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14:30 ~ 16:00

A note on CI-moves

Kokoro TANAKA (Gakushuin University)

16:30 ~ 18:00

Finiteness conditions and mapping tori

Jonathan HILLMAN (University of Sydney)

Abstract

A note on CI-moves (Kokoto Tanaka)
A chart is an oriented labelled graph in a $2$-disk satisfying some conditions. This notion was introduced by S. Kamada in order to describe a surface braid, which is a $2$-dimensional analogue of a classical braid. He also defined moves for charts, called C-moves, which consist of three classes of moves: a CI-, CII- and CIII-move. He proved that there exists a one-to-one correspondence between the equivalence classes of surface braids and the C-move equivalence classes of charts.
Both CII- and CIII-move are local moves, but a CI-move is a global move. Thus it is natural to try to decompose a CI-move into some local moves. Actually J. S. Carter and M. Saito stated that any CI-move can be realized by a finite sequence of seven types of local CI-moves. This is a fundamental result of surface braid theory (and also of surface-knot theory), but it has been known that there are some ambiguous arguments in their proof. In this talk, we give an outline of a precise proof for their assertion by a different approach from theirs.

Finiteness conditions and mapping tori (Jonathan Hillman)
This is an account of joint work with D.Kochloukova (Brazil), on conditions under which infinite covering spaces of PD complexes are again PD complexes. The most important special case is the case of infinite cyclic covers.

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