### Seminars

### on knot theory and related topics

# Scanning diffeomorphisms

Date:

Time:

2021 07 28

07:30 (GMT+0)

Speaker:

Ryan Budney

Moscow-Beijing Topology Seminar

Cerf gave a beautiful homotopy-equivalence, a map we call scanning, between the space of diffeomorphisms of the n-disc (fixing boundary) Diff(D^n) and the loop space of embeddings of a co-dimension one disc ΩEmb(D^{n−1},D^n). The "barbell manifold" is our term for the boundary connect-sum of two copies of S^{n−1}×D^2. We will use a weak form of Cerf's scanning map to show a family of diffeomorphisms of the barbell manifold is non-trivial. We then proceed to embed the barbell manifold in S^1×D^n and check the extensions of the barbell diffeomorphism families to Diff(S^1×D^n) are homotopically non-trivial, using another scanning map. This allows us to show π_{n−3} Diff(S^1×D^n) is not finitely generated for all n≥3. One way of restating this result is that the component of the unknot in the space of smooth embeddings Emb(S^{n−1},S^{n+1}) has a not-finitely-generated (n−2)-nd homotopy group, for all n≥3. Our invariants of diffeomorphism groups can be thought of as (low order) Vassiliev invariants, generalizing previous work of Budney-Conant-Sinha-Scannell on quadrisecants. If there is time I will sketch recent generalizations where we compute subgroups of \pi_0 Diff(S^1 x D^n) for all n >= 3, connecting the work of Hatcher-Wagoner to our own.

Meeting ID: 831 5020 0580

Password: 141592

Zoom link: https://zoom.com.cn/j/83150200580?pwd=Z3VaRzhreko1TUhnWnJXQ05mRHNrQT09