Embedding lens spaces in definite 4-manifolds

TL;DR

This paper investigates the embedding properties of lens spaces into complex projective planes, establishing limitations on their smooth embeddings and providing new insights into 4-manifold topology.

Contribution

It proves that while every lens space can embed in a connected sum of 8 complex projective planes, there is no universal n for smooth embeddings into n copies of the plane.

Findings

Every lens space embeds in 8 copies of CP^2

No single n allows all lens spaces to smoothly embed in n copies

Differentiates between locally flat and smooth embedding capabilities

Abstract

Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.

Equations

$$a_1$$

$$(H_2(X;\mathbb{Z})/\mathrm{Tors},Q_X)\hookrightarrow (H_2(P(p,q);\mathbb{Z}),Q_{P(p,q)})\oplus ({\mathbb{Z}}^{b_2(X)-m},\mathrm{Id}).$$

$$a_1$$

$$2$$

$$x_i=e_i+e_{i+1}\text{ for }1\le i\le a_1-2.$$

$$x_{a_1-1}=e_{a_1-1}+\sum _{a_1-1<j}{c}_j{e}_j.$$

$$2$$

$$(H_2(X;\mathbb{Z})/\mathrm{Tors},Q_X)\oplus (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q})\hookrightarrow (H_2(W;\mathbb{Z})/\mathrm{Tors},Q_W).$$

$$(H_2(W;\mathbb{Z})/\mathrm{Tors},Q_W)\cong ({\mathbb{Z}}^{b_2(W)},\mathrm{Id}).$$

$${\Gamma }_{\varphi }^{\perp }:=\{x\in {\Gamma }^{\prime }\mid x\cdot \varphi (y)=0\text{ for all }y\in \Gamma \}.$$

$$\varphi :(\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q})\hookrightarrow ({\mathbb{Z}}^N,\mathrm{Id}),$$

$$\varphi :(\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q})\hookrightarrow ({\mathbb{Z}}^{m+\mathrm{rk}\left(\mathbb{Z}{\Gamma }_{p,p-q}\right)},\mathrm{Id})\oplus ({\mathbb{Z}}^{N-m-\mathrm{rk}\left(\mathbb{Z}{\Gamma }_{p,p-q}\right)},\mathrm{Id}),$$

$$(\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{\varphi }^{\perp }\cong (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{\pi \circ \varphi }^{\perp }\oplus ({\mathbb{Z}}^{N-m-\mathrm{rk}\left(\mathbb{Z}{\Gamma }_{p,p-q}\right)},\mathrm{Id}).$$

$$\psi :(\mathbb{Z}{\Gamma }_{p,q},Q_{p,q})\oplus (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q})\hookrightarrow ({\mathbb{Z}}^{m+\mathrm{rk}\left(\mathbb{Z}{\Gamma }_{p,p-q}\right)},\mathrm{Id}).$$

$$(\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{\pi \circ \varphi }^{\perp }\cong (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{{\psi }_{p,p-q}}^{\perp }.$$

$${\psi }_{p,q}:(\mathbb{Z}{\Gamma }_{p,q},Q_{p,q})\hookrightarrow (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{{\psi }_{p,p-q}}^{\perp }.$$

$$c_1=\cdots =c_{a_1-1},c_{n_{\ell }}=\cdots =c_{n_{\ell }+a_{\ell }-3},\text{ for }2\le \ell \le m-1,\text{ and }{c}_{n_m}=\cdots =c_{n_m+a_m-2}$$

$$c_{n_{\ell +1}-2}+c_{n_{\ell +1}-1}+c_{n_{\ell +1}}=0,\text{ for }1\le \ell \le m-1.$$

$$x_1$$

$$x_{\ell }$$

$$x_m$$

$$\psi :(H_2(X;\mathbb{Z})/\mathrm{Tors},Q_X)\oplus (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q})\hookrightarrow ({\mathbb{Z}}^{b_2(X)+\mathrm{rk}(\mathbb{Z}{\Gamma }_{p,p-q})},\mathrm{Id}).$$

$$\psi {|}_{H_2(X;\mathbb{Z})/\mathrm{Tors}}:(H_2(X;\mathbb{Z})/\mathrm{Tors},Q_X)\hookrightarrow (\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{{\psi }_{p,p-q}}^{\perp }.$$

$$(\mathbb{Z}{\Gamma }_{p,p-q},Q_{p,p-q}{)}_{{\psi }_{p,p-q}}^{\perp }\cong (\mathbb{Z}{\Gamma }_{p,q},Q_{p,q}).$$

Full text

1. Introduction

Every closed $3$-manifold embeds in $S^{5}$ [Hir61, Roh65, Wal65]. On the other hand, there are strong restrictions on which closed $3$-manifolds can embed in $S^{4}$. It was shown by Hantzsche [Han38] that if a rational homology $3$-sphere $Y$ embeds in $S^{4}$, then $H_{1}(Y;\mathbb{Z})\cong G\oplus G$. In particular, no lens space (other than $S^{3}$ and $S^{1}\times S^{2}$) embed in $S^{4}$. Further, a punctured lens space $L(p,q)_{0}$ admits an embedding in $S^{4}$ if and only if $p$ is odd [Zee65, Eps65]. Also, we have a complete understanding of which connected sums of lens spaces can be smoothly embedded in $S^{4}$ by Donald [Don15] (see also [KK80, GL83, FS87]). There are also various interesting results on embedding other $3$-manifolds in $S^{4}$ [Kaw77, CH81, Hil96, CH98, BB08, Hil09, IM18].

Even though the embedding problem of lens spaces in $S^{4}$ is completely solved, there are many interesting generalizations. In this paper, we focus on the embedding problem of lens spaces in definite $4$-manifolds (the embedding problem of lens spaces in spin $4$-manifolds has been studied in [AGL17]). In [EL96], Edmonds and Livingston showed that every lens space smoothly embeds in $\#_{n}\mathbb{C}\mathbb{P}^{2}$ for some positive integer $n$. Further, they showed that there is a family of lens spaces that do not have locally flat embeddings in $\#_{4}\mathbb{C}\mathbb{P}^{2}$. Later, Edmonds [Edm05] showed that every lens space has a locally flat embedding in $\#_{8}\mathbb{C}\mathbb{P}^{2}$ using independent works of Boyer [Boy93] and Stong [Sto93] which extend Freedman’s [Fre82] realization result. In contrast, we show that there is no $n$ such that every lens space smoothly embeds in $\#_{n}\mathbb{C}\mathbb{P}^{2}$. Our main argument relies on Donaldson’s diagonalization theorem [Don87] and is based on the combinatorics of integral lattices.

Theorem 1.1**.**

Let $L(p,q)$ be a lens space bounded by the canonical positive definite plumbed manifold $P(p,q)$ with the plumbing graph

[TABLE]

Suppose that $a_{i}\geq 6$ for all $i$. If $L(p,q)$ smoothly embeds in a definite $4$-manifold $W$ with $b_{1}(W)=0$, then $b_{2}(W)>m$. In particular, if $L(p,q)$ smoothly embeds in $\#_{n}\mathbb{C}\mathbb{P}^{2}$, then $n>m$.

Furthermore, we show that if a lens space $L(p,q)$ as in Theorem 1.1 bounds a smooth, positive definite $4$-manifold, there is a strong restriction on its intersection form. This also reflects the big discrepancy between the smooth and the topological category in dimension $4$ since for every $3$-manifold $Y$ and a $\mathbb{Z}$-valued symmetric bilinear form $Q$ that presents the linking form of $Y$, $Q$ is realized as the intersection form of a simply connected, topological $4$-manifold bounded by $Y$ [Boy93, Sto93]. For instance, every lens space bounds a simply connected, positive definite, topological $4$-manifold $X$ with $b_{2}(X)\leq 6$ [Edm05]. Recall that an integral lattice is a pair $(G,Q)$, where $G$ is a finitely generated free abelian group and $Q$ is a $\mathbb{Z}$-valued symmetric bilinear form defined on $G$. The integral lattice with the standard positive definite form is denoted by $(\mathbb{Z}^{N},\operatorname{Id})$. A morphism of integral lattices is a homomorphism of abelian groups which preserves the form. An embedding is an injective morphism. To a given $4$-manifold $X$, we associate the integral lattice $(H_{2}(X;\mathbb{Z})/{\rm Tors},Q_{X})$, where $Q_{X}$ is the intersection form on $X$.

Theorem 1.2**.**

Let $L(p,q)$ be a lens space that satisfies the assumption of Theorem 1.1. If $L(p,q)$ bounds a smooth, positive definite $4$-manifold $X$, then $b_{2}(X)\geq m$ and there is an embedding

[TABLE]

*Moreover, $b_{2}(X)=m$ if and only if $X$ and $P(p,q)$ have isomorphic intersection forms. *

Acknowledgments

This project started when both authors were at the Max-Planck-Institut für Mathematik. We would like to thank the MPIM for providing an excellent environment for research.

2. proof of Theorem $1.1$ and $1.2$

We work in the smooth category and all manifolds are oriented. Recall that the lens space $L(p,q)$ is the result of $-p/q$ Dehn surgery on the unknot. Up to orientation preserving diffeomorphism, we may assume that $p>q>0$. For the rest of this article we only consider lens spaces $L(p,q)$ that bound the canonical positive definite plumbded manifolds $P(p,q)$ with the plumbing graph $\Gamma_{p,q}:=$

[TABLE]

where $a_{i}\geq 6$ for all $i$.

Let $-L(p,q)$ be the lens space $L(p,q)$ with the reversed orientation, then there is an orientation preserving diffeomorphism between $-L(p,q)$ and $L(p,p-q)$. Using Riemenschneider’s point rule [Rie74] (see also [Lis07, Lec12]), we see that $L(p,p-q)$ bounds a canonical positive definite plumbed manifold $P(p,p-q)$ with the plumbing graph $\Gamma_{p,p-q}:=$

[TABLE]

We denote the integral lattice associated to $P(p,q)$ as $(\mathbb{Z}\Gamma_{p,q},Q_{p,q})$ and call it the integral lattice associated with $L(p,q)$. Similarly, we also have a dual positive definite integral lattice $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ associated with $L(p,p-q)$.

Proposition 2.1**.**

If there is an embedding from $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ to $(\mathbb{Z}^{N},\operatorname{Id})$, then $N\geq m+\operatorname{rk}(\mathbb{Z}\Gamma_{p,p-q})$.

Proof.

We label the first $a_{1}$ vertices of $\Gamma_{p,p-q}$ as follows

[TABLE]

Let $\{e_{1},\ldots,e_{N}\}$ be the standard basis for $(\mathbb{Z}^{N},\operatorname{Id})$. By abuse of notation we identify $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ with its image in the standard lattice. It is straightforward to see that a chain of $2$’s with length longer than $3$ has a unique embedding. Hence up to reordering and changing sign of the standard basis elements we may write

[TABLE]

Further, since $x_{a_{1}-1}$ intersects with $x_{a_{1}-2}$ once and has norm $3$,

[TABLE]

Lastly, $x_{a_{1}}$ has a trivial intersection with $e_{a_{1}-1}$ since it is disjoint from the first chain of $2$’s and it has norm $2$. Therefore, $x_{a_{1}-1}-e_{a_{1}-1}$ is disjoint from the first chain of $2$’s and intersects $x_{a_{1}}$ once. Now, if we only consider $x_{a_{1}-1}-e_{a_{1}-1}$ and all the vertices that reside on the right hand side of $x_{a_{1}-1}$, we get to the same situation as we have started with. Hence we can repeat the same argument to get the following identifications

[TABLE]

for each chain of $2$’s where $n_{\ell}=\sum\limits_{k=1}^{\ell-1}a_{k}+(3-\ell)$ and $2\leq\ell\leq m-1$, and

[TABLE]

for the $m$-th chain of $2$’s where $n_{m}=\sum\limits_{k=1}^{m-1}a_{k}+(3-m)$. The $\ell$-th vertex with weight $3$ shares one coordinate from its left chain and one coordinate from its right chain. Further, it needs an extra coordinate, which we denote it by $e_{n_{\ell+1}-1}$.

In total, we have used $\sum\limits_{k=1}^{m}a_{k}-m+1$ coordinates which implies that $N\geq\sum\limits_{k=1}^{m}a_{k}-m+1$. The proof is complete by observing that $m+\operatorname{rk}(\mathbb{Z}\Gamma_{p,p-q})=\sum\limits_{k=1}^{m}a_{k}-m+1$.∎

Remark 2.2**.**

In fact, the proof of Proposition 2.1 shows that there is a unique embedding up to change of basis from $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ to $(\mathbb{Z}^{N},\operatorname{Id})$ when $N\geq m+\operatorname{rk}(\mathbb{Z}\Gamma_{p,p-q})$.

Proposition 2.3**.**

If $L(p,q)$ bounds a positive definite $4$-manifold $X$, then $b_{2}(X)\geq m$.

Proof.

Let $W$ be the closed $4$-manifold obtained by gluing $X$ with $P(p,p-q)$ along $L(p,q)$. We obtain the following embedding

[TABLE]

Further, by Donaldson’s diagonalization theorem [Don87] we have

[TABLE]

Combining Proposition 2.1 with $b_{2}(W)=b_{2}(X)+\operatorname{rk}(\mathbb{Z}\Gamma_{p,p-q})$ completes the proof. ∎

Proof of Theorem 1.1.

Suppose $L(p,q)$ smoothly embeds in a definite $4$-manifold $W$. Since $L(p,q)$ embeds in $W$ if and only if $L(p,q)$ embeds is $-W$, we may assume that $W$ is positive definite. Then $L(p,q)$ separates $W$ into two positive definite components, the closures of which we denote by $X_{1}$ and $X_{2}$. Note that by the Mayer–Vietoris sequence we have $b_{2}(X_{1})+b_{2}(X_{2})=b_{2}(W)$ and the result follows from Proposition 2.3 and the fact that $L(p,q)$ does not bound a rational ball (see [Lis07]). ∎

The rest of the section is devoted to proving Theorem 1.2. Suppose there is an embedding of an integral lattice $\phi\colon\Gamma\hookrightarrow\Gamma^{\prime}$. Then the orthogonal complement of $\Gamma$ in $\Gamma^{\prime}$ with respect to $\phi$ is defined as follows,

[TABLE]

Proposition 2.4**.**

Suppose there is an embedding

[TABLE]

then $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})^{\perp}_{\phi}\cong(\mathbb{Z}\Gamma_{p,q},Q_{p,q})\oplus(\mathbb{Z}^{N-m-\operatorname{rk}\left(\mathbb{Z}\Gamma_{p,p-q}\right)},\operatorname{Id})$. In particular, if $N=m+\operatorname{rk}(\mathbb{Z}\Gamma_{p,p-q})$, then $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})^{\perp}_{\phi}\cong(\mathbb{Z}\Gamma_{p,q},Q_{p,q})$.

Proof.

From Proposition 2.1 and Remark 2.2, we know that there is a unique embedding up to change of basis from $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ to $(\mathbb{Z}^{N},\operatorname{Id})$. Hence we may decompose $\phi$ as follows

[TABLE]

where the image of $\phi$ on the second summand is trivial. Let $\pi$ be the projection map from $(\mathbb{Z}^{N},\operatorname{Id})$ to $(\mathbb{Z}^{m+\operatorname{rk}\left(\mathbb{Z}\Gamma_{p,p-q}\right)},\operatorname{Id})$, then we have the following identification

[TABLE]

Let $W$ be the closed $4$-manifold obtained by gluing $P(p,q)$ with $P(p,p-q)$ along $L(p,q)$. Using Donaldson’s diagonalization theorem [Don87], we have an embedding

[TABLE]

Again, since there is a unique embedding up to change of basis from $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$ to $(\mathbb{Z}^{m+\operatorname{rk}\left(\mathbb{Z}\Gamma_{p,p-q}\right)},\operatorname{Id})$, we may assume that the embedding $\psi$ restricted to $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})$, denoted by $\psi_{p,p-q}$, coincides with $\pi\circ\phi$. In particular,

[TABLE]

Further, we can use the coordinates from the proof of Proposition 2.1.

By restricting $\psi$ to $\mathbb{Z}\Gamma_{p,q}$, denoted by $\psi_{p,q}$, we have

[TABLE]

Now, suppose $x=\sum\limits_{i=1}^{N}c_{i}e_{i}\in(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})^{\perp}_{\psi_{p,p-q}}$. Since $x$ needs to have trivial intersections with all the chain of $2$’s, we have

[TABLE]

where $n_{\ell}=\sum\limits_{\ell=1}^{i-1}a_{\ell}+(3-\ell)$ for $2\leq\ell\leq m$. Further, $x$ has trivial intersections with vertices with weight $3$. This implies

[TABLE]

From the above relations, we see that $\{x_{\ell}\}$ forms a basis for $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})^{\perp}_{\psi_{p,p-q}}$, where

[TABLE]

Finally, it is straightforward to check that the matrix, denoted by $M$, that represents the intersection form of $(\mathbb{Z}\Gamma_{p,p-q},Q_{p,p-q})^{\perp}_{\psi_{p,p-q}}$ with respect to the basis $\{x_{\ell}\}$ coincides with the matrix, denoted by $M_{p,q}$, that represents $Q_{p,q}$ with respect to the obvious basis for $\mathbb{Z}\Gamma_{p,q}$. Note that we have $M=P^{\top}M_{p,q}P$ where $P$ is a matrix that represents $\psi_{p,q}$. This implies that $P$ is unimodular and $\psi_{p,q}$ is an isomorphism. Then the result follows from (2.1) and (2.2). ∎

Proposition 2.4 is motivated by [ACP18, Proposition 4.1]. By restricting to a smaller family of lens spaces, Proposition 2.4 gives the same conclusion as [ACP18, Proposition 4.1] with a weaker assumption. We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2.

Let $W$ be the closed $4$-manifold resulting from gluing $X$ and $P(p,p-q)$ along $L(p,q)$. Again, by Donaldson’s theorem [Don87] we have an embedding

[TABLE]

Let $\psi_{p,p-q}$ be the embedding obtained by restricting $\psi$ to $\mathbb{Z}\Gamma_{p,p-q}$. Further, by restricting $\psi$ to $H_{2}(X;\mathbb{Z})/{\rm Tors}$ we have the following embedding

[TABLE]

Then the first part of the theorem follows from Proposition 2.3 and Proposition 2.4.

Suppose now that $m=b_{2}(X)$, then by Proposition 2.4 we have

[TABLE]

Let $M_{X}$ and $M_{p,q}$ be matrices that represent $Q_{X}$ and $Q_{p,q}$, respectively. Then $M_{X}=P^{\top}M_{p,q}P$ where $P$ is a matrix that represents $\psi|_{H_{2}(X;\mathbb{Z})/{\rm Tors}}$. Recall that $M_{X}$ presents a subgroup of $H_{1}(L(p,q);\mathbb{Z})$ (see, for instance, [OS06, Section 2]) and $M_{p,q}$ presents $H_{1}(L(p,q);\mathbb{Z})$. In particular, $\det(M_{X})\leq p$ and $\det(M_{p,q})=p$, which implies that $P$ is unimodular and concludes the proof.∎