A universal ribbon surface in $B^{4}$

Abstract

We construct an orientable ribbon surface F ⊂ B 4 , which is universal in the following sense: any orientable 4-manifold M ∼ = B 4 ∪ 1-handles ∪ 2-handles can be represented as a cover of B 4 branched over F .

Lefschetz fibrations over the disc

“…When this kink is positive, the diagrams of N t and N t are directly related by move s 20 if (x i , t i ) is a local maximum point for η and the kink is appearing, or if (x i , t i ) is a local minimum point for η and the kink is disappearing. All the other cases of a positive kink can be reduced to the previous ones, by means of move s 19 . The case of a negative kink is symmetric, we can just use movess i in place of the moves s i .…”

“…If x i is a flat vertex, then the tangent plane to S t at H(x i , t) is vertical for t = t i and its projection reverses the orientation when t passes from t to t . Moves s 24 ands 24 (modulo moves s 9 and s 19 ) describe the effect on the diagram of such a reversion of the tangent plane. If x i is a singular vertex, then N t changes into N t by one move s 17 ,s 17 , s 18 ors 18 .…”

“…Finally, if t i is of type 4, a pair of cancelling half-twists is appearing or disappearing along a band, just as in move s 19 .…”

“…But first we need the following lemma (cf. [3,19]). Figure 4), modulo flat isotopy moves in Figure 7.…”

“…The moves in Figure 40 together with r 2 and r 3 , generate the moves s 5 to s 18 in Figures 6 and 7 and the move s 27 in Figure 27. The only non-trivial facts in this respect are the following: (1) move s 9 , which does not have an explicit representation in Figure 40, can be obtained as a composition of moves r 2 , r 8 and r 10 ; (2) r 14 and r 15 completed with the right terminations, give s 15 and s 16 (after contracting the extra tongue, by using r 2 , r 4 , r 5 , r 8 and r 9 ); (3) similarly, r 16 , r 17 , r 18 and r 19 give s 5 , s 6 , s 17 and s 18 .…”