Sigma$ in a $4$-manifold $X$ is the fundamental group of its complement, and it is reasonable to ask what groups can occur, especially if $X$ is simply connected. With no further restrictions on the embedding, there is a simple characterisation of such groups: A finitely presented group $G$ is $\pi_1(X-\Sigma)$ for some $\Sigma \subset X$ if and only if $H_1(G)$ is cyclic and $G$ is the normal closure of a single element.

Previous work of Kim and Ruberman provides a surface with self-intersection $0$, but the problem is more challenging if one requires that the self-intersection is non-zero.  The motivation is an observation of Kronheimer and Mrowka that if the self-intersection number were non-zero and square-free, $\pi_{1}(X-\Sigma)$ would have no non-trivial representations in $\mathrm{SO}(3)$. 

Today I would like to discuss the following result of Hughes and Ruberman:

Theorem 1: For any non-zero $n$, there is a simply connected $4$-manifold $X$ and surface $\Sigma$ smoothly embedded in $X$ where $\Sigma \cdot \Sigma = n$ and $\pi_1(X-\Sigma) \neq \{1\}$. 

The reason is that the proof involves (what I find) a surprising instance of the cohomology of various simple groups being useful in geometric topology, especially those of Higman-Thompson groups which at first glance have nothing to do with the question at hand.

Reduction to group theory

Thompson groups

The groups that we will end up using are Higman-Thompson groups, but since the literature is a bit of a mess and I find it a bit daunting, I thought I'd take this opportunity to jot down some notes on such things.

Definition: Thompson's group F is the group (under composition) of homeomorphisms of the interval $[0,1]$ which satisfy the following conditions:

1. . they are piecewise linear and orientation-preserving,
2.  in the pieces where the maps are linear, the slope is always a power of 2, and
3.  the breakpoints are dyadic, i.e., they belong to the set $D \times D$, where $D = [0, 1] \cap \mathbb{Z}[\frac{1}{2}].$

If we identify the endpoints of $[0, 1]$ to form a circle $S^1$, then $F$ acts on $S^1$. We obtain Thompson's group $T$ by adding one more generating homeomorphism $Y : S^1 \to S^1$, namely, $x \mapsto x + 1/2$ modulo 1.

The third of the Thompson groups, $V$, is the group of self-maps of $S^1$ generated by $T$ and one final discontinuous generator $Z$ that fixes the half-open interval $[0, 1/2)$ and interchanges the half-open intervals $[1/2, 3/4)$ and $[3/4, 1)$.

These three groups have led to hundreds of papers and a head-spinning number of different generalisations, some of which we will see below, due to their mysterious and wonderful properties. Let me briefly describe some of these.

Properties of $F$:

1.  $F$ has a classifying space with 2 cells in every positive dimension, so is of type $F_{\infty}$  
2. $F$ is locally indicable, i.e. any non-trivial finitely generated subgroup admits a surjection to $\mathbb{Z}$. In particular, $F$ is torsion-free and satisfies all the Kaplansky group ring conjectures 
3. $F$ is not elementary amenable and contains a free sub-semigroup but contains no non-abelian free subgroup. Whether it is amenable or not has been the subject of much controversy.
4. $F$ has a 2-generator, 2-relator presentation. Another instance of just how wild things can get when one adds another relator to a one-relator group (which aren't exactly well-behaved, but are sort of tractable).
5. The derived subgroup $F'$ is an infinitely generated simple group. Note there is no contradiction with the first item!

For more details the reader is invited to consult either the Wikipedia page on Thompson groups, which mostly focusses on $F$, and the references therein, or this book.


In contrast to the above, $T$ has torsion and non-abelian free subgroups. It is however also finitely presented, and is simple, making it the first known example of a finitely presented simple group. $T$ and $V$ have many similar properties, as we see now:

Properties of $V$: 

1. $V$ is a simple group; 
2. $V$ is acyclic, which follows from a difficult paper of Szymik-Wahl that uses homological stability methods;
3. $V$ contains every finite group as a subgroup; 
4. $V$ is finitely presented;
5. $V$ is type $\mathsf{FP}_\infty$, which combined with the previous item implies that $V$ is type $\mathsf{F}_\infty$
6. $V$ has solvable word problem (hence so do $F$ and $T$) and solvable conjugacy problem.
7. $V$ is $\frac{3}{2}$ generated: this means that for any non-trivial element $x$ there exists some $y$ such that $V$ is generated by $x$ and $y$. $T$ also has this property, as do all finite simple groups. Note that another important class of groups with this property is the class of Tarski monsters.
   

 
I now quote directly Anthony Genevois' description of general Higman-Thompson groups from MO:

 Given a tree $\mathcal{T}$, a quasi-automorphism is a bijection $\mathcal{T}^{(0)} \to \mathcal{T}^{(0)}$ that preserves adjacency and non-adjacency for all but finitely many pairs of vertices. Now, $\mathrm{QAut}(\mathcal{T})$ contains a natural normal subgroup: the subgroup $\mathrm{FSym}(\mathcal{T})$ of finitely supported bijections $\mathcal{T}^{(0)} \to \mathcal{T}^{(0)}$. The group of almost-isometries $\mathrm{AIsom}(\mathcal{T})$ is the quotient $\mathrm{QAut}(\mathcal{T})/ \mathrm{FSym}(\mathcal{T})$.  

Equivalently, an almost-isometry can be described as a triple $(A,B,f)$, where $A,B$ are finite subtrees and where $f : \mathcal{T} \backslash A \to \mathcal{T} \backslash B$ is an isometry; two almost-isometries $(A,B,f)$ and $(C,D,g)$ being identified whenever $f$ and $g$ agree on some cofinite subset. In practice, this means that you remove two finite subtrees from two copies of $\mathcal{T}$ and permute the connected components by isometries.

   

From such a picture, you can impose restrictions on the permutations of components and on the isometries between components. If $\mathcal{T}$ is a rooted binary tree (thought of as drawn on the plane), we obtain

 - Thompson group $F$ if permutations and isometries preserve the left-right order induced by the plane;
 - Thompson group $T$ if the permutations preserve the cyclic order on the components and the isometries the left-right order on each component;
 - Thompson group $V$ if there is no restriction on the permutations and the isometries preserve the left-right order on each component.

But it is possible to modify the tree $\mathcal{T}$ and to obtain quite similar groups. Let $\mathcal{T}_{n,r}$ denote the tree with one vertex of degree $r$ while all the other vertices have degree $n+1$. Then the same definitions as above respectively give Thompson groups $F_{n,r}$, $T_{n,r}$, and $V_{n,r}$ (sometimes also denoted by $G_{n,r}$, following Higman). 

Seifert fibred 3-manifolds and Thompson's group $V$

Proposition 4:  Let $Y$ be a Seifert fibred space and let $H=\pi_1 Y$.  Let $\mu\in Y$ be a generator of the centre of $Y$.  Then there exists a homomorphism $\psi\colon H\to V$ such that $\psi(\mu)$ normally generates $V$.

Proof: Since $V$ is acyclic $\Omega_3(V)=0$ and $\Omega^{spin}_3(V)=0$. All 3-manifold groups are residually finite so we can find a finite quotient $h\colon H\twoheadrightarrow Q$ such that $h(\mu)$ is non-trivial.  Since every finite group is a subgroup of $V$ we may embed $Q$ into $V$ via some homomorphism $i\colon Q\to V$.  We define $\psi$ to be $i\circ h$.  Now, as $V$ is simple, every non-trivial element of $V$ normally generates it. Thus, $\psi(\mu)$ normally generates $V$. $\blacksquare$
 

This completes the proof of theorem 1. In fact we can build more examples out of the Higman--Thompson groups $V_{m,r}$.  In the previously mentioned paper of Szymik and Wahl the authors show $H_\ast(V_{m,r};\mathbb{Z})\cong H_\ast(\Omega^\infty_0\mathbf{M}_{m-1};\mathbb{Z})$, where the second object is the homology of the zeroth component of the infinite loop space of the mod $m-1$ Moore spectrum.  The relevance for us is Propositions 6.1 and 6.2 of that paper; there it is shown that when $m$ is even and $p$ is the smallest prime dividing $m-1$ we have $\widetilde H_d(V_{m,r};\mathbb{Z})=0$ for $d<2p-3$ and $H_{2p-3}(V_{m,r};\mathbb{Z})=;\mathbb{Z}/p$.  In particular, these groups also apply to the above construction with one caveat: if $3|m-1$ then one must map to a finite group $Q$ with $|H_3(Q;\mathbb{Z})|$ coprime to $3$ to ensure the bordism class vanishes.

Hughes and Ruberman also produce examples where the complement has fundamental group a suitable finite simple group, and to show that something satisfies the conditions to check, one need only look into the vast literature on finite simple groups.