Relations amongst twists along Montesinos twins in the 4-sphere

Abstract

Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain twists along submanifolds analogous to Dehn twists in dimension two. The subgroup of the smooth mapping class group of the 4-sphere coming from loops of 5-dimensional handles of index 1 and 2 coincides with the subgroup generated by twists along Montesinos twins (pairs of 2-spheres intersecting transversely twice) in which one of the two 2-spheres in the twin is unknotted. In this paper we show that this subgroup is in fact trivial or cyclic of order two.

Figures (7)

• Figure 1: An illustration of $W(3) = (R(3),S)$, with the generalization to $W(i)$ being to wrap $i$ times around instead of three times around. The red and blue disks in $R(3)$ ("ear holes" of the snake) are pushed forward and backwards in time to avoid self-intersection. We only show the equator of $S$, with the hemispheres lying in the past and future. The two intersection points are colored pink and green.
• Figure 2: An alternate illustration of $W(3)$, involving two disjoint, embedded $2$--spheres in $S^4$ (the two thick circles capped off with hemispheres in past and future) and an arc connecting them. Pushing a finger from one of the spheres out along this arc and then doing a finger move when one encounters the other sphere, creating a pair of transverse intersections, gives $W(3)$. To recover the illustration in Figure \ref{['F:snake']}, push the finger from $R(3)$ until it meets $S$. However, this description is more "balanced" between $R(3)$, allowing the user to decide which sphere they prefer to draw as the complicated one.
• Figure 3: Isotoping $W(1)$ into a symmetric position so as to see that $W(1) = (R(1),S)$ is isotopic to $\overline{W(1)} = (S,R(1))$.
• Figure 4: An isotopy of $W(3)$. The final frame should be interpreted as a diagram of an embedded torus in $S^1 \times S^3$, the result of surgering along $R(3)$. The interpretation of this diagram is made clearer in Figure \ref{['F:SnakeBarInS1XS3']}.
• Figure 5: The embedded torus $\overline{T(3)}$ in $S^1 \times S^3$. The top is glued to the bottom, and horizontal slices are $S^3$'s, with the "time" coordinate indicated in red/blue shading, as in Figure \ref{['F:snake']}.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof : Proof of Lemma \ref{['L:TwinsAndTori']}
• Corollary 7
• Theorem 8