Hatcher-Wagoner invariants for half-unknotted barbell diffeomorphisms

Xiayu Tan (Tsinghua University)

Monday 15th June 15:00-16:00
Maths 110

Abstract

In this talk, we briefly recall the general definition of implanted (i,n-i)-barbell diffeomorphisms and try to detect the non-triviality of a half-unknotted barbell diffeomorphism by associating it with a specific pseudo-isotopy (denoted as standard barbell pseudo-isotopy) and compute its Hatcher-Wagoner invariants explicitly. We give explicit formulae for (2, n-2)-barbells and a special class of (3,n-3)-barbells. Using these we show that:

 (1) For any smooth manifold with dimension n≥ 6, every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies with i=2 or 3. 

 (2) In the n=4 case, we further generalize the constructions and computations to half-unknotted immersed barbell diffeomorphisms and show that, for all elements in the codomain of the second Hatcher-Wagoner invariants satisfying Singh's condition, it can be realized by a pseudo-isotopy which results in a half-unknotted immersed barbell diffeomorphism.

(Remark: We will recall all the knowledge we need from Cerf theory and Hatcher-Wagoner invariants in the pre-seminar. So don't hesitate to join!)

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