The four dimensional smooth Poincare conjecture from the viewpoint of Neumann algebra representations

Abstract: In this talk we outline the construction and basic properties of a new smooth 4-manifold invariant obtained by the aid of the rich representation theory of the hyperfinite II_1 factor von Neumann algebra. This invariant gives rise to a unital Abelian semigroup homomorphism from (the category of) connected compact oriented smooth 4-manifold equipped with the connected sum operation into the semi-open interval [0,1) with Abelian semigroup operation $(s,t)\mapsto s+t-st$. This invariant has the interesting property that its range is appropriately restricted by the possible values of Jones' indices of subfactors within the II_1 hyperfinite factor hence consists of a discrete and a continuous part. It is then observed that (i) the invariant is not injective on its continuous range part; (ii) when evaluated on the standard 4-sphere its value falls within the discrete part of the range and its injectivity at this specific value is equivalent to the validity of the 4 dimensional smooth Poincare conjecture. Moreover, as the punch line of this talk, it is expected that this invariant possesses a sort of continuity hence non-invertability at its specifec values in the continuous range will imply non-invartability at nearby values; however such argument cannot be applied to study the 4 dimensional smooth Poincare conjecture because of the aforementioned discreteness hence the conjecture's difficulty might be related with the isolation of the 4-sphere in this sense from the rest of smooth 4-manifolds.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative geometry in NYC

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