On the number of singularities of a generic surface with boundary in a 3-manifold

Abstract

We consider a c\infty generic map f : Marrow N of a compact surface M with boundary into a 3-manifold N with boundary which is neat (i.e., f^{-1}(\partial N)=\partial M ). The isolated singularities of the image f(M) are triple points, cross caps and boundary double points. Under certain homological conditions, we give some formulae relating the numbers of these singularities. We also obtain some geometrical applications of these results.

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

“…We remark that we can also apply other formulae involving the number of swallowtails and triple points on singular surfaces in a 3-manifolds (cf., [43,45,49]) to our situation in order to get further relations among invariants of the lightlike differential geometry of spacelike surfaces in Minkowski 4-space.…”

Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications

“…* = 0, 0 0, 1 1, 1 0, 2 1, 2 2, 2 3 4 5 6 7 a In fact, items (1)-( 11) of the above proposition correspond to the relations obtained by combining the relations obtained from II 0,0 o (f ) and II 0,0 e (f ), II e (f ), II 0,1 o (f ) and II 0,1 e (f ), respectively. We note that item (11) is also obtained from the two graphs II a o (f ) and II a e (f ).…”

“…Remark 2.6. For stable maps f of 4-manifolds into 3-manifolds, the triple points of f | S(f ) correspond to the singular fibre of types III 0,0,0 , III 0,0,1 , III 0,1,1 , III 1,1,1 , III 0,0,2 , III 0,2,2 , III 1,1,2 , III 1,2,2 , III 0,1,2 , III 2,2,2 , III 0,3 , III 0,4 , III 0,5 , III 0,6 , III 0,7 , III 1,3 , III 1,4 , III 1,5 , III 1,6 , III 1,7 , III 2,3 , III 2,4 , III 2,5 , III 2,6 , III 2,7 , III 8 , III 9 , III 10 , III 11 , III 12 , III 13 , III 14 , III 15 , III 16 , III 17 , III 18 , III 19 , III 20 , III 21 , III 22 , III 21 , III 22 , III 23 , III 24 , III 25 and III 26 of f . Thus the number of triple points of f | S(f ) coincides with the total number of singular fibre of types as above.…”

Complementary Regions for Maps of Surfaces

“…This shows that the Izumiya-Marar equality is equivalent in this setting to χ(V ) = χ(F )+N + 1 2 K, which is the sum of our two equalities k (1−k)a k = 1 2 (χ+N +C a ) and k (1 − k)b k = 1 2 (χ + N + C b ). Another consequence of the two equalities is that for any i, χ + N + C a and χ + N + C b are even, (compare [2,6,8]).…”