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Online: Rasmussen's s-invariant in $\sharp^r S^1 \times S^2$ and applications to some four-manifolds

Posted in
Marco Marengon
Peter Teicher, Arunima Ray, Tobias Barthel
Mon, 2020-10-05 14:00 - 14:20
Parent event: 
MPIM Topology Seminar

Zoom Meeting ID: 916 5855 1117
For password see the email or contact: Arunima Ray or Tobias Barthel

Given a knot $K$ in $S^3$, i.e. a closed connected 1-submanifold up to isotopy, Rasmussen defined an even integer $s(K)$ associated to it, called its s-invariant.
Crucially, one can use $s(K)$ to give a lower bound to the genus of an embedded surface $\Sigma \subset B^4$ with boundary $\partial \Sigma = K \subset S^3 = \partial B^4$.
I will discuss a generalisation of Rasmussen's s-invariant for null-homologous knots in $\sharp^r S^1 \times S^2$, together with some applications, including:
1) A genus bound for embedded surfaces in $\natural^r D^2 \times S^2$ and $\natural^r S^1 \times B^3$ (both of them have boundary $\sharp^r S^1 \times S^2$).
2) An adjunction inequality for surfaces in $\sharp^r \overline{\mathbb{CP}^2}$.
3) That the $s$-invariant cannot detect exoticness of a certain family of potential counterexamples to the smooth 4-D Poincaré conjecture.
This is all joint work with Ciprian Manolescu, Sucharit Sarkar, and Michael Willis.
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