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Topology/Geometry Seminar

Extending periodic maps on surfaces over the 4-sphere

Shicheng Wang (Peking University)

Location:  Hill Center, Room 705
Date & time: Tuesday, 31 January 2023 at 4:00PM - 5:00PM

Let \(F_g\) be the closed orientable surface of genus \(g\).
We address the problem to extend torsion elements of the mapping class group of \(M(F_g)\) over the 4-sphere \(S^4\).
Let \(w_g\) be a torsion element of maximum order in \(M(F_g)\).
Results including:

(1) For each \(g\),
\(w_g\) is periodically extendable over \(S^4\) for some non-smooth embedding \(e: F_g\to S^4\), and not periodically extendable over \(S^4\) for any smooth embedding \(e: F_g\to S^4\).

(2) For each \(g\), \(w_g\) is extendable over \(S^4\) for some smooth embedding \(e: F_g\to S^4\)
if and only if \(g=4k, 4k+3\).

(3) For infinitely many primes, each periodic map of order \(p\) on \(F_g\) is extendable over \(S^4\) for some smooth embedding \(e: F_g\to S^4\).

This is a joint work with Zhongzi Wang.

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