Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let : F 1 → F 2 be a homeomorphism, and M = M 1 ∪ M 2 , F = F 2 = (F 1 ). Then it ...Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let : F 1 → F 2 be a homeomorphism, and M = M 1 ∪ M 2 , F = F 2 = (F 1 ). Then it is known that g(M ) g(M 1 ) + g(M 2 ) - g(F ). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i1 , Q i ) in (M i1 , F i ) satisfying χ(Q i ) 】 3 - 2g(M i ), i = 1, 2. Then g(M ) = g(M 1 ) + g(M 2 ) - g(F ).展开更多
Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.
For a closed orientable surface Sg of genus not smaller than 2,C(Sg) is the curve complex on S g whose vertices consist of the isotopy classes of nontrivial circles on Sg. It has been showed that any two vertices in C...For a closed orientable surface Sg of genus not smaller than 2,C(Sg) is the curve complex on S g whose vertices consist of the isotopy classes of nontrivial circles on Sg. It has been showed that any two vertices in C(Sg) can be connected by an edge path,and C(Sg) has an infinite diameter. We show that for 0 ≤i≤3g-5,two i-simplices can be connected by an(i +1)-path in C(Sg),and the diameter of C(Sg) under such a distance is infinite.展开更多
Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the dec...Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim（Cy ∩ Cw）.展开更多
基金supported in part by a Key Grant of National Natural Science Foundation of China (Grant No. 10931005)
文摘Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let : F 1 → F 2 be a homeomorphism, and M = M 1 ∪ M 2 , F = F 2 = (F 1 ). Then it is known that g(M ) g(M 1 ) + g(M 2 ) - g(F ). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i1 , Q i ) in (M i1 , F i ) satisfying χ(Q i ) 】 3 - 2g(M i ), i = 1, 2. Then g(M ) = g(M 1 ) + g(M 2 ) - g(F ).
基金supported by National Natural Science Foundation of China(Grant Nos.11329101 and 11101058)
文摘Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.
基金supported by National Natural Science Foundation of China(Grant Nos.10931005 and 11101058)the National Science Foundation for Post-doctoral Scientists of China(Grant No.2011M500049)
文摘For a closed orientable surface Sg of genus not smaller than 2,C(Sg) is the curve complex on S g whose vertices consist of the isotopy classes of nontrivial circles on Sg. It has been showed that any two vertices in C(Sg) can be connected by an edge path,and C(Sg) has an infinite diameter. We show that for 0 ≤i≤3g-5,two i-simplices can be connected by an(i +1)-path in C(Sg),and the diameter of C(Sg) under such a distance is infinite.
基金supported by National Natural Science Foundation of China(Grant Nos.10931005 and 11101058)the National Science Foundation for Post-doctoral Scientists of China(Grant No.2011M500049)
文摘Let Ⅴ∪S W be a reducible Heegaard splitting of genus g = g（S） ≥ 2. For a maximal prime connected sum decomposition of Ⅴ∪S W, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim（Cy ∩ Cw）.
