Tuesday Seminar on Topology

Date, time & place	Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s)	HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Shicheng Wang (Peking University)
Extending surface automorphisms over 4-space

[ Abstract ]
Let

$e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding
from a closed orientable manifold to the Euclidean space and

$E_e$ be the subgroup of

${\\cal M}_M$ , the mapping class group
of

$M$ , whose elements extend over

$R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure
on

$M$ derived from the embedding which is preserved by each

$\\tau \\in E_e$ .

Some applications: (1) the index

$[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding

$e:F_g\\to R^4$ , where

$F_g$
is the surface of genus

$g$ . (2)

$[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$ , where

$T^p$ is the

$p$ -dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

Graduate School of Mathematical Sciences, The University of Tokyo

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