0-concordance of 2-knots

Geometry Topology Seminar
Monday, February 13, 2017 - 14:00
1 hour (actually 50 minutes)
Skiles 006
Calvin College
A 2-knot is defined to be an embedding of S^2 in S^4. Unlike the theory of concordance for knots in S^3, the theory of concordance of 2-knots is trivial. This talk will be framed around the related concept of 0-concordance of 2-knots. It has been conjectured that this is also a trivial theory, that every 2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes. We'll in particular point out how the concept of 0-concordance is related to understanding smooth structures on S^4. The proof will involve invariants coming from Heegaard-Floer homology, and we will furthermore see how these invariants can be used shed light on other properties of 2-knots such as amphichirality and invertibility.