Title: K-theoretic Torsion Invariants in Symplectic and Contact Topology.
Abstract: Torsion invariants can be used to detect nontrivial geometric phenomena when homological methods fail to do so — for example Reidemeister torsion was invented to distinguish lens spaces which are homotopy equivalent but not homeomorphic. These torsion invariants are K-theoretic in nature and can be understood from the viewpoint of Morse theory. Higher torsion invariants are their parametric analogues and can be understood from the viewpoint of parametrized Morse theory.
In this talk I will describe a program joint with K. Igusa to develop Morse-theoretic torsion invariants in the context of symplectic and contact topology — more precisely to develop torsion invariants for Legendrian (resp. exact Lagrangian) submanifolds of 1-jet spaces (resp. cotangent bundles). Our invariants come in various flavors (Reidemeister torsion, Turaev torsion, Whitehead torsion) and also have parametric analogues. My main focus for this talk will be on higher Whitehead torsion, an invariant for Legendrians in 1-jet spaces whose nontriviality is guaranteed by the literature on Waldhausen’s algebraic K-theory of spaces.
After describing its definition, I will discuss work in progress to prove that the higher Whitehead torsion of exact Lagrangians in cotangent bundles must always vanish. We secretly hope our proof will fail, since a single example of an exact Lagrangian in a cotangent bundle with nontrivial higher Whitehead torsion would disprove the nearby Lagrangian conjecture — one of the outstanding open problems in symplectic topology.
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