Erratum to: Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

1 Erratum to: Math. Z. (2007) 256:913–923 DOI 10.1007/s00209-007-0113-8

In the original publication, the proof of Lemma 4 was incorrect and it is now corrected.

In [1, Lemma 4], we constructed two homotopic pinches between closed surfaces, such that the boundary circles of the two regions we pinched are not homotopic. However, as pointed out by Tao Li, our construction in Case 1 in the original proof is not correct. In this note, we will provide a new proof of this lemma.

We start by recalling some notations from [1]. If $C$ is a subpolyhedron of a manifold $M$, let $N(C)$ be a regular neighborhood of $C$ in $M$. If $M$ is a compact manifold, let $\mathrm {int}(M)$ be the interior of $M$. Suppose that $g_s>g_t\ge 1$ are two integers. Let $F_s,F_t$ be two closed oriented surfaces with genus $g_s,g_t$, respectively. Fix a disk $D\subset F_t$, let $V=F_t-\mathrm {int}(D)$. Suppose that $p_0, p_1 : F_s\rightarrow F_t$ are two pinches. Namely, there exist two compact subsurfaces $V_0,V_1\subset F_s$, such that $p_j$ maps $V_j$ homeomorphically to $V$, and $p_j$ maps $W_j=F_s-\mathrm int (V_j)$ into $D$, $j=0,1$. Let $e_j:V\hookrightarrow F_s$ be the inverse of $p_j$.

Lemma 4

With the notation above, there exist two pinches $p_0, p_1 : F_s\rightarrow F_t$ such that

• (i) $p_0$ and $p_1$ are homotopic;

• (ii) $e_0(\partial D)$ is not homotopic to $e_1(\partial D)$.

Proof

Let $h=g_s-g_t$, $S$ be a closed oriented surface of genus $h+1$. As in [1, Figure 2], we can choose three simple closed curves $\alpha ,\beta _0,\beta _1\subset S$, such that $\alpha$ intersects $\beta _j$ transversely at exactly one point, $j=0,1$, and $\beta _0$ is homologous to $\beta _1$. Moreover, let $\gamma _j=\partial N(\alpha \cup \beta _j)$, then $\gamma _0$ is not freely homotopic to $\gamma _1$. We may further assume that $\alpha \cap \beta _0=\alpha \cap \beta _1$, and a singular two-chain $\Phi$ connecting $\beta _0$ and $\beta _1$ is disjoint from $\alpha$ except in a small neighborhood of $\alpha \cap \beta _0$.

Let $S^{\prime }$ be the surface obtained by removing an open disk about $\alpha \cap \beta _0$ from $S$, and let $\alpha ^{\prime }=\alpha \cap S^{\prime }$, $\beta ^{\prime }_j=\beta _j\cap S^{\prime }$. Clearly, $\beta ^{\prime }_0$ and $\beta ^{\prime }_1$ are homologous in $S^{\prime }$ relative to $\partial S^{\prime }$. So we can find a properly embedded surface $B\subset S^{\prime }\times [0,1]$, such that $B\cap (S^{\prime }\times \{j\})=\beta ^{\prime }_j\times \{j\}, j=0,1$. We may assume that $B\cap (\alpha ^{\prime }\times [0,1])=\emptyset$, since $B$ can be chosen to be a lift of $\Phi \cap S^{\prime }$, and we have assumed that $\Phi$ is disjoint from $\alpha$ except in a small neighborhood of $\alpha \cap \beta _0$.

Let $T$ be a torus, $\xi ,\eta \subset T$ be two simple closed curves which intersect transversely at exactly one point. Let $T^{\prime }$ be the surface obtained by removing an open disk about $\xi \cap \eta$ from $T$, and let $\xi ^{\prime }=\xi \cap T^{\prime }$, $\eta ^{\prime }=\eta \cap T^{\prime }$.

Now we construct a map $Q: S^{\prime }\times [0,1]\rightarrow T^{\prime }$ in three steps:

Step 1. Construct

$$\begin{aligned} Q_1 : \left( \partial S'\times [0,1]\right) \cup \left( \alpha ^{\prime }\times [0,1]\right) \rightarrow \partial T^{\prime }\cup \xi ^{\prime } \end{aligned}$$

by first projecting to $\partial S^{\prime }\cup \alpha ^{\prime }$, then map $\partial S^{\prime }\cup \alpha ^{\prime }$ homeomorphically to $\partial T^{\prime }\cup \xi ^{\prime }$.

Step 2. Extend $Q_1$ to a map

$$\begin{aligned} Q_2: \left( \partial S^{\prime }\times [0,1]\right) \cup \left( \alpha ^{\prime }\times [0,1]\right) \cup B\rightarrow \partial T^{\prime }\cup \xi ^{\prime }\cup \eta ^{\prime }. \end{aligned}$$

This can be achieved by first mapping $\beta _j^{\prime }\times \{j\}$ homeomorphically to $\eta ^{\prime }, j=0,1$, then send the interior of $B$ to $\eta ^{\prime }$ by using the contractibility of $\eta ^{\prime }$.

Step 3. Extend $Q_2$ to a map

$$\begin{aligned} Q_3: N\left( \left( \partial S^{\prime }\times [0,1]\right) \cup \left( \alpha ^{\prime }\times [0,1]\right) \cup B\right) \rightarrow N\left( \partial T^{\prime }\cup \xi ^{\prime }\cup \eta ^{\prime }\right) , \end{aligned}$$

then send the closure of

$$\begin{aligned} \left( S^{\prime }\times [0,1]\right) \setminus N\left( \left( \partial S^{\prime }\times [0,1]\right) \cup \left( \alpha ^{\prime }\times [0,1]\right) \cup B\right) \end{aligned}$$

to the closure of

$$\begin{aligned} T^{\prime }\setminus N\left( \partial T^{\prime }\cup \xi ^{\prime }\cup \eta ^{\prime }\right) \end{aligned}$$

using the contractibility of the target, and we can get the map $Q$ we want.

Let $\varSigma$ be a compact surface of genus $g_t-1$ with exactly one boundary component. Let $P_0$ be the projection from $\varSigma \times [0,1]\rightarrow \varSigma$. Gluing the two maps $P_0$ and $Q$ together, we get a map $P: F_s\times [0,1]\rightarrow F_t$, which is a homotopy connecting two pinches $p_0, p_1 : F_s\rightarrow F_t$. Clearly, Condition(ii) in the statement of this lemma is also satisfied.