Release Time:2019-03-10  Hits:

Indexed by: Journal Article

Date of Publication: 2009-01-01

Journal: ALGEBRAIC AND GEOMETRIC TOPOLOGY

Included Journals: SCIE

Volume: 9

Issue: 4

Page Number: 2041-2054

ISSN: 1472-2739

Abstract: Let M be a compact, orientable, partial derivative-irreducible 3-manifold and F be a connected closed essential surface in M with g. (F) >= 1 which cuts M into M(1) and M(2). In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary. (Q(i), partial derivative Q(i)) in (M(i), F) satisfying chi(Q(i)) > 2 + g(F) - 2g(M(i)), i = 1; 2. Then g (M) = g(M(1)) + g(M(2)) - g(F). As a consequence, we further show that if M(i) has a Heegaard splitting V(i)boolean OR(Si) W(i) with distance D(S(i)) >= 2g(M(i)) - g(F), i = 1; 2, then g(M) = g(M(1)) + g(M(2)) - g(F).
   The main results follow from a new technique which is a stronger version of Schultens' Lemma.