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    Surjective homomorphisms between surface braid groups

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    Let PBn(Sg,p)PB_n(S_{g,p}) be the pure braid group of a genus g>1g>1 surface with pp punctures. In this paper we prove that any surjective homomorphism PBn(Sg,p)β†’PBm(Sg,p)PB_n(S_{g,p})\to PB_m(S_{g,p}) factors through one of the forgetful homomorphisms. We then compute the automorphism group of PBn(Sg,p)PB_n(S_{g,p}), extending Irmak, Ivanov and McCarthy's result to the punctured case. Surprisingly, in contrast to the n=1n=1 case, any automorphism of PBn(Sg,p)PB_n(S_{g,p}), n>1n>1 is geometric.Comment: 12 pages, changes on writin

    Partitions of the polytope of Doubly Substochastic Matrices

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    In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given
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